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+XdA0T4oBgHgl3EQfFP_h/content/2301.02031v1.pdf filter=lfs diff=lfs merge=lfs -text +BNAyT4oBgHgl3EQfRvex/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +U9AyT4oBgHgl3EQfuvmH/content/2301.00619v1.pdf filter=lfs diff=lfs merge=lfs -text +E9AzT4oBgHgl3EQfG_uu/content/2301.01038v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/0NE1T4oBgHgl3EQfkwRo/content/tmp_files/2301.03277v1.pdf.txt b/0NE1T4oBgHgl3EQfkwRo/content/tmp_files/2301.03277v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..491c29c0964026d4f5840cfa8a3db9633477a365 --- /dev/null +++ b/0NE1T4oBgHgl3EQfkwRo/content/tmp_files/2301.03277v1.pdf.txt @@ -0,0 +1,1704 @@ +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: popovici@math.univ-toulouse.fr +and +Soheil.Erfan@math.univ-toulouse.fr +21 + diff --git a/0NFAT4oBgHgl3EQfCBxS/content/tmp_files/2301.08407v1.pdf.txt b/0NFAT4oBgHgl3EQfCBxS/content/tmp_files/2301.08407v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..79c2f915a13272c250b1115745db7179e1c7cb3a --- /dev/null +++ b/0NFAT4oBgHgl3EQfCBxS/content/tmp_files/2301.08407v1.pdf.txt @@ -0,0 +1,858 @@ +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: h7wong@ucsd.edu +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. 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D +77, 043512 (2008), arXiv:0712.0618 [astro-ph]. + diff --git a/0NFAT4oBgHgl3EQfCBxS/content/tmp_files/load_file.txt b/0NFAT4oBgHgl3EQfCBxS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2081b3633dc1195d60c61e418142708c07b992f4 --- /dev/null +++ b/0NFAT4oBgHgl3EQfCBxS/content/tmp_files/load_file.txt @@ -0,0 +1,729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf,len=728 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' In this Letter, we propose a methodology to constrain the Hubble constant H0 by incorporating the TDE parameters measured by EM observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', stellar mass, black hole (BH) mass and spin, and other orbital parameters) into the observed TDE GW waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' It was only until recently has this technique been implemented in the binary neu- tron star merger event GW170817 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' By incorporating the multi-messenger information of the event, the viewing-angle-dependent features of various EM emission models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' One would naturally question whether compact object mergers remain the sole astrophysical sources to mea- sure H0 in a multi-messenger approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' As long as mas- sive objects revolve around each other, GW emissions are guaranteed, therefore tidal disruption of stars by massive ∗ Email: h7wong@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='edu 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Tidal disruption events (TDEs) could only generate GW bursts as the star is often disrupted within an or- bital timescale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' the intact star does not survive an entire orbit to produce a full period of GW waveform [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' An open comprehensive living catalog of TDE GW wave- forms has been built to explore a wide range of parame- ters [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Given their relatively long orbital timescale prior to disruption (∼ 102−4 s), the characteristic GW burst frequency is approximately in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='1 − 10 mHz, which corresponds to the designed sensitivities of the up- coming space-borne GW detectors [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='08407v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The pessimistic observed rate by LISA inspires us to explore the possibility of using a very limited number of events to independently measure H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' In Section III, we discuss possible ways to further improve the precision of H0 determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' In order to estimate an accurate Hubble con- stant H0, the problem lies in constraining the luminosity distance DL from their GW counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' (3) which depends mainly on three parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' We hereby show the parameter inter-dependencies (summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='2) and how they are expected to yield a constrained H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' EM Constraints on TDE Parameters 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The β de- pendency on TDE light curves is investigated with hydro- dynamical simulations [21], resulting in co-dependency on both masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' A relatively weaker constraint on β is through analyzing the probability distribution among the EM-observed TDE population [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Both works provided corresponding analytical formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' For the high signal- to-noise TDE detections by LISA, a lower bound on β (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', βmin ≳ 10 for MBH = 106 M⊙) can be imposed [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', all stellar materials are swallowed at pericenter passage thus EM signal detection is unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' It is important to note that these three parameters are not completely independent, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', a less massive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', more compact, star could not be tidally disrupted by a very massive BH, but will instead be swallowed whole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' rT ≤ rSch, where rSch is the Schwarzschild radius of the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Some regions in the parameter space are thus automatically ruled out for any EM-observable event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Schematic illustration of how a typical η orbit would evolve into an accretion disk of specific ori- entation given a BH spin offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The z-axis is set to be the line of sight (los).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' However, when GW is included as part of the multi-messenger analysis, η is of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', increase in viewing angle of the disk generally augments the optical-to-X-ray lumi- nosity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='1 and indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Al- beit the mild sensitivity in e, implying the insignificant contribution to the uncertainty of DL, EM constraints are still possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The common assumption is that most TDE stars have a roughly parabolic (e ≈ 1) flyby orbit [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' and near-circular orbits are highly unlikely based on loss cone dynamics [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The eccen- tricity is then essentially constrained to some values in between given by the fallback rate [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' This angle can hardly be constrained by EM observations nor TDE models and shall not possess any prior in the waveform fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' (All angles discussed θ, η, and φ are defined identically to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=') B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Luminosity Distance DL and H0 Estimate 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Note that some parameter values are prohibited (rT ≤ rSch), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', large MBH for disruption of small m⋆, high β for specific MBH and m⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The PDF of DL would translate directly to the PDF of H0 us- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=" Setting this PDF as the likelihood in Bayesian 𝑴𝐁𝐇 𝒎⋆ 𝜷 𝑒 𝑎$% 𝜃 𝜂 𝜙 𝐷& ℎ'(,*+,-." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=" (𝑡) EM constraints ℎ'(,+/0(𝑡) fitting 𝐷& 𝑧+/0 𝐻1 To be fitted 𝐻1 𝐻1 prior posterior likelihood EM constraint FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The main TDE parameters (MBH, m⋆, β) are indi- cated by blue circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Dashed arrows illustrate how param- eters are obtained via other parameters, which involve some simulations/models and additional EM observables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=', light curves and spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The H0 likelihood and cosmo- logically determined prior then give rise to the posterior (the final determination of H0 by a single TDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The filled boxes indicate the outputs of each procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The graphical description is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Bottleneck in H0 Measurement Uncertainty As long as the parameters are entangled in such a complicated manner (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' TDE parameters that dominate the depen- dency on GW waveform amplitude |hGW|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' All waveforms are centered at tburst, the time when the peak amplitude is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' The other parameters are chosen as follows: aBH = 0, θ = 0, e = 1, η = 0, DL = 20 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Plotted from simulation results [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='2), thus their uncer- tainties are cumulative and are projected onto the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Amongst them, we believe the penetration parameter β ought to contribute the largest uncertainty of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' σH0 is found to be dominated by the distance-inclination degeneracy [2, 4], implying that ση should dominate over the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Given that β is used to determine η, σβ should in turn dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' σMBH would then be the next dominating un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' If a uniform search in parameter space is implemented, the number of waveforms generated would skyrocket as the number of parameters increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Adaptive resolution on which parame- ters to explore should precede uniformly increasing the total number of waveforms across all parameter spaces in the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Ultimately, the goal of the multi-messenger analysis is to better constrain DL, not finding the best-fit TDE parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' more β- sensitive observable could be found with improved mod- eling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' 6 ACKNOWLEDGMENTS I thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Li, Paul C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFAT4oBgHgl3EQfCBxS/content/2301.08407v1.pdf'} +page_content=' Lai, Lars L.' metadata={'source': 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https://git-lfs.github.com/spec/v1 +oid sha256:9780b158d8daf050b874d1120be91a01ca66e5db031bcf1f5a373bc81d452082 +size 95962 diff --git a/2NAzT4oBgHgl3EQfDfro/content/tmp_files/2301.00979v1.pdf.txt b/2NAzT4oBgHgl3EQfDfro/content/tmp_files/2301.00979v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4fdb42fc2f64986c1de36ea2b3f168c6c35e5453 --- /dev/null +++ b/2NAzT4oBgHgl3EQfDfro/content/tmp_files/2301.00979v1.pdf.txt @@ -0,0 +1,1325 @@ +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: yang@xmu.edu.cn +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. +References +Ba, J. L.; Kiros, J. R.; and Hinton, G. E. 2016. 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In Proceedings of the 29th ACM In- +ternational Conference on Information & Knowledge Man- +agement, 1893–1902. + diff --git a/2NAzT4oBgHgl3EQfDfro/content/tmp_files/load_file.txt b/2NAzT4oBgHgl3EQfDfro/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..03f9fc1cc7eb86d22c9a550b63e4451db7b28ff2 --- /dev/null +++ b/2NAzT4oBgHgl3EQfDfro/content/tmp_files/load_file.txt @@ -0,0 +1,1089 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf,len=1088 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='cn, {zengfeng, yang}@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='cn Abstract Increasing research interests focus on sequential recom- mender systems, aiming to model dynamic sequence repre- sentation precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' However, the most commonly used loss function in state-of-the-art sequential recommendation mod- els has essential limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To name a few, Bayesian Per- sonalized Ranking (BPR) loss suffers the vanishing gradi- ent problem from numerous negative sampling and prediction biases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To avoid these limitations, in this paper, we propose to calculate Cumulative Cross-Entropy (CCE) loss over the se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' CCE is simple and direct, which enjoys the virtues of painless deployment, no negative sampling, and effective and efficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We conduct extensive experiments on five benchmark datasets to demonstrate the effectiveness and effi- ciency of CCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The results show that employing CCE loss on three state-of-the-art models GRU4Rec, SASRec, and S3-Rec can reach 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='63%, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='90%, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='24% average improve- ment of full ranking NDCG@5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For instance, GRU4Rec (Hidasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2016), GRU4Rec+ (Hidasi and Karatzoglou 2018), and NARM (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2017) are based on RNN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SASRec (Kang and McAuley 2018), BERT4Rec (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2019), S3-Rec (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020), and NOVA-BERT (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2021) are based on transformer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SR-GNN (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2019) and Caser (Tang and Wang 2018) are based on GNN and CNN, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Email-address: yang@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='cn 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' However, existing loss functions used in sequential recommendation have their own defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For example, one of the popular methods, GRU4Rec utilizes BPR (Rendle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2009) or TOP1 loss as the ob- jective function, which suffers from the gradient vanishing problem (Hidasi and Karatzoglou 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We focus on two rarely discussed issues about loss func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 1 gives an illustrative example, where Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Second, negative sampling is a widely-used approach to improve performance for sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Cor- respondingly, the loss functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='00979v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (a) GRU4Rec model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (c) Simplified experimental results of GRU4Rec with different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Figure 1: The architectures of GRU4Rec & SASRec and performance comparison of three loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We displays the average NDCG@10 scores of GRU4Rec using three loss functions, at each timestamp on the Yelp dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The sequence length is fixed to 50, with the 49th and 50th timestamps represent the validation and test item respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In addition, CCE sufficiently covers the gra- dient of the item embedding matrix by each item’s softmax score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Furthermore, the proposed model employs the mask- ing strategy for the varied length of user sequence to guar- antee the training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We validate our method on three typical sequential recom- mendation models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', GRU4Rec, SASRec, and S3-Rec) on five benchmark datasets from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Exper- imental results show that our method obtain average im- provements of 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='63%, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='90%, and 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='24% in terms of full ranking NDCG@5 for GRU4Rec, SASRec, and S3-Rec, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Specifically, GRU4Rec trained with CCE loss can markedly improve the NDCG@5 score by 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='67% over vanilla GRU4Rec on the Toys dataset (McAuley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The main contributions are threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' First, we identify limitations in the existing loss function used by sequen- tial recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To the best of our knowledge, this issue has not received much attention in existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Last Timestamp Loss Family It refers to the loss function that only involves the last timestamp of the training sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Generally, most neural network-based sequential recommendation models belong to this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The first sequential recommendation method based on RNN is GRU4Rec (Hidasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Most recent works that are influenced by GRU4Rec directly adopt or adapt BPR loss, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', the hierar- chical gating networks HGN (Ma, Kang, and Liu 2019), the GNN-based model MA-GNN (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020) and STEN (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Besides, some models use CE loss as the objective function, such as NARM (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2017), STAMP (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2018), SMART SENSE (Jeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2022), and SR- GNN (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To alleviate the item cold-start prob- lem, Mecos (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2021) uses CE loss to optimize a meta-learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Besides, a recent work (Petrov and Mac- donald 2022) utilizes the LambdaRank (Burges 2010) loss function, which still belongs to the last timestamp family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Masked Language Model Loss Family The Masked Language Model (MLM) (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Recent work adopted the idea of MLM and employs MLM loss in sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For example, BERT4Rec (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2019), utilizes BERT (De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2018) to model user behavior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' NOVA-BERT (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2021) introduces an attention mechanism that suf- ficiently leverages side information to guide and preserve item representations invariant in its vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' However, the item masking methods sacrifice much training time to achieve good performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' All Timestamp Loss Family As the name suggests, it considers all timestamps of the training sequence in loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To the best of our knowledge, the BCE loss is the mainly member in this fam- ily besides CCE loss proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020), ELECRec (Chen, Li, and Xiong 2022) and CAFE (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2022), and the state-of-the-art self-supervised learning model S3-Rec (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' There is a paucity of discus- sions on the training objective of BCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' However, the BCE loss is inevitably affected by negative sampling, and the number of negative samples will affect its perfor- mance and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' BPR, TOP1, BCE, and CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Problem Statement Suppose that there are a set of users U = � u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', u|U| � and a set of items I = � i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', i|I| � , where |U| and |I| denote the the number of users and items, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In the sequential recommendation, we mainly focus on the user’s historical interaction records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Therefore, we formulate a user sequence S1:n = (S1, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Rn = � rn,1, rn,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', rn,|I| � denotes the outputs of all items at timestamp n, where rn,t is the prediction score of item it at timestamp n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' R1:n = (R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', Rn) is the result on all timestamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Next, we define the embedding layer and prediction layer, which are the typical operations in the sequential recommen- dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Given a sequence input with the fixed-length l, the input sequence of the embedding layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', S1:l) is trans- formed to the embedding vector E1:l = (e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', el) ∈ Rl×e by the embedding matrix We ∈ R|I|×e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We now proceed to inntroduce the GRU4Rec and SASRec models, as well as the corresponding loss fuctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Given three components of the GRU, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (3) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', rl,|I| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Loss Function There are three loss function of vanilla GRU4Rec, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', BPR loss (Rendle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2009), TOP1 loss, and CE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' rl,pos, rl,neg are the scores of the positive item and negative item at the last timestamp l, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Note that we omit the regular- ization term for readability since it has nothing to do with the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We then examine their gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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−,+)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (10) Obviously, the vanishing gradient problem will occur for both loss functions when the number of negative samples Ns increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In addition, the prediction bias b+,− for BPR (or b1,+ for TOP1) that tends to infinity also induces the vanish- ing gradient problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In practice, due to the huge size of the negative set, the case of prediction bias occurs frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We also examine their gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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−,+)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (14) Through the softmax score sneg, the new loss can miti- gate the vanishing gradient problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Meanwhile, the sampling operation may skip informative negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SASRec Model Architecture The SASRec model is the first to in- troduce Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2017) into sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SASRec stacks two layers of transformer encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For readability, we only introduce the single layers of the transformer encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Before the transformer en- coder, SASRec adds a position vector P1:l ∈ Rl×e, thereby the final input is ˆE = E1:l + P1:l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Then it use Multi-Head Self-Attention (MH) layer to learn the asymmetric interac- tions and make the model more flexible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' which consists of multiple independent self-attention layers (SA) and trans- form by a weight matrix WO ∈ Re×e: MH( ˆE) = [SA1( ˆE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SA2( ˆE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SAH( ˆE)]WO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (15) SAj( ˆE) = attention( ˆEWQj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' ˆEWKj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' ˆEWV j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (16) attention(Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' V ) = softmax �QKT √e � V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (17) where WQj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' WKj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' WV j ∈ Re×e/H are the linear projection matrix that scales the input ˆE into a small space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Note that in the case of self-attention, the queries Q, keys K, and values V all equal to the input ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To satisfy the nature of sequence data, SASRec cut off the connection of Qi and Kj(j > i) in the attention calculation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The multi-head self- attention layer aggregate all previous item embedding with adaptive weights and is still a linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (18) To avoid overfitting, dropout (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2014) and layer normalization (Ba, Kiros, and Hinton 2016) are used for the input of both modules (MH and F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Further, to sta- bilize training, a residual connection (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2016) is ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' g(x) = x + Dropout(g(LayerNormalization(x))), (19) where g(x) is the multi-head self-attention layer or point- wise feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Finally, through the prediction layer, the result of SASRec is R1:l = F( ˆE)W T e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Loss Function SASRec adopts the binary cross-entropy (BCE) loss as the objective function, and here we use mask to simplify the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Lbce = − l � t=1 MASK [log σ(rt,pos) + log σ(1 − rt,neg)] , (20) where MASK = (mask1, mask2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', maskl) is the mask vector, maskt is False when St in the sequence S1:l is the mask item, and True otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We can find that the main difference in loss function between GRU4Rec and SASRec is the cumulative term of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Intuitively, this loss allows more positive samples to participate in the optimization pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' However, it depends on the negative sampling oper- ation, and randomly generates only one negative item for each timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Further, we give the gradient w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (22) As we can see, the gradient coincides with the objective of the sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Therefore, BCE is essentially prone to lose information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Intuitively, adding more negative examples can alleviate this problem, but it would spend much more time on sampling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In this sense, the Cross-Entropy (CE) loss is a natural choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (25) Note that without sampling, CE loss aggregates the predic- tion score of the whole item size, which contains the whole negative example set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2) The gradient of CE loss can cover the whole item set in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 3) CE avoids negative sampling, and hence refrains from difficulties aris- ing therefrom, such as the additional time cost of sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Therefore, CE can improve the training efficiency and re- duces the risk of information loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' However, the current form of CE loss used in sequential recommendation only focuses on the last timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (26) The idea of CCE is simple and direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Further, it avoids the negative sampling operation in BCE, and calculates gradient on the entire item set like CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Extensive experiments verify the effectiveness of the CCE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' RQ1: How does the CCE loss perform when em- ployed in the state-of-the-art models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' RQ2: How efficient is the training of the models using the CCE loss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' RQ3: How does the CCE loss perform across all timestamps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2015): Beauty, Sports and Toys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' a business recommendation dataset Yelp2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' and a music artist recommendation dataset LastFM3 (Cantador, Brusilovsky, and Kuflik 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Note that we only use the transaction records after January 1st, 2019 in Yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 1http://jmcauley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='edu/data/amazon/links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='html 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='yelp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='com/dataset 3https://grouplens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='14 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='06 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='80 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='40 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='41 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='05% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='07% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='05% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='07% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='32% Data Processing Following the recent and state-of-the- arts in sequential recommendation (Kang and McAuley 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Tang and Wang 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2019), we divide a given dataset into train, validation, and test sets according to the leave-one-out strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In addi- tion, to reproduce the pre-training model S3-Rec, we pre- processed the original datasets as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (1) We remove users and items with less than five interaction records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (2) We group the interaction records by users and sort them chronologically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (3) We keep the user sequence with the fixed-length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' After preprocessing, the statistics of the five datasets are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' which is the first to apply GRU to model user interaction sequence for the session- based recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' SASRec (Kang and McAuley 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' which is a transformer-based model, using a multi-head attention mechanism to learn the asymmetric interactions and make the model more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' S3-Rec (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' which is the first to introduce self-supervised learning to the sequential recommenda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Besides, we use BCE as the representative of all timestamp loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Note that we ignore the masked language model loss due to its large training cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The hyperparameters of these models are set as suggested in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For each dataset, the fixed length of the input sequence is set to 50, the size of the item em- beddings is 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Besides, we use the Adam optimizer with the default learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='001, parameters β1 and β2 are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='999, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We train models for 150 epochs with the early stop strategy6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We save the optimal model based on the evaluation metrics on the validation set 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='com/RUCAIBox/CIKM2020-S3Rec 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='com/RUCAIBox/RecBole 6We terminate the training when the evaluation metric does not improve for ten consecutive epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Table 2: Comparing three loss functions with respect to the performance of GRU4Rec, SASRec, and S3-Rec on five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Best results are in boldface, and the best one between Lbce and Lce is indicated by underline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' “Improve” denotes the improvement over the best performance of Lbce (or Lce), while the degradation cases are marked with ↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Dataset Metric GRU4Rec SASRec S3-Rec Lbce Lce Lcce Improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Lbce Lce Lcce Improve.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='32% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='0491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='0346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='0519 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='70% at the training stage, then report their performances on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' All experiments are conducted us- ing 10-cores of an Intel i9-10900K CPU, 24GB of memory and an NVIDIA GeForce RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Kang and McAuley 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The details of the metrics can be found in (Krichene and Rendle 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Recent work on sampling strategies (Dallmann, Zoller, and Hotho 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' To avoid inconsistency, we report the full ranking metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Experimental Results Overall Results (RQ1) Table 2 shows the performance of the GRU4Rec, SASRec, and S3-Rec using CCE, BCE, and CE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We observe that the CCE loss improves the best performance of BCE (or CE) for all models in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='03%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For GRU4Rec, compared with BCE and CE, the pro- posed CCE loss greatly promotes the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The average improvements on five datasets in terms of HR@5, HR@10, HR@20, NDCG@5, NDCG@10, NDCG@20 are 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='99%, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='90%, 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='66% 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='63%, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='05% and 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='76% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Interestingly, the CCE loss brings an astonishing 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='67% improvement at NDCG@5 on Toys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For SASRec, our CCE loss achieves an overall increase in terms of all metrics on five datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Specifically, the av- erage improvements in terms of HR@5, HR@10, HR@20, NDCG@5, NDCG@10, NDCG@20 are 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='82%, 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='41%, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='38%, 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='90%, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='05%, and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='09%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', the average improvements are 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='75% (HR@5), 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='96% (HR@10), 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='73% (HR@20), 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='24% (NDCG@5), 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='24% (NDCG@10), and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='83% (NDCG@20), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Training Efficiency (RQ2) We evaluate the training effi- ciency of our approach from two aspects, as suggested in (Kang and McAuley 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For example, the SASRec+CCE takes about 100 seconds to reach a much higher value of NDCG@10 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='035) on Sports, while spends 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='24 sec- onds for one epoch, which is close to BCE (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='21s/epoch) and CE (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='62s/epoch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' In summary, we argue that the CCE loss can effectively and efficiently help model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Performance on All Timestamps (RQ3) In this section, we extend the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 1c to five benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 3, the vertical axes represent NDCG@10 scores of GRU4Rec, while the horizontal axes represent the whole timestamp of the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' The last two times- tamps refer to the validation and test item, respectively, where the model performance drops drastically in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' For the LastFM dataset, CCE has only a slight improvement over BCE in the training sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' This result may explain why it does not show great advantages on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Intuitively, a loss function that is able to guar- antee the accuracy for all timestamps of training sequence can effectively improve the recommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Conclusion In this paper, we address the issue of loss function design in sequential recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We point out that the whole training sequence should be considered when calcu- lating the loss, rather than the last timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Meanwhile, avoiding negative sampling can improve the training effi- ciency and accuracy of recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' We propose a novel cumulative cross-entropy loss and apply it to three typical models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=', GRU4Rec, SASRec, and S3Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Exper- iments on five benchmark datasets demonstrate its effective- ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' References Ba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAzT4oBgHgl3EQfDfro/content/2301.00979v1.pdf'} +page_content=' Kiros, J.' 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diff --git a/2NFST4oBgHgl3EQfXTj3/content/tmp_files/2301.13784v1.pdf.txt b/2NFST4oBgHgl3EQfXTj3/content/tmp_files/2301.13784v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..076f803a80ceef906bdd399459a041238842c089 --- /dev/null +++ b/2NFST4oBgHgl3EQfXTj3/content/tmp_files/2301.13784v1.pdf.txt @@ -0,0 +1,1349 @@ +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 +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +β +� +1 +�♠ +♠ +♠ +♠ +♠ +♠ +♠ +♠ +♠ +♠ +♠ +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +Z +ǫ +� Ω +X +γ +�❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +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. 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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). +[Vat] +Vincent Vatter. “Permutation classes” in “Handbook of enumerative combinatorics” ed. by Mikl´os +B´ona. CRC Press, 2015. arXiv:1409.5159 + diff --git a/2NFST4oBgHgl3EQfXTj3/content/tmp_files/load_file.txt b/2NFST4oBgHgl3EQfXTj3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff06428552aebfcaf2b6b4c529085406d37fffa3 --- /dev/null +++ b/2NFST4oBgHgl3EQfXTj3/content/tmp_files/load_file.txt @@ -0,0 +1,1604 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf,len=1603 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13784v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='RT] 31 Jan 2023 PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM NATE HARMAN AND ANDREW SNOWDEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Galois categories can be viewed as the combinatorial analog of Tannakian cat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We introduce the notion of pre-Galois category, which can be viewed as the combi- natorial analog of pre-Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Given an oligomorphic group G, the category S(G) of finitary smooth G-sets is pre-Galois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Our main theorem (approximately) says that these examples are exhaustive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' this result is, in a sense, a reformulation of Fra¨ıss´e’s the- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This work is motivated by certain applications to pre-Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Oligomorphic groups 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Combinatorial tensor categories 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Pre-Galois categories 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Categories of atoms 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fra¨ıss´e theory 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Examples from relational structures 22 References 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The famous Tannakian reconstruction theorem says that an algebraic group can be recovered from its representation category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To be a bit more precise, fix an algebraically closed field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A pre-Tannakian category is a k-linear abelian category equipped with a symmetric tensor structure satisfying some axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A Tannakian category is a pre- Tannakian category C equipped with a fiber functor ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', a faithful exact tensor functor to finite dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The motivating example of a Tannakian category is the category Repk(G) of finite dimensional representations of an algebraic group G/k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' the fiber functor is simply the forgetful functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' See [DM] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A major problem in the field of tensor categories is to better understand the pre-Tannakian landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Date: January 31, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1 2 NATE HARMAN AND ANDREW SNOWDEN There is a combinatorial analog of Tannakian reconstruction, in the form of Grothendieck’s Galois theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A Galois category is a category C equipped with a functor ω to finite sets satisfying certain axioms (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The motivating example of a Galois category is the category of finite G-sets, for a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Grothendieck applied this theorem to construct the ´etale fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Conspicuously absent from the combinatorial side is an analog of pre-Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Pre-Galois categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following is our combinatorial analog of pre-Tannakian categories: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A category B is pre-Galois if the following conditions hold: (a) B has finite co-products (and thus an initial object 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Every object of B is isomorphic to a finite co-product of atoms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', objects that do not decompose under co-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (d) B has fiber products and a final object 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (e) Any monomorphism of atoms is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (f) If X → Z and Y → Z are maps of atoms then X ×Z Y is non-empty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', not 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (g) The final object 1 is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (h) Equivalence relations in B are effective (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ The above axioms are motivated by properties of the category of finite G-sets, for a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' “Atoms” should be thought of as transitive G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The first three axioms basically say that objects admit a finite “orbit decomposition” which behaves in the expected manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define a B-category to be one satisfying axioms (a)–(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This turns out to be a very nice class of categories already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For example, we show that every B-category is balanced (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13) and has finite Hom sets (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Axiom (e) is somewhat subtle, but these nice properties of B-categories depend on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that a B-category is non-degenerate if it satisfies (f) and (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Non-degeneracy implies a number of nice properties, such as existence of co-equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Axiom (h) ensures that quotients are well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One can match properties of pre-Galois categories and pre-Tannakian categories, to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Axiom (a) corresponds to additivity on the pre-Tannakian side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Both pre-Galois and pre-Tannakian categories are finitely complete and co-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Axiom (h) corresponds to the first isomorphism theorem on the pre-Tannakian side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Axiom (b) corresponds to the finite length condition on the pre-Tannakian side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The co-product and product in a pre-Galois category correspond to the direct sum and tensor product in a pre-Tannakian category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Axiom (g) corresponds to the pre-Tannakian axiom that the unit object is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Finally, (f) corresponds to the fact that in a pre-Tannkain category the tensor product of non-zero objects is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For any group G, the category S(G) of finite G-sets is a pre-Galois category, and this is the motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One might try to construct other examples by considering (possibly infinite) G-sets with finitely many orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This does not work in general since a product of two such G-sets need not have finitely many orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It turns out that the above idea can be made to work in at least one situation, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Recall that an oligomorphic group is a permutation group (G, Ω) such that G has finitely many orbits on Ωn for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The simplest example of an oligomorphic group is the infinite symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Model theory, and the theory of Fra¨ıss´e limits in particular, provides many more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' See [Cam1] for general background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It turns out that this category is closed under products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' this is a consequence of the oligomorphic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is not hard to show that S(G) is in fact pre-Galois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' From the topological perspective, the key finiteness property of G is Roelcke pre-compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We review this theory in §2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' a more detailed treatment can be found in [HS1, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3 we give a non-trivial example of a degenerate B-category using a non- Fra¨ıss´e class of relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It would be interesting if one could give a more direct construction of such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following is our main result on pre-Galois categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2 (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following are equivalent: (a) B is pre-Galois and has countably many isomorphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) B is equivalent to S(G) for some first-countable admissible group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The countability hypotheses above should not be necessary, but we impose them to make the proof and exposition easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15 in fact does a bit more than the above theorem, in that it accommodates all (countable) non-degenerate B-categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' in other words, we still obtain a classification result when we do not impose Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The non-degeneracy condition seems essential, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Overview of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a B-category, and let A be the full subcategory of Bop spanned by atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The key point in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This theorem produces a universal homogeneous ind-object Ω in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We show that G = Aut(Ω) is naturally an admissible group, and that B is equivalent to S(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We follow this plan in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 4 NATE HARMAN AND ANDREW SNOWDEN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' As stated above, a major problem in tensor category theory is under- standing pre-Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Our construction recovers Deligne’s interpolation categories in certain cases, and leads to new categories (like the Delannoy category) in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Some con- structions and results in [HS1] hold for more general B-categories, and this was our original motivation for developing the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In forthcoming work [HS3], we give an application of this paper, which we now briefly describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be a pre-Tannakian tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Define Frob(C) to be the category whose objects are special commutative Frobenius algebras in C, and whose morphisms are co-algebra homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define the oligomorphic component group of C to be the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is an interesting invariant of the category C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Using this, we classify pre-Tannakian categories with enough Frobenius algebras, which (we hope) is a step towards a general classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In §2, we review oligomorphic and admissible groups and the associated cat- egories S(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' these are the motivating examples of pre-Galois categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In §3, we define B-categories and establish some of their basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In §4, we introduce pre-Galois cat- egories, and establish some of their special features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In §5, we study the category of atoms in a B-category, which leads to the notion of A-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In §6, we review Fra¨ıss´e theory and prove our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Finally, in §7, we give some examples of A- and B-categories coming from relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We list some of the important notation here: 0 : the initial object of a B-category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', the empty set) 1 : the final object of a B-category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1) B(A) : the B-category associated to A (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Oligomorphic groups In this section, we review oligomorphic and admissible groups, and recall the category S(G) of finitary G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' These categories are the motivation for the general notion of pre-Galois category we study in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Oligomorphic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An oligomorphic group is a permutation group (G, Ω) such that G has finitely many orbits on Ωn for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Here are a few concrete examples: The infinite symmetric group S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', the group of all permutations of Ω = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The infinite general linear group over a finite field F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', the group of all linear automorphisms of F⊕∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The group of all order-preserving self-bijections of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 5 Many more examples can be obtained from Fra¨ıss´e limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We refer to Cameron’s book [Cam1] for general background on oligomorphic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Admissible groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fix an oligomorphic group (G, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For a finite subset A of Ω, let G(A) be the subgroup of G fixing each element of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The groups G(A) form a neighborhood basis of the identity for a topology on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This topology has the following properties [HS1, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2]: It is Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is non-archimedean: open subgroups form a neighborhood basis of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is Roelcke pre-compact: if U and V are open subgroups then V \\G/U is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define an admissible group to be a topological group with these three properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus every oligomorphic group gives rise to an admissible group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We also note that any finite group is admissible (with the discrete topology), and any profinite group is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G be an admissible group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that an action of G on a set X is smooth if all stabilizers are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An important property of S(G) is that it is closed under products and fiber products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' see [HS1, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is a variant of the category S(G) that will play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A stabilizer class in G is a collection E of open subgroups of G satisfying the following conditions: (a) E contains G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) E is closed under conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) E is closed under finite intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (d) E forms a neighborhood basis for the identity of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that a G-set is E -smooth if its stabilizers all belong to E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We let S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) be the full subcategory of S(G) spanned by the E -smooth sets, and analogously define T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The category S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) is also closed under products and fiber products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let S be the infinite symmetric group, let S(n) ⊂ S be the subgroup fixing each of 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , n, and let Sn be the symmetric group on n letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then E and Y are stabilizer classes in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Combinatorial tensor categories In this section, we introduce the class of B-categories, which we view as combinatorial analogs of tensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' All categories in this section are essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 6 NATE HARMAN AND ANDREW SNOWDEN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a category with finite co-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We write 0 for the initial object and refer to it (or any object isomorphic to it) as empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' that is, given an isomorphism X ∼= Y ∐ Z either Y or Z is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now introduce our combinatorial analog of tensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A B-category is an essentially small category B satisfying the following conditions: (a) B has finite co-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Every object of B is isomorphic to a finite co-product of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (d) B has fiber products and a final object 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (e) Any monomorphism of atoms is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ The following proposition establishes the motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G be an admissible group and let E be a stabilizer class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then the category S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) is a B-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) The co-product is given by disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Atoms are transitive E -smooth G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Every finitary E -smooth G-set is clearly a finite disjoint union of transitive E -smooth G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) Suppose X is an atom and Y and Z are arbitrary objects of S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y ∐Z be a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus axiom (c) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (d) The ordinary fiber product of sets is the fiber product in S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The final object is the one-point G-set (which is E -smooth since E is required to contain G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (e) Suppose f : X → Y is a monomorphism of atoms in S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' As in any category with fiber products, this implies that the projection map X ×Y X → X is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since the set underlying X ×Y X is just the usual fiber product of sets, we see that f is an injective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since f is an injective map of transitive G-sets, it is bijective, and thus an isomorphism in the category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We mention a few simple ways of producing new B-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) Let B be a B-category and let X be an object of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let Σ be the class of all atomic objects appearing as a summand of Xn for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B′ be the full subcategory of B spanned by objects that are co-products of objects in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then B′ is a B-category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we call this the subcategory generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Let B be a B-category and let S be an object of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then the category B/S of objects over S is a B-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If B = S(G) and S = G/U for an open subgroup U then B/S = S(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) Suppose B1 and B2 are B-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then the product category B1 ⊞ B2 is a B- category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we call it the sum category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then B is naturally equivalent to B/S1 ⊞ · · · ⊞ B/Sn, and each B/Si has an atomic final object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Properties of B0-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Although we are mostly interested in B-categories, some results hold in greater generality, and this additional generality is useful in later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In this spirit, we now prove some basic results about B0-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We fix a B0-category B for §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If X is non-empty then there are no maps X → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It suffices to treat where X is atomic, so we assume this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Write X = X1 ∐ · · · ∐ Xn and Y = Y1 ∐ · · · ∐ Ym where each Xi and Yi is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There exists a unique function a: [n] → [m] such that the restriction of f to Xi factors uniquely through Ya(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' let fi : Xi → Ya(i) be the induced map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f is uniquely determined by a and the fi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Moreover, every choice of a and the fi’s comes from some f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For each i, the natural map m � j=1 Hom(Xi, Yj) → Hom(Xi, Y ) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For m = 0, this is Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4, for m = 1 it is obvious, and for m ≥ 2 it follows from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(c) inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This proves the existence of a and the fi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' That they determine f, and that every choice arises, follows from the definition of co-product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f, f ′: X → X′ and g, g′: Y → Y ′ be morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f ∐ g = f ′ ∐ g′ if and only if f = f ′ and g = g′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → X′ and g : Y → Y ′ be morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then: (a) f ∐ g is monomorphic if and only if f and g are monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) f ∐ g is epimorphic if and only if f and g are epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) First suppose that f is not monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let h, h′: W → X be distinct maps such that fh = fh′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then h∐idY and h′ ∐idY are maps W ∐Y → X ∐Y , which are distinct by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6, but have the same composition with f ∐g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus f ∐g is not monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now suppose that f and g are monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let h, h′ : W → X ∐ Y be maps that have equal composition with f ∐ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We show h = h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus assume W is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then W maps into exactly one of X or Y under h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' without loss of generality, say X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then W maps into X′ under (f ∐ g) ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that W also maps into X′ under (f ∐ g) ◦ h′, and so must map into X under h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since f is monomorphic, we conclude h = h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus f ∐ g is monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 8 NATE HARMAN AND ANDREW SNOWDEN (b) First suppose that f is not epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let h, h′ : X′ → Z be distinct maps such that hf = h′f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then h ∐ idY ′ and h′ ∐ idY ′ are maps X′ ∐ Y ′ → X ∐ Y ′, which are distinct by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6, but have the same composition with f ∐ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus f ∐ g is not epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now suppose that f and g are epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let h, h′ : X′ ∐ Y ′ → Z be maps having equal composition with f ∐g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Restricting h and h′ to X′, we see that they have equal composition with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since f is epimorphic, this means h and h′ have equal restriction to X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Similarly, they have equal restriction to Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By the definition of co-product, this means h = h′, and so f ∐ g is epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For any objects X and Y , the natural map X → X ∐Y is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let i be the identity map of X, and let j : 0 → Y be the unique map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Clearly, i and j are monomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The map in question is (isomorphic to) i ∐ j, and is thus a monomorphism by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fiber products distribute over co-products, in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X, X′, and Y be objects of B equipped with morphisms to another object Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose that the fiber products X ×Z Y and X′ ×Z Y exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since P has natural maps to X ∐ X′ and Y that agree when composed to Z, there is a natural transformation Hom(−, P) → Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It suffices to show that this is an isomorphism, for then P will represent the fiber product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Properties of B-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now prove some general results on B-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We fix a B-category B for the duration of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The only subobjects of an atom X are 0 and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose that Y is a non-empty subobject of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Write Y = Y1 ∐ · · · ∐ Yn with each Yi an atom and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since Yi → Y is monic by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8, it follows that Yi → X is monic, and thus an isomorphism by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(c), but have equal composition to X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Any map of atoms is epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a map of atoms, and let g, h: Y → Z be maps such that g◦f = h◦f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since B has finite limits, the equalizer Eq(g, h) of g and h exists, and is naturally a subobject of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since f factors through Eq(g, h) and X is non-empty, it follows that Eq(g, h) is non- empty (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus Eq(g, h) is equal to Y (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='10), and so g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Write X = X1 ∐ · · · ∐ Xn and Y = Y1 ∐ · · · ∐ Ym where each Xi and Yi is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let a: [n] → [m] and fi : Xi → Ya(i) be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) f is epimorphic if and only if a is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) f is monomorphic if and only if a is injective and each fi is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For j ∈ [m], let Xj = � a(i)=j Xi, and let f j : Xj → Yj be the restriction of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f is the co-product of the f j, and so by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7, f is monomorphic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' epimorphic) if and only if each f j is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) Suppose that f is monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then each f j is monomorphic, and so by Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='10 either Xj is empty or f j is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that a is injective and each fi is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Conversely, suppose that a is injective and each fi is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then each f j is clearly monomorphic, and so f is too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Suppose that f is epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then each f j is epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that Xj is non-empty, as 0 → Yj is not epimorphic (the two maps Yj → Yj ∐ Yj are distinct by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(c) but have the same restriction to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus a is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Conversely, suppose that a is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then for each j ∈ [m] there is some i with a(i) = j, and then map fi is epimorphic by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that f j is epimorphic too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since this holds for each j, we find that f is epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The category B is balanced: a morphism that is both monomorphic and epimorphic is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X = X1 ∐ · · · ∐ Xn with each Xi atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For a subset S of [n], let XS = � i∈S Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then every subobject of X is one of the XS, and XS ⊂ XT if and only if S ⊂ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This follows immediately from the structure of monomorphisms given in Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism, and use notation as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) im(f) exists, and is equal to � j∈im(a) Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) f is an epimorphism if and only if im(f) = Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) The map X → im(f) is an epimorphism, and a monomorphism if and only if f is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This follows from the structure of f given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5, the characterization of monomorphisms and epimorphisms in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12, and the classification of subobjects in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism, and let ∆: X → X ×Y X be the diagonal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following are equivalent: (a) f is monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 10 NATE HARMAN AND ANDREW SNOWDEN (b) ∆ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) ∆ is epimorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In any category, (a) and (b) are equivalent, and (b) implies (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In a balanced category (such as a B-category), (c) implies (b) since ∆ is always monomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For any objects X and Y , the set Hom(X, Y ) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Consider a map f : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since idX × f is a monomorphism, it follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15 that the natural map X → Γf is both a monomorphism and an epimorphism, and is thus an isomorphism by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' its inverse is clearly p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus see that f = q ◦ p−1, and so f can be recovered from Γf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' As X × Y has only finitely many subobjects (by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='14), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Any self-map of an atom is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → X be a map with X an atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f is an epimorphism (Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11), and so f ∗: Hom(X, X) → Hom(X, X) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since Hom(X, X) is finite (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='17), it follows that f ∗ is bijective, and so there exists g ∈ Hom(X, X) such g ◦ f = idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus f is a monomorphism, and hence an isomorphism (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose G is an admissible group and X is a finitary G-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One can then form the orbit space G\\X, which is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Passing to orbits is often an important idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is an analog of this construction in our more general categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a B0- category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define the orbit set of X, denoted Xorb, to be the set of atomic subobjects of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This construction is natural: it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5 that a map f : X → Y naturally induces a function f orb: Xorb → Y orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We therefore have a functor B → FinSet, X �→ Xorb, where FinSet is the category of finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now show how one can read off some properties of a morphism from how it behaves on orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose B is a B-category and f : X → Y is a morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) f is epimorphic if and only if f orb is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) f is monomorphic if and only if Xorb → (X ×Y X)orb is surjective (or bijective);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' in this case, f orb is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now prove (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let ∆: X → X ×Y X be the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If f is monomorphic then ∆ is an isomorphism (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='16), and so ∆orb is a bijection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' conversely, if ∆orb is surjective then ∆ is epimorphic by (a), and so f is monomorphic (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If f is monomorphic then f orb is injective by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a B1-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One can sometimes modify B to produce a B- category, as we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We make the following definitions: f is a pre-monomorphism if the map Xorb → (X ×Y X)orb is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' f is a pre-epimorphism if the map Xorb → Y orb is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' f is a pre-isomorphism if it is a pre-monomorphism and pre-isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 11 Suppose that the class of pre-isomorphisms is stable under base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then this class forms a right multiplicative system, as defined in [Stacks, Tag 04VC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Motivated by these observations, we introduce the class of pre-Galois categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We also discuss how they relate to the existing notion of Galois category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' All categories in this section are assumed to be essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Non-degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We begin with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a B1-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following are equivalent: (a) If X → Z and Y → Z are maps of atoms then X ×Z Y is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) A base change of an epimorphism is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) A product of epimorphisms is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) ⇒ (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We show that f ′ is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since fiber products distribute over co-products, it suffices to treat the case where X, Y , and Y ′ are atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By assumption, X′ is then non-empty, and so f ′ is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) ⇒ (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X → Y and X′ → Y ′ be epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Consider the composition X × X′ → Y × X′ → Y × Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The first map is the base change of the epimorphism X → Y along the map X′ → 1, and is thus an epimorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' similarly, the second map is the base change of the epimorphism X′ → Y ′ along the map Y → 1, and is thus an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that the composition X × X′ → Y × Y ′ is an epimorphism, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) ⇒ (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X → Y and Y ′ → Y be maps of atoms, and let X′ = X ×Y Y ′ be the fiber product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since X → Y is an epimorphism, by assumption X′ → Y ′ is also an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus X′ is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Motivated by the above proposition, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A B1-category is non-degenerate if the equivalent conditions of Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1 hold, and the final object 1 is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ It is clear that the category S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) is non-degenerate, for any admissible group G and stabilizer class E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3, we give an interesting example of a degenerate B-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Implications of non-degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fix a non-degenerate B-category B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now ex- amine some consequences of the non-degeneracy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We note that these results can be deduced from the classification of such categories (provided by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15), but we find it instructive to give direct proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For a morphism f : X → Y , we define the kernel pair of f to be Eq(f) = X ×Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is a subobject of X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y and g : X → Z be epimorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f factors through g if and only if Eq(g) ⊂ Eq(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 12 NATE HARMAN AND ANDREW SNOWDEN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is clear that if f factors through g then Eq(g) ⊂ Eq(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now prove the converse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' thus assume Eq(g) ⊂ Eq(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' note that f = p ◦ h and g = q ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have Eq(h) = Eq(f × g) = Eq(f) ∩ Eq(g) = Eq(g), where f × g denotes the map X → Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The right map is an epimorphism since h is an epimorphism and the category B is non-degenerate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that the bottom map is an epimorphism, and so q is an monomorphism (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='16), and thus an isomorphism (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus have f = p ◦ q−1 ◦ g, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For X fixed, there are finitely many epimorphisms X → Y up to isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By the proposition, an epimorphism f : X → Y is determined up to isomorphism by Eq(f), which is a subobject of X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since X × X has finitely many subobjects (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='14), the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A non-degenerate B-category B is finitely co-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since B has finite co-products, it suffices to show that it has co-equalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f, g : X → Y be parallel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let {qi : Y → Zi}i∈U be representatives of the isomor- phism classes of epimorphisms out of Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' this set is finite by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let V be the set of indices i ∈ U such that qi ◦ f = qi ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Define I to be the image of the map Y → � i∈V Zi, and let h: Y → I be the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We claim that h is a co-equalizer of (f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To see this, suppose that a: Y → T is a morphism with a ◦ f = a ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The morphism a factors as c◦b, where b is an epimorphism and c is a monomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we may as well assume b = qi for some i ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since c is a monomorphism, it follows that qi ◦ f = qi ◦ g, and so i ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let pi : I → Zi be the projection onto the ith factor, so that qi = pi ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Composing with c, we have a = c ◦ pi ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus see that a factors through h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The factorization is unique since h is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The above proof actually shows that any B-category satisfying Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4 is finitely co-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' All B-categories we know (including the degenerate ones) satisfy this corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Effective equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G be an admissible group and let E be a stabi- lizer class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5, the category S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) is finitely co-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is somewhat surprising, since every smooth G-set is a quotient of some E -smooth G-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The explanation here is that co-equalizers in S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) do not agree with co-equalizers in S(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In fact, S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) is a reflective subcategory of S(G), and co-equalizers in S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) are obtained by computing in S(G) and then applying the reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now give an example to illustrate the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 13 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G = S be the infinite symmetric group acting on Ω = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let p: Ω[2] → Ω(2) be the natural surjection, and let R = Eq(p) be the kernel-pair of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In the category S(G), the co-equalizer of R ⇒ Ω[2] is Ω(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now, let E be the stabilizer class consisting of subgroups conjugate to some S(n), as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The G-sets Ω[2] and R are E -smooth, while Ω(2) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The reflector Φ: S(G) → S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have Ω(2) ∼= G/U, where U = S2 × S(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' From the classification of open subgroups of S (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', [HS1, Proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1]), we see that the only subgroup in E containing U is S itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ The following terminology is useful for explaining this situation: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be a finitely complete category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that C has effective equivalence relations if all equivalence relations in C are effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ With this terminology, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7 can be summarized as follows: R is an effective equivalence relation in S(S), but not in the subcategory S(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following proposition gives the general statement in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G be an admissible group and let E be a stabilizer class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) The category S(G) has effective equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) If S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) has effective equivalence relations then S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) = S(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', E contains all open subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) The category of sets has effective equivalence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This property passes to S(G) since finite limits and co-limits here are computed on the underlying sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Let U be an open subgroup of G, and let V be a member of E with V ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since Y × Y belongs to S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ), so does the subobject R, and so R defines an equivalence relation on Y in the category S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus, by assumption, there is a map π′: Y → X′ in S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) with kernel pair R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' of course, we may as well assume π′ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since the inclusion of S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) into S(G) preserves fiber products, it follows that R is the kernel pair of π′ in S(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus π and π′ are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In particular, G/V is E -smooth, and so V belongs to E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Pre-Galois categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now introduce this class of categories: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A pre-Galois category is a non-degenerate B-category with effective equiv- alence relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ This definition is equivalent to the one given in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' As the preceding dis- cussion shows, if G is an admissible group then S(G) is a pre-Galois category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Comparison with Galois categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now discuss the relation between the clas- sical notion of Galois category and our notion of pre-Galois category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We begin by recalling the former: 14 NATE HARMAN AND ANDREW SNOWDEN Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', X → I is the co-equalizer of X ×I X ⇒ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) ω is exact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', it commutes with finite limits and co-limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (d) ω is conservative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', ω(ϕ) is an isomorphism if and only if ϕ is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We note there are other axiomizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' this one comes from [Cad, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ The following is the main result we are after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a category and ω : B → FinSet a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following are equivalent: (i) (B, ω) is a Galois category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (ii) B is a pre-Galois category and ω is exact and conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By the main theorem of Galois categories [Cad, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8], up to equivalence, B is the category of finite G-sets, for some pro-finite group G, and ω is the forgetful functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since G is an admissible group and B = S(G), it follows that B is pre-Galois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now suppose (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We verify the conditions of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Conditions (c) and (d) hold by assumtion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Any B-category is finitely complete by definition, and a non-degenerate one is finitely co-complete by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' thus (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus to complete the proof of (b), it suffices to show that every epimorphism is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be an epimorphism, and let R = Eq(f) be its kernel pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since equivalence relations are effective, the quotient g : X → X/R exists, and R = Eq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3, we see that g and f are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since g is the co-kernel of R, so is f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', f is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ The proposition can be summarized as: “Galois = pre-Galois + fiber functor.” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Categories of atoms A B-category is completely determined by its atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The A-category perspective is useful since it provides a bridge between B-categories and finite relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' All categories in this section are assumed to be essentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The A and B constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a B0-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define A(B) to be the full subcategory of Bop spanned by the atoms of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For example, if B = S(G) then A(B) = T(G)op is the opposite of the category of transitive G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be an essentially small category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define a category B(A) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An object of B(A) is a finite sequence X• = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) where Xi is an object of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A morphism (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) → (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Ym) consists of a function a: [n] → [m] together with a morphism Xi → Ya(i) in Aop for each i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Composition is defined in the obvious manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For any B0-category B, we have an equivalence Φ: B(A(B)) → B given on objects by Φ((X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn)) = X1 ∐ · · · ∐ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This follows from the basic properties of B0-categories established in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For any category A, the category B(A) is a B0-category and we have a natural equivalence A ∼= A(B(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) It is clear that co-products in B(A) are given on objects by (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xm) ∐ (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Yn) = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xm, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Yn), with the obvious structure maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We note that the zero object of B(A) is the empty sequence ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Suppose that X• = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) and Y• = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Ym) are isomorphic objects of B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since the composition is the identity, it follows that b ◦ a and a ◦ b are the identity maps of [n] and [m];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' thus n = m and a and b are inverse permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Moreover, fi : Xi → Ya(i) is an isomorphism with inverse ga(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus see that every object of B(A) is a finite co-product of atomic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ We thus see that there is a correspondence between B0-categories and all (essentially small) categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In the remainder of this section, we refine this correspondence, and determine what B1- and B-categories correspond to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To this end, we begin with one simple observation: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism in the category A, and let f ′: (Y ) → (X) be the corresponding morphism in B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f is an isomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' monomorphism, epimorphism) if and only if f ′ is an isomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' epimorphism, monomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The statement for isomorphisms is clear, as an inverse to one of f or f ′ gives an inverse to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Similarly, it is clear that if f is not an epimorphism then f ′ is not a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now suppose that f ′ is not a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then there exist distinct morphisms g′, h′: (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Zn) → (Y ) such that f ′ ◦ g′ = f ′ ◦ h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let g′ i and h′ i be the components of g′ and h′, and let gi and hi be the corresponding morphisms in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since g′ ̸= h′ there is some i such that g′ i ̸= h′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus gi and hi are distinct morphisms in A with gi ◦ f = hi ◦ f, and so f is not an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Finally, suppose that f ′ is not an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then there exist distinct morphisms g′, h′: (X) → (W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Wn) such that g′ ◦ f ′ = h′ ◦ f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By definition, g′ corresponds to a morphism g : Wi → X for some i, and h′ to a morphism h: Wj → X for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The equality g′ ◦ f ′ = h′ ◦ f ′ exactly means that i = j and g ◦ f = h ◦ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since g′ ̸= h′ we have g ̸= h, and so f is not a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Initial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose A has an initial set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then AI has I as an initial object, and A is the disjoint union of the AI’s (as a category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Conversely, if A is a (set-indexed) disjoint union of categories with initial objects, then A has an initial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be a category, and let B = B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) A has a finite initial set if and only if B has a final object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) A has an initial object if and only if B has an atomic final object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) Suppose that {I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , In} is an initial object of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We claim that I• = (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , In) is a final object of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Indeed, let X• = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xm) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The map a together with f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , fn define a morphism X• → I• in B, and it is clearly the unique such map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus I• is a final object of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This reasoning is reversible too: if I• is a final object of B then {I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , In} is an initial set of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) This is clear from the proof of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Amalgamations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A pre-amalgamation in A is a pair of morphisms (b: A → B, c: A → C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An amalgamation set for (b, c) is an initial set of this category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we call the elements of this set amalgamations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be a category and let B = B(A) be the corresponding B0-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following are equivalent: (a) Every pre-amalgamation in A has a finite amalgamation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) The category B has fiber products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let (b, c) be a pre-amalgamation in A, where b: A → B and c: A → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) be the fiber product of (B) with (C) over (A) in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The map (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) → (B) in B corresponds to morphisms fi : B → Xi in A, for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Similarly, the map (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) → C corresponds to morphisms gi : Xi → C in A, for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Clearly, fi ◦ a = gi ◦ b, so each (fi, gi) is an object of Amalg(b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We claim that S = {(fi, gi)}1≤i≤n is an amalgamation set for (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus let (f : B → Y, g : C → Y ) be an arbitrary object of Amalg(b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then f defines a morphism (Y ) → (B) in B, and similarly, g defines a morphism (Y ) → (C) in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The two composition to (A) agree, and so there is a unique morphism (Y ) → (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) that composes with the projections to the given morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This proves the claim, and so (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now suppose (a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let (B) → (A) and (C) → (A) be morphisms of atoms in B, corresponding to maps b: A → B and c: A → C in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let {(fi, gi)}1≤i≤n be an amalgamation set for (b, c), where fi and gi map to Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then, reversing the above reasoning, we see that (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , Xn) is naturally the fiber product of (B) and (C) over (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus find that the fiber product of morphisms of atoms in B always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9 that all fibers products exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This means that every diagram B � D A b � c � C � can be filled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', one can find D and the dotted arrows making the square commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be a category in which all pre-amalgamations have a finite amal- gamation set, and let B = B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is clear from the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We are finally ready to introduce the main concept of this section: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An A-category is an essentially small category A satisfying the following conditions: (a) The category A has a finite initial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Every pre-amalgamation has a finite amalgamation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) Every epimorphism in A is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An A1-category is an essentially small category satisfying conditions (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ The following is the main result of this section: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be a category and put B = B(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) B is a B1-category ⇐⇒ A is an A1-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) B is a B-category ⇐⇒ A is an A-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) B is a non-degenerate B-category ⇐⇒ A is an A-category with an initial object and the amalgamation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) follows from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4(a), and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) then follows from Proposi- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' and (c) then follows from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4(b) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' All morphisms in an A-category are monomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This follows from Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Any endomorphism in an A-category is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='18 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A category in which all endomorphisms are isomorphisms is called an EI- category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus the above corollary shows that every A-category is an EI-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Repre- sentations of EI-categories have received some attention in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', [GL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ We now discuss the condition Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7(c) in a bit more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The contrapositive of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' As Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9 suggests, when working on the “A side,” morphisms will in some sense be embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' From this perspective, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 18 NATE HARMAN AND ANDREW SNOWDEN There is one other perspective on Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7(c) that is sometimes useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let f : X → Y be a morphism in an A1-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We refer to objects in the amalgamation set of (f, f) as self-amalgamations of Y over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is always a trivial self-amalgamation, namely Y itself, or more precisely, the pair (idY , idY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One easily sees that f is an epimorphism if and only if this is the only self-amalgamation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus the contrapositive of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A1 and A2 be A1-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This motivates the following construction: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B1 and B2 be B1-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We define the tensor product category to be the B1-category B1 ⊠ B2 = B(A(B1) × A(B2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If B1 and B2 are both B-categories then so is B1 ⊠ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G1 and G2 be admissible groups with stabilizer classes E1 and E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One can then show S(G1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E1) ⊠ S(G2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E2) ∼= S(G1 × G2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E1 × E2), where E1 × E2 denotes the set of open subgroups of the product of the form U1 × U2 with Ui ∈ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Note that if E1 and E2 each contain all open subgroups then the same need not be true for E1 ×E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus one is essentially forced to confront stabilizer classes when considering the tensor product construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Classical Fra¨ıss´e theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now recall the classical formulation Fra¨ıss´e’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' While we will not apply this version of the theorem, it serves as motivation for the categorical form discussed in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2 that we do use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We will also use the language of relational structures in §7 to construct examples of A-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We refer to [Cam1] and [Mac] for more complete discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fix a signature Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A (relational) structure for Σ is a set X equipped with for each i ∈ I an ni-ary relation Ri on X (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', a subset of Xni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Given a structure X and a subset Y , there is an induced structure on Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we call structures obtained in this manner substructures of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An embedding of structures X → Y is an injective function that identifies X with a substructure of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The age of a structure Ω, denoted age(Ω), is the set of all finite structures that embed into Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The set |C| of isomorphism classes in C is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 19 C satisfies the amalgamation property, as defined in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' here we treat C as a category with morphisms being embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For a class C of structures, let Cn denote the subclass consisting of structures with n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose Ω is a homogeneous structure and C = age(Ω) has the property that |Cn| is finite for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then one easily sees that G = Aut(Ω) acts oligomorphically on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In this way, Fra¨ıss´e limits provide a powerful mechanism for constructing oligomorphic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We give a few examples of Fra¨ıss´e limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) Take the signature to be empty, so that a structure is simply a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The oligomorphic group G = Aut(Ω) is the infinite symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) Take the signature to consist of a single binary relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) Again, take the signature to consist of a single binary relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be the class of all finite simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is a Fra¨ıss´e class, and the limit is the Rado graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Categorical Fra¨ıss´e theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Given a class C of relational structures, one can regard C as a category with morphisms being embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fra¨ıss´e’s theorem is thus a statement about a certain class of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It turns out that the theorem actually holds for a much broader class of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We follow the treatment in the appendix to our recent paper [HS2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fix a category C in which all objects are monomorphisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we often refer to morphisms in C as embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' An ind-object in C is a diagram X1 → X2 → · · · in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is possible to consider ind-objects indexed by more general posets, but we will only need this simple version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is a natural notion of morphism between ind-objects, and between an ordinary object and an ind-object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' see [HS2, §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let Ω be an ind-object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that Ω is universal if every object of C embeds into Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that Ω is homogeneous if every isomorphism of finite subobjects is induced by an automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Precisely, this means the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then there must exist an automorphism σ of Ω such that σ ◦ α = β ◦ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that C is a Fra¨ıss´e category if it admits a universal homogeneous ind-object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We note that any two universal homogeneous ind-objects are isomorphic [HS2, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fra¨ıss´e’s theorem gives a characterization of Fra¨ıss´e categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To state it, we will need the amalgamation property (AP) defined in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following is the categorical Fra¨ıss´e theorem (in one form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 20 NATE HARMAN AND ANDREW SNOWDEN Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2 ([HS2, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Suppose that C has an initial object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then C is a Fra¨ıss´e category if and only if (RCC) and (AP) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Here is an example where the categorical Fra¨ıss´e theorem applies while the classical one does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A cubic space is a complex vector space V equipped with a linear map Sym3(V ) → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is a natural notion of embedding for cubic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In [HS2], we show that the category of finite dimensional cubic spaces is a Fra¨ıss´e category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' we give many other related examples as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fra¨ıss´e theory for A-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following is our main Fra¨ıss´e-like theorem for A-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be an A-category satisfying the following conditions: A has an initial object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A satisfies the amalgamation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A has countably many isomorphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then there exists an admissible group G and a stabilizer class E for G such that A is equivalent to T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We will actually prove a slightly more precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be any category satisfying the three conditions of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2, the category A is Fra¨ıss´e, and thus admits a universal homogeneous ind-object Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G be its automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For an object X, we let Φ(X) be the set of all embeddings X → Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' note that this is non-empty since Ω is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The group G naturally acts on Φ(X), via its action on Ω, and this action is transitive by homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Give α ∈ Φ(X), we let Gα be the stabilizer of α in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let E be the set of all subgroups of G of the form Gα, for some α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A be an A1-category satisfying the three conditions of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4, and let Ω, G, E , and Φ be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (a) The family E is a neighborhood basis for a first-countable admissible topology on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (b) The family E is a stabilizer class for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) The construction Φ defines a faithful and essentially surjective functor A → T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (d) If A is an A-category then the functor in (c) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4, we give an example of an A1-category (that is not an A-category) where the functor in (c) is not an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is a notion of completeness for admissible groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4, there is in fact a unique (up to isomorphism) complete group satisfying the concluding statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The group G constructed following the statement of the theorem is thie complete group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ We now prove the theorem, in a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We fix A, Ω, G, E , and Φ as in the theorem statement in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We also write 1 for the initial object of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X and Y be objects of A, and let α: X → Ω and β : Y → Ω be embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then there is a unique (up to isomorphism) diagram Y δ �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ β � 1 �♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ Z ǫ � Ω X γ �❧ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have Gǫ = Gα ∩ Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The existence and uniqueness of the diagram follow from the definition of A1-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have α = ǫγ, and so for σ ∈ G we have σα = σǫγ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' thus Gǫ ⊂ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Of course, the same holds with β, and so Gǫ ⊂ Gα ∩ Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now prove the reverse containment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus let σ ∈ Gα ∩ Gβ be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then the above diagram commutes with ǫ changed to σǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By uniqueness of the above diagram, it follows that ǫ = σǫ, and so σ ∈ Gǫ, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then we have a natural bijection E ∼= F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' in particular, E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Given α ∈ Φ(X) and β ∈ Φ(Y ), let (Z, γ, δ) be the amalgamation from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is clear that if (α, β) is modified by an element of G then the amalgamation is unchanged (up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This construction therefore yields a well-defined map E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This provides a map F → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One readily verifies the two maps are inverse to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since F is finite by the definition of A1-category, it follows that E is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The set E is a neighborhood basis for an admissible topology on G, and E is a stabilizer class for the admissible group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If α is the unique embedding of the trivial object into Ω then Gα = G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' thus G belongs to E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is clear that E is closed under conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8 shows that E is closed under finite intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that E is a neighborhood basis for a topology on G, and the E is a stabilizer class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It remains to show that the topological group G is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is non-archimedean by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now verify that it is Hausdorff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus suppose σ belongs to � U∈E U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then for any embedding α: X → Ω we have σα = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Finally, we show G is Roelcke pre-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It suffices to show Gα0\\G/Gβ0 is finite for two embeddings α0 : X → Ω and β0 : Y → Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This set is in bijection with G\\(G/Gα0 × G/Gβ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since G acts transitively on Φ(X) with stabilizer Gα0, the set G/Gα (with its G-action) is identified with Φ(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' similarly, G/Gβ0 is identified with Φ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus finiteness follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ We have thus proved Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Now, the action of G on Φ(X) is smooth, by definition of the topology on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' If α: X → Y is a morphism in A then there is an induced morphism α∗: Φ(Y ) → Φ(X) of G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It follows that we have a functor Φ: A → T(G)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To complete the proof of the theorem, we study properties of this functor in the next sequence of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The functor Φ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let α and β be two morphisms X → Y in C such that α∗ = β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Choose an embedding γ : Y → Ω, which is possible since Ω is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By assumption, we have γ ◦ α = γ ◦ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since γ is a monomorphism, it follows that α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus Φ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 22 NATE HARMAN AND ANDREW SNOWDEN Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The essential image of Φ is T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' For an object X of A, the G-set Φ(X) is isomorphic to G/Gα, where α ∈ Φ(X) is any element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ We have thus proved Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We now turn our attention to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In what follows, we assume that A is an A-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The functor Φ is conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since A is an A-category, it is enough to show that α is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus suppose that β and γ are maps Y → Z such that β ◦ α = γ ◦ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus have α∗β∗ = α∗γ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since α∗ is an isomorphism, it follows that β∗ = γ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since Φ is faithful, we find β = γ, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The functor Φ is full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X and Y be objects of C, and let ϕ: Φ(Y ) → Φ(X) be a map of G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Choose an element β ∈ Φ(Y ), and let α = ϕ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Note that since ϕ is G-equivariant, we have Gβ ⊂ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have Gǫ = Gα ∩ Gβ = Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus γ∗: Φ(Z) → Φ(Y ) is an isomorphism of G-sets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By the Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='13, it follows that γ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since the diagram in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8 is only defined up to isomorphism, we may as well suppose that Z = Y , γ = idY , and β = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus see that δ∗: Φ(Y ) → Φ(X) is a map of G-sets carrying β to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since Φ(Y ) is transitive, it follows that ϕ = δ∗, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fra¨ıss´e theory for B-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The following is our main theorem on B-categories, and contains Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2 as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B be a B-category that is non-degenerate and has countably many isomorphism classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then there is a first-countable admissible group G and a stabilizer class E such that B is equivalent to S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Moreover, if equivalence relations in B are effective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', B is pre-Galois) then B is equivalent to S(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let A = A(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='8, this is an A-category satisfying the three conditions of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus by that theorem, we have A ∼= T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) for some first-countable admissible group G and stabilizer class E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have equivalences B = B(A) and S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ) = B(T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op), and so we obtain an equivalence B ∼= S(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The second statement follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Examples from relational structures We now look at some examples of A-categories and B-categories coming from classes of relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' See §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1 for basic definitions on relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' General comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be a non-empty class of finite relational structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We assume throughout this section that C is hereditary and that |Cn| is finite for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Recall that we can regard C as a category, with morphisms being embeddings of structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The category C is an A1-category, and the following are equivalent: (a) C is an A-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (c) Given Y ∈ C and a proper substructure X ⊂ Y , there exists a non-trivial self- amalgamation of Y over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The class C contains the empty structure since it is non-empty and hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is clear that the empty structure is the initial object of C, and so C has an initial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This verifies Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let (β, γ) be a pre-amalgamation, where β : A → B and γ : A → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Consider an object (δ, ǫ) of Amalg(β, γ), where δ: B → D and ǫ: C → D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We say that (δ, ǫ) is minimal if δ and ǫ are jointly surjective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', D = im(δ) ∪ im(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Every object of Amalg(β, γ) admits a unique (up to isomorphism) map from a minimal object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Indeed, in the above notation, let D′ = im(δ) ∪ im(ǫ), regarded as a substructure of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then D′ is a minimal, with structure maps δ and ǫ, and the inclusion D′ → D is a map in Amalg(β, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' a key point here is that D′ still belongs ot the class C since C is hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let S be a set of isomorphism class representatives for the minimal objects of Amalg(β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The above argument shows that S is an amalgamation set for (β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This verifies Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have already explained (at the end of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4) how the remaining three conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ Suppose that C is indeed an A-category and that it is also satisfies the amalgamation property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' then C is a Fra¨ıss´e class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let Ω be the Fra¨ıss´e limit, and let G = Aut(Ω), which acts oligomorphically on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4 gives an equivalence of A with T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op, where E is the set of subgroups of G of the form G(A) where A ⊂ Ω is a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (Recall that G(A) is the subgroup of G fixing each element of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=') 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be the class of all finite sets (the signature in this case is empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is an A-category by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The amalgamation property holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The Fra¨ıss´e limit is the countable set Ω = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='} and its automorphism group is the infinite symmetric group S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let E be the stabilizer class consisting conjugates of S(n), for variable n (see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then we have an equivalence of categories C ∼= T(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We can also describe the A-category T(S)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Define a category C′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then T(S)op is equivalent to C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Total orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be the class of finite totally ordered sets (the signature consists of a single binary relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is an A-category by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1, and the amalgamation property holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The Fra¨ıss´e limit Ω is the set of rational numbers, with its usual order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let G = Aut(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We thus have an equivalence C ∼= T(G)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The Delannoy category studied in [HSS] is associated to this group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The countable matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be the class of all simple graphs in which each vertex belongs to at most one edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' the signature consists of a single binary relation (the edge relation on vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is a Fra¨ıss´e class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The limit Ω is a perfect matching on a countable vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Its automorphism group G is the wreath product Z/2 ≀ S, where S is the infinite symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The category C is an A1-category by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1, but it is not an A-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To see this, let Y be a single edge, and let X ⊂ Y be one of the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then any map X → Z admits at most one extension to Y , and so Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(b) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Alternatively, the only self-amalgamation of Y over X is the trivial one, and so Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1(c) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5 does produce a faithful and essentially surjective functor Φ: C → T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op, for an appropriate stabilizer class E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The inverse map does not come from a map Y → X in C, as there are no such maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C0 be the (non-hereditary) subclass of C consisting of graphs in which each vertex belongs to exactly one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then Φ restricts to an equivalence C0 → T(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' E )op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Permutation classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let P be the class of all finite sets equipped with a pair of total orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let X be a structure of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Label the elements of X as 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , n according to the first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We can then enumerate the elements of X under the second order to get a string in the alphabet {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , n} in which each letter appears once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This string exactly determines the isomorphism type of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We can thus view structures in P as permutations, and thus typically use symbols like σ for its members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The embedding order on P is the so-called containment order on partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A permutation class is a non-empty hereditary subclass C of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is an extensive literature on permutation classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' for an overview, see [Vat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let σ be a permutation of length n, and let α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , αn be other permutations of lengths m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' There is then a permutation σ[α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , αn] of length m = m1 + · · · + mn, called inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We refer to [Vat, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2] for the definition, and just give an example here: 231[12, 321, 3412] = 56 987 3412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' We have inserted spaces into the result to make the operation more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The three com- ponents on the right correspond to the three permutations in the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Each uses an interval of numbers, and the order of the intervals is determined by the outside permuta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A permutation class C is substitution closed if σ[α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , αn] belongs to C whenever σ, α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , αn all belong to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let C be a substitution closed permutation class containing some permu- tation of length ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then C is an A-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let τ → σ be a non-isomorphism in C, and suppose the embedding misses i ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let n be the length of σ, and consider the inflation σ′ = σ[α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' , αn] where αj = 1 for j ̸= i, and αi has length length 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Note that C contains the permutation 1 and some permutation of length 2 since it is hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' One easily sees that σ′ is a non-trivial self-amalgamation of σ over τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Thus C is an A-category by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ PRE-GALOIS CATEGORIES AND FRA¨ISS´E’S THEOREM 25 Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' A permutation is separable if it can be built from the permutation 1 with sums and skew-sums;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' the empty permutation is also separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' (Given two permutations α and β their sum is 12[α, β] and their skew-sum is 21[α, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=') Equivalently, a permutation is separable if the permutations 2413 and 3142 do not embed into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The class C of all separable permutations is a substitution closed permutation class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' It is thus an A-category by the above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The class C does not have the amalgamation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' To see this, regard 123 as a subper- mutation of 1342 (using the first three positions) and 3124 (using the last three positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In the class of all permutations, there is a unique amalgamation, namely 41352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' This is not separable, since when the middle 3 is deleted we obtain 3142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Let B = B(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Then B is a B-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Since C has an initial object, the final object 1 of B is atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' However, since C has the joint embedding property, it follows that X × Y is non-empty for all atoms X and Y of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' □ References [Cad] Anna Cadoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' “Galois categories” in Arithmetic and geometry around Galois theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 304, Birkh¨auser/Springer, Basel, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 171–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1007/978-3-0348-0487-5 3 [Cam1] Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Cameron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Oligomorphic permutation groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' London Mathematical Society Lecture Note Series, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 152, Cambridge University Press, Cambridge, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' [Cam2] Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Cameron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Homogeneous Permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 9 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='37236/1674 [Car] Olivia Caramello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fraisse’s construction from a topos-theoretic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Univers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 8 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 2, 261–281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1007/s11787-014-0104-6 arXiv:0805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='2778 [Del] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Deligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' La cat´egorie des repr´esentations du groupe sym´etrique St, lorsque t n’est pas un entier naturel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In: Algebraic Groups and Homogeneous Spaces, in: Tata Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Res.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 131 (Part 3) 1992, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 49–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1090/conm/131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 237 (1953), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 540–542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' [GL] Wee Liang Gan, Liping Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Noetherian property of infinite EI categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' New York J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='04526 [HS2] Nate Harman, Andrew Snowden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Ultrahomogeneous tensor structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='09626 [HS3] Nate Harman, Andrew Snowden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Oligomorphic component groups of pre-Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' In preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' [HSS] Nate Harman, Andrew Snowden, Noah Snyder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' The Delannoy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='15392 26 NATE HARMAN AND ANDREW SNOWDEN [Irw] Trevor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Irwin.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' On symmetric fusion categories in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' Selecta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' 26 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='1007/s00029-020-00567-5 arXiv:1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='01492 [Stacks] Stacks Project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' http://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='edu (accessed 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' [Vat] Vincent Vatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' “Permutation classes” in “Handbook of enumerative combinatorics” ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' by Mikl´os B´ona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' CRC Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content=' arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} +page_content='5159' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFST4oBgHgl3EQfXTj3/content/2301.13784v1.pdf'} diff --git a/4dAyT4oBgHgl3EQfcPej/content/tmp_files/2301.00279v1.pdf.txt b/4dAyT4oBgHgl3EQfcPej/content/tmp_files/2301.00279v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..968278404a192f25b7226bd0a44524e5e9924096 --- /dev/null +++ b/4dAyT4oBgHgl3EQfcPej/content/tmp_files/2301.00279v1.pdf.txt @@ -0,0 +1,537 @@ +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. +□ +References +[1] F. A. Almahdi, M. Tamekkante and R. A. K. Assaad, On the right orthogonal complement +of the class of w-flat modules, J. Ramanujan Math. Soc. 33 No.2 (2018) 159–175. 2, 3 +[2] H. Kim and F. Wang, On LCM-stable modules, J. Algebra Appl. 13, no. 4 (2014), 1350133, +18 pages. 2 +[3] B. H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967) 155–158. 2 +[4] L. Mao and N. Ding, FP-projective dimension, Comm. in Algebra. 33 (2005) 1153–1170. 2, +5 +[5] C. Megibben, Absolutely pure modules, Proc. Am. Math. Soc. 26 (1970) 561-566. 2 +[6] A. Mimouni, Integral domains in which each ideal is a w-ideal, Commun. Algebra 33 (2005), +1345–1355. 4 +[7] B. Stenstr¨om, Coherent rings and FP-injective modules, J. Lond. Math. Soc. 2(2) (1970) +323–329. 2 +[8] Y. Y. Pu, W. Zhao, G. H. Tang, and F. G. Wang, w∞-projective modules and Krull domains, +Commun. Algebra, Vol. 50, No. 8, (2022), 3390–3402. 2, 5 +[9] M. Tamekkante, R. A. K. Assaad and E. Bouba, Note On The DW Rings, Inter. Elec. J. of +Algebra. VO. 25 (2019). 5 +[10] M. Tamekkante, M. Chhiti and K.Louartiti, Weak Projective Modules and Dimension, Int. +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 +[18] F. Wang and L. Qiao, A homological characterization of Krull domains II, Comm. in Algebra. +(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 +[20] F. Wang and L. Qiao, A new version of a theorem of Kaplansky. arXiv: 1901.02316. 2, 7 +[21] F. G. Wang and D. C. Zhou, A homological characterization of Krull domains, Bull. Korean +Math. Soc. 55 (2018), no. 2, 649–657. 2 +[22] S. Xing and F. Wang, Purity over Pr¨ufer v-multiplication domains, J. of Algebra Appl. Vol. +16, No. 5 1850100 (2018). 7 +[23] H. Y. Yin, F. G. Wang, X. S. Zhu and Y. H. Chen, w-modules over commutative rings, J. +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: refat90@hotmail.com + diff --git a/4dAyT4oBgHgl3EQfcPej/content/tmp_files/load_file.txt b/4dAyT4oBgHgl3EQfcPej/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8de69ad0e11b3c7fde1986cd402923c8ffa7c28 --- /dev/null +++ b/4dAyT4oBgHgl3EQfcPej/content/tmp_files/load_file.txt @@ -0,0 +1,577 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf,len=576 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='00279v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='AC] 31 Dec 2022 A NOTE ON WEAK w-PROJECTIVE MODULES REFAT ABDELMAWLA KHALED ASSAAD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In this paper, we in- troduce and study some properties of weak w-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' And we use these modules to characterize some classical rings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' we will prove that a ring R is a DW -ring if and only if every weak w-projective is projective,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Introduction In this paper, all rings are considered commutative with unity and all modules are unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring and M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='dim(R) and gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='dim(R) to denote, respectively, the weak and global homological dimensions of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Now, we review some definitions and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let J be an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Note that the set GV (R) of GV -ideals of R is a multiplicative system of ideals of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is Defined torGV (M) = {x ∈ M | Jx = 0 for some J ∈ GV (R)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is clear that torGV (M) is submodule of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' M is said to be GV -torsion (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', GV -torsion-free) if torGV (M) = M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', torGV (M) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A GV -torsion-free module M is called a w-module if Ext1 R(R/J, M) = 0 for any J ∈ GV (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, projective modules and reflexive modules are w-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In the recent paper [23], it was shown that flat modules are w-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let w − Max(R) denote the set of w-ideals of R maximal among proper integral w-ideals of R (maximal w-ideals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [23, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8], every maximal w-ideal is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any GV -torsion free module M, Mw := {x ∈ E(M) | Jx ⊆ M for some J ∈ GV (R)} 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 13D05, 13D07, 13H05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' projective modules , weak w-projective modules, w-flat, GV -torsion, finitely presented type, DW -rings, coherent rings, w-coherent rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 1 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is clear that a GV -torsion-free module M is a w-module if and only if Mw = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M and N be R-modules and let f : M → N be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [12], f is called a w-monomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', w-epimorphism, w-isomorphism) if fm : Mm → Nm is a monomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=', an epimorphism, an isomorphism) for all m ∈ w − Max(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In recent years, homological theoretic characterization of w-modules has received attention in several papers the literature (for example see [[1], [21], [19], [18]]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In [14], F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Wang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Kim generalized projective modules to w-projective modules by the w-operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by Pw the class of all w-projective R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by Sw the class of all w-split R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4], every w-split module is w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by Fw the class of all w-flat R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Following [18], Wang and Qiao introduce the notions of the weak w-projective dimension (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pd) of a module and the global weak w-projective dimension (gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-dim) of a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by W∞ the class of all strong w- modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then all GV -torsion-free injective modules are strong w-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Clearly, P†∞ w ⊆ W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Denote by FP the class of all FP-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, an absolutely pure module is precisely a FP-injective module in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A NOTE ON WEAK w-PROJECTIVE MODULES 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' results In this section, we introduce a characterize of some classical ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' But we need the following lemma Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ([18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' It is obvious that, for the class of modules { projective } ⊆ { w-split } ⊆ { w-proective } ⊆ {weak w-proective } ⊆ { w-flat }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[1], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5], if R is a perfect ring, then the five classes of modules above coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In the following proposition, we will give some characterizations of weak w- projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then the following are equivalent: (1) M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) M ⊗ F is weak w-projective for any projective R-module F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) HomR(F, M) is weak w-projective for any finitely generated projective R- module F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let F be a projective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is a weak w-projective, Ext1 R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, Ext1 R(F ⊗ M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, F ⊗ M is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (1) and (3) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Follow by letting F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is weak w-projective, so Ext1 R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Ext1 R(HomR(F, M), N) = 0, which implies that HomR(F, M) is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is weak w-projective and A ∈ P†∞ w , so Ext1 R(M, A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, 0 → HomR(M, A) → HomR(M, B) → HomR(M, C) → 0 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → L → E → M → 0 be a w-exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any R- module N ∈ P†∞ w so N ∈ W∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[18], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1], we have the exact sequence 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD 0 → HomR(M, N) → HomR(E, N) → HomR(L, N) → Ext1 R(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is weak w-projective, so Ext1 R(M, N) = 0, and (5) is holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) ⇒ (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → L → E → M → 0 be an exact sequence with E is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall from [12], that a ring is said to be w-coherent if every finitely generated ideal of R is of finitely presented type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, if M is weak w-projective R-module, so TorR n (M, Hom(N, E)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5] and since every R{x}-module in P†∞ w by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Henace, by [[16], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='13(6)], we have TorR n (M, Hom(N, E)) ∼= Hom(Extn R(M, N), E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Which implies that, TorR n (M, Hom(N, E)) = 0 □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Every weak w-projective of finite type is of finitely presented type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a weak w-projective R-module of finite type, so by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9] M is w-projective of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, by [[13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='22], we have M is finitely presented type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) Mw/M is a weak w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) M is a weak w-projective if and only if so is Mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' So, by [[13], Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5] we have Mw/M a GV -torsion module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3(2)], we have Mw/M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N be an R-module in P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since M is GV -torsion-free, we have by (1) Mw/M is weak w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Consider the following exact sequence 0 → M → Mw → Mw/M → 0 which is w-exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5], M is weak w-projective if and only if Mw is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Examples of DW-rings are Pr¨ufer domains, domains with Krull dimension one, and rings with Krull dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' We note that if R is DW-ring, then every R-module in P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' In the following proposition, we will give a new characterizations of DW-rings which are the only rings with these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following statements are equivalent: A NOTE ON WEAK w-PROJECTIVE MODULES 5 (1) Every weak w-projective R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every w-projective R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Every GV -torsion R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) Every GV -torsion-free R-module is strong w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) Every finitely presented type w-flat is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) Every weak w-projective R-module is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (7) R is DW-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2) and (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a strong w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[12], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8] since every strong w-module is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Trivial, since every projective R-module is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a finitely presented type w-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9], we have M is a finite type w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a projective R-module by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) ⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a finitely presented w-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, M is finitely presented type w-flat, so M is projective by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[9], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1], we have R is a DW-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (6) ⇒ (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5, we have Mw/M is weak w-projectiveis and so w-module by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, Mw/M is a GV -torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Mw/M = 0 and so Mw = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, M is w-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, R is a DW-ring by [[12], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (7) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For any R-module N, we have Ext1 R(M, N) = 0 because N ∈ P†∞ w (since R is DW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Note that the equivalence (1) ⇔ (7) in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6 was given in [[8], Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4] for the domain case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Mao and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Ding in [[4]], proved that a ring R is a Von Neumann regular if and only if every FP-projective R-module is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Next, we will give new characterizations of a Von Neumann regular rings by weak w-projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, the following statements are equivalent: (1) Every FP-projective R-module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every finitely presented R-module is weak w-projectiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Every finitely presented R-module is w-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) R is a Von Neumann regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Follows from the fact that every finitely presented R-module is FP-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a finitely presented R-module, so M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is w-flat by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let I be a finitely generated ideal of R, then R/I is finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' So R/I is w-flat by (3), then w − fdR(R/I) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, w − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='gl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='dim(R) = 0 by [[19], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, R is Von Neumann regular by [[15], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a FP-projective, so M is projective by [[4], Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Next, we will give an example of FP-projective module which is not weak w- projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 6 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, (X) is FP-projective R-module which is not weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since R is a Quasi-Frobenius ring, then every absolutely pure R-module is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, for any absolutely pure R-module N, we have Ext1 R((X), N) = 0, so (X) is FP-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' But, (X) is not projective by [[10], Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='2], and so not weak w-projective, since R is DW-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Recall from [[15]] that a ring R is said to be w-semi-hereditary if every finite type ideal of R is w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following are equivalent: (1) R w-semi-hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every finite type submodule of a free module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Every finite type ideal of R is a weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) Every finitely generated submodule of a free module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) Every finitely generated ideal of R is a weak w-projective Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let J be a finite type submodule of a any free module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, J is w-projective by [[15], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then J is weak w-projective by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3) ⇒ (5) and (2) ⇒ (4) ⇒ (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' These are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (5) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let J be a finite type ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then J is w-isomorphic to a finitely generated subideal I of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence J is weak w-projective by hypothesis and [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Every GV -torsion-free weak w-projective module is torsion- free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be a GV -torsion-free weak w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, M is a GV - torsion-free w-flat by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='Thus, by [[13], Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6], we have M is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ In the next example we will prove that a weak w-projective module need not to be torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be an integral domain and J be a proper GV -ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then M := R ⊕ R/J is a weak w-projective module but not torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let R be a ring and M be a finitely presented R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, the following statements are equivalent: (1) M is w-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) M is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Trivial, since every w-split R-module is weak w-projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → A → B → C → 0 be a w-exact sequence of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, for any maximal w-ideal m of R, 0 → Am → Bm → Cm → 0 is exact sequence of Rm-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, since Mm is free by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[20], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let C be a finitely presented type R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, the follow- ing statements are equivalent: (1) C is a weak w-projective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Every w-exact sequence of R-modules 0 → A → B → C → 0 is w-pure exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since every weak w-projective is w-flat by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[22], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6], we have the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let 0 → A → B → C → 0 be a w-exact sequence, so is a w-pure exact by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, C is w-flat by [[22], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence C is a weak w-projective by [[18], Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following are equivalent for a finite type R-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) M is a w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Ext1 R(M, B) = 0 for any B ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) Ext1 R(M, N) = 0 for any R{x}-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) M{x} is a projective R{x}-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' This is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' By [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (3) ⇒ (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N be an R{x}-module, we have by [[16], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5], Extn R{x}(M{x}, N) ∼= Extn R(M, N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, M{x} is a projective R{x}-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (4) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M{x} be a projective R{x}-module, so M{x} is finitely generated by [[13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='24] and since M is of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, by [[13], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='18], M is w-projective module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M be an R-module and any integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) Extm R (M, D) = 0 for all P†∞ w -divisible R-module D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 8 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' ASSAAD Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (1) ⇒ (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let N ∈ P†∞ w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then there exists an exact sequence of R-modules 0 → N → E → H → 0, where E is a GV -torsion-free injective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, D is P†∞ w -divisibl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The left term is zero by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Extm+1 R (M, N) = 0, which implies that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM ≤ m by [[18], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' (2) ⇒ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM ≤ m and D be a P†∞ w -divisible R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, we have the exact sequence 0 = Extm R (M, E) → Extm R (M, H) → Extm+1 R (M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The right term is zero by [[18], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Therefore, Extm R (M, H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Let M and N be two R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Then, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdR(M ⊕ N) = sup{w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRN} Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' The inequality w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdR(M ⊕ N) ≤ sup{w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRN} follows from the fact that the class of weak w-projective modules is closed under direct sums by [[18], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='5(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' For the converse inequality, we may assume that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdR(M ⊕ N) = n is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Thus, for any R-module X ∈ P†∞ w , Extn+1 R (M ⊕ N, X) ∼= Extn+1 R (M, X) ⊕ Extn+1 R (N, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Since Extn+1 R (M ⊕ N, X) = 0 by [[18], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' Hence, Extn+1 R (M, X) = Extn+1 R (N, X) = 0, which implies that, sup{w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRM, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='w-pdRN} ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' □ References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content=' A.' metadata={'source': 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refat90@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQfcPej/content/2301.00279v1.pdf'} diff --git a/5NE3T4oBgHgl3EQfpArI/vector_store/index.pkl b/5NE3T4oBgHgl3EQfpArI/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..2579c906f9f1a6739baf39850b259ccb9d1609c8 --- /dev/null +++ b/5NE3T4oBgHgl3EQfpArI/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:31d697d382c5913a13d8c7f99df84f60c743e4848e42c402ec70c32bd8bab464 +size 14933 diff --git a/89E3T4oBgHgl3EQfqwpo/content/tmp_files/2301.04654v1.pdf.txt b/89E3T4oBgHgl3EQfqwpo/content/tmp_files/2301.04654v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..edc80aacdb4dcb0a16fb08eb7153a29e0d2a02ee --- /dev/null +++ b/89E3T4oBgHgl3EQfqwpo/content/tmp_files/2301.04654v1.pdf.txt @@ -0,0 +1,1127 @@ +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: tiago.araujol@ufrpe.br +† ricardo.carmo@ifal.edu.br +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. +[1] J. R. Dorfman, An Introduction to Chaos in Nonequilib- +rium Statiscal Mechanis, 1st ed. (Cambridge University +Press, 1999). +[2] Y. G. 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Bhosale, Superposition and higher-order spacing +ratios in random matrix theory with application to com- +plex systems, Physical Review B 104, 054204 (2021). + diff --git a/89E3T4oBgHgl3EQfqwpo/content/tmp_files/load_file.txt b/89E3T4oBgHgl3EQfqwpo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab82925cfb845f130a4965c76d6b9e75cf5398e0 --- /dev/null +++ b/89E3T4oBgHgl3EQfqwpo/content/tmp_files/load_file.txt @@ -0,0 +1,783 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf,len=782 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04654v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='CD] 11 Jan 2023 Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Ara´ujo Lima1, ∗ and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The results reported here seek to shed light on the quantization of classically chaotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Both are elliptical perturbations of chaotic billiards with originally circular sectors on their borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the E-C3B family, the eigenstates can be split into singlets and doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' BRB describes p(s) for singlets as the previous family in the mixed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We observed that as ρc decreases, the p(s)’s tend to move away simultaneously from the GOE (singlets) and GUE (doublets) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Keywords: Billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In particular, these particles, constituents of gases, are asso- ciated with the concept of ergodicity, then called molecu- lar chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The so-called Ergodic Hypothesis only gained the rigor of a theorem with the work of the Russian mathematician Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Sinai in the 60s- 70s for an ideal gas of only two particles [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A system is chaotic if two neighboring trajectories in the phase space separate exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Suppose the distance in phase space between such trajectories is proportional to eλt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The λ parameter is called the Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In re- ality, λ represents the greatest of Lyapunov’s exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Therefore, the existence of at least one positive Lyapunov exponent characterizes a chaotic system [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Billiards sys- tems are prototypes in the study of chaos and describe ∗ Corresponding author: tiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='araujol@ufrpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='br † ricardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='carmo@ifal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='br the free movement of a point particle in a closed domain Ω with elastic reflections on the boundary ∂Ω of the do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The nature of this conservative dynamical system depends exclusively on the shape of the border ∂Ω, vary- ing from entirely regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', ellipses and annular con- centric regions) to completely chaotic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', Sinai billiard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Without loss of generality, we consider that the particle has mass m = 1 and velocity of module |v| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A primor- dial example that deserves to be mentioned here is the Bunimovich stadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This billiard can present λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Its shape consists of two semicircles joined by two finite- size segments 2t, forming a stadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' It is chaotic for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In a pure circular billiard, collisions keep the angular momentum in relation to its center (focus) con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The Bunimovich stadium does not present this property in its dynamics, known as a defocusing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Quantum mechanics has been one of the best-tested physical theories since its emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Nothing is more natural than whether there is an effect analogous to chaos in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The term quantum chaos is generally understood as studying the quantum behav- 2 ior of classically chaotic systems [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In billiards, obtaining the energy spectrum is an essen- tial step for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Where k2 n = 2mEn/ℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This approach be- came relevant after two important conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Namely, the Berry-Tabor (BT) conjecture [9] and the Bohigas- Giannoni-Schmit (BGS) conjecture [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' As a result, the semiclassical nns distribution p(s) must obey Poisson: pP(s) = exp(−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Several recent works have improved the turnover of the BGS conjecture in a theorem [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These proofs still have controver- sies and limitations pointed out by some authors [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The terminology ”BGS conjecture” fits the current arti- cle for quantized billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' More recently, in [17], the conjecture was extended to purely ergodic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Later, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Dembowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' used microwave bil- liards with C3 symmetry to check experimentally the re- sult predicted by LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Besides, they showed that singlets follow GOE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, we propose to shed light on the quantum properties of bil- liards with mixed classical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In sequence, we introduce a perturbation of the C-C3B, replacing the circumferences with ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Recently, billiards with el- liptical borders have been studied in other contexts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', in singular potentials [30], in relativistic limits [31], and flows that move around chaotic cores [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The first one consists of a perturbation of the Bunimovich Stadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Based on these previous works, we focus our analysis on this last interval and t ∈ (0, 1/ √ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The E-C3B is based on C-C3B, but ellipses instead of circumferences curve the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The larger (smaller) ellipse has Ae (ae) and Be (be) semi-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In all cases described here, the relations Ae = 2ae and Be = 2be are maintained, with ae and be in the range (0, √ 3/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The LSS billiard is reproduced with ae = be = √ 3/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 A Fundamental Domain (FD) is a neighborhood in Ω that contains only one image for any point in the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Besides the boundary of the ∂Ω, there are addi- tional boundaries between adjacent FDs, which are the symmetry lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Classically, billiard dynamics can always be reduced to a FD by assuming specular reflections at the symmetry lines [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For this, we use the FD of each billiard in our calculations on the classical dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1, we graph billiards in families indicating the parameters and their respective FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (a) Original Boundaries of Elliptical Stadium Billiard and Elliptical-C3 Billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For ESB, the symmetry lines are referent to reflections on vertical and horizontal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' While E-C3B are referent to 120◦ rotational axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (b) Fundamental Domains of ESB and E-C3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The symmetry lines are replaced by additional boundaries forming the planar region where we analyze these billiards’ classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' An additional part of the boundary dictates this edge and does not change with the variation of parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The small gray dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We plot one example of a stable trajectory in red for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The phase plane is partitioned into Nc small disjoint cells to measure these quantities [27, 29, 35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The relative measure r(n) is defined as the fraction of visited cells averaged over a set of ICs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', r(n) = ⟨N(n)⟩/Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In our numerical approach, we consider Nc = 106, n = 2 · 107, and averages in 20 random ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the billiards with mixed phase space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 2, ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='991884 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='857184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 shows a numerical diagram of ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ergodic property (ρc = 1) is numerically guaranteed in black regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' y y vy (a) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15 (b) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper Panels: ESB boundaries for some values of pa- rameters (a, t) with stable trajectories in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panels: corresponding phase portraits for 105 collisions with the ver- tical boundary from the IC (y0, vy0) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0) (small gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The red plots correspond to the trajectories in the up- per panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed phase spaces present ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='991884 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='857184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1, which does not change with the vari- ation of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The small gray dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The values of chaotic fraction are ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='935152 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='800792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 shows a numerical diagram of ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ergodic property (ρc = 1) is numerically guaranteed in black regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This map 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 ae be FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Left Panel: diagram of the chaotic fraction of the phase space ρc for the ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The tiny green line is the critical line t(a) = √ a2 − 1 studied in [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Right Panel: same diagram for the E-C3B showing a distinguished phase space behavior depending on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ergodic property (ρc = 1) is numerically guaranteed in black regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These maps will guide us in exploring quantum properties, where these values will be relevant parameters to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' will guide us in exploring quantum properties described in the next section, where these values will be relevant parameters to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' G x x (a) ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784 (b) ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088 vx be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper Panels: E-C3B boundaries for some values of parameters (ae, be) with stable trajectories in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panels: corresponding phase portraits for 2·107 collisions with the horizontal boundary from the IC (x0, vx0) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0) (small gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The red plots correspond to the trajectories in the upper panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed phase spaces present ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='935152 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='800792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' QUANTIZATION AND EIGENVALUES SHORT CORRELATIONS All Energy spectra {En} of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (1) were calculated with an algorithm based on the scaling method intro- duced by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Vergini and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Saraceno (VS) in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Our results are based on sets of approximately 70,000 eigenvalues for a given pair of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The Singlets Case Initially, we focused on results for ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Some pro- poses have been made to describe these distributions for systems whose present mixed-phase space on its classi- cal counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here we focus on two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' They result in intermediate formulas between Poisson and GOE statistics through parameters variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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) \uf8f1 \uf8f2 \uf8f3 ρ2 r (β + 1)Γ � β+2 β+1 �Q � 1 β + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' aβ(ρcs)β+1 � + [2ρrρc + (β + 1)aβρβ+2 c sβ] exp[−aβ(ρcs)β+1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6) As in the Brody distribution, aβ = � Γ � β+2 β+1 ��β+1 and Q(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' x) is the Incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This dis- tribution can go through other distributions varying the free parameters ρc and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For β = 0, pBRB(s) = pP(s) and for β = 1 it recovers the distribution of Berry-Robnik (BR) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' If ρc = 0, pBRB(s) = pP(s) again, while for ρc = 1, pBRB = pB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The nns for ESB were previously studied in [22] with around 3,000 eigenvalues of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We use the VS method to obtain around 65,000 eigenvalues beyond the first 5,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BRB distribution can fit all p(s) ob- tained for all parameters tested on ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We have two independent parameters for this distribution, ρc, and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' However, we fixed ρc at the value obtained in the diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The upper panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5 shows representa- tive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='020, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='978 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='018 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='014, intermediate distributions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These results go in the direction of the quantum localization, previously studied in other billiards systems [42, 43] and discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 �� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 ��� p(s) p(s) s s s ( ESB E-C3B E-C3B E-C3B ESB ESB (b) \x0e \x0f \x10 \x11 (e) \x12\x13 \x14 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='287 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Representative results for BRB distributions fits for p(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper Panels: results on ESB with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 and some values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='287 (ρc = 1) presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='020, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed cases t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='978 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='018 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='014 respectively, intermediate distributions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panels: results on E-C3B with some values of (ae, be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case, (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25) (ρc = 1) presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='097, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed cases, (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256) and (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522) (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='057 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='073 respectively, in the range of intermediate distributions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The fits with the Brody formula and BRB distribution are indistinguishable in both billiards families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The phenomenon can be reviewed in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We follow closely [42] in the short description below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' From [42] the ratio is expressed as α = kL πNT , (7) where L is the perimeter of the boundary and k2 ∼ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The condition for quantum dynamical localization in a given energy spectrum, α ⩽ 1, can then be written as k ⩽ kc = πNT/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Sat- uration of ⟨p2⟩ occurs at different times NT depending on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the ESB family, all calculated spec- tra have kmax ≲ kc as the largest eigenvalue, equivalent to the 70,000th level at least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These facts are in agree- ment with the intermediate statistics well fitted with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6) as in [23, 27, 40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The same occurs for the sin- glets in the E-C3B family, where the condition kmax ≲ kc is equivalent to the 70,000th level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The representative results are in the lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='097, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='057 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='073, in the range of intermediate distribu- tions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the next section, we discuss the doublets subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The Doublets Case Consider a classically chaotic system with time- reversal (TR) invariance and a point-group (PG) symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For example, consider a billiard in the xy plane with the C3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Let Θ be the time reversal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Θ is an antiunitary operator that commutes with the Hamil- tonian H, which has eigenvalue Em, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', Hϕm = Emϕm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' It follows that HΘϕm = ΘHϕm = EmΘϕm (Θϕm is also an eigenfunction of H with the same eigenvalue Em).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Are 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 \x15 \x16\x17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0 \x18 \x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 log10 n Elliptical-C3 Billiard Elliptical Stadium Billiard 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lines are guides for the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper panel: results for members of the ESB family with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The red dots are for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='287 and present saturation at NT ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Blue dots are for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15, and saturation at NT ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='103, and black dots are for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 presenting NT ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panel: same calcula- tions for the E-C3B, the red dots are for (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25) and present saturation at NT ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Blue dots are for (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256) and saturation at NT ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102, and, black dots are for (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522) presenting NT ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' ϕm and Θϕm the same eigenstate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For this subspace one may write Θϕm = (−1)mϕ−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Thus, Θϕ0 = ϕ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', ϕ0 is a singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The top panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7 show cases of the probability density |varphi0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' On the other hand, ϕ1 and ϕ−1 must correspond to distinct states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' One refers to this doublet state as a Kramers degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mid- dle panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7 show the real and imaginary parts of the member ϕ1 of a doublet, say (ϕ1, ϕ−1), in the same billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The probability density |ϕ1|2 recovers the C3 symmetry (rightmost middle panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The bottom panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7 show the same state under the application under rotation operator R(2π/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A complex conjugation of the shown state obtains the other mem- ber ϕ−1 of the doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the E-C3B, the degenerate states remain invariant to TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Thus, it is necessary to use new inter- mediate formulas to study the distribution of doublets in billiards with mixed classical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The following formulas we derived in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Following the same steps in [39] led to the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (5), a Brody-like formula for the transition between the Poisson and GUE distributions is FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Top panels: Density plots of squared eigenfunctions corresponding to singlet states in the LSS billiard, exhibiting the underlying C3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the color scale, |ϕ1,a|2 is the maximum probability in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Middle panels: Real and imaginary parts of a member ϕ1 of a doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the color scale, ±ϕ1,a is the minimum and maximum of the wave func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The probability density recovers the C3 symmetry (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Bottom panels: Same state in the middle under the application of the rotation operator R(2π/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For η = 0, pB,2(s) reduces to the Poisson distribution, whereas for η = 1, the Wigner distribution for the GUE is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In [40], the dynamical local- ization of chaotic eigenstates was taken into account and their coupling with the regular ones through tunneling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The so-called BRB distribution previously dis- cussed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (9) and Q(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' x) is the incom- plete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, pBRB,2(s) = pP(s) if ρr = 1 or if γ = 0, and pBRB,2(s) = pB,2(s) if ρr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In [27], the above formula was widely tested only in the regime of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='507 full ergodicity (polygonal cases) and in a single case with ρc < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, we detail a non-polygonal billiards family that produces a wide variability of ρc values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In these cases, pBRB,2 well-fitted distributions of nns for ρc < 1 for all investigated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The representative results are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 E-C3B E-C3B E-C3B (a) (b) \x1a\x1b ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522 p(s) p(s) p(s) s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Representative results for BRB-like distributions, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10), fits for p(s) in doublets subspace for same members of E- C3B family of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In panel (a), the chaotic case (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25) (ρc = 1) presents γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='960 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='050, in the range of a GUE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In panels (b) and (c), the mixed cases (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256) and (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522) (0 < ρc < 1) present γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='032 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='13 respectively, in the range of intermediate distributions between Poisson and GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Fits with Brody-like formula (8), and BRB-like distribution (10), are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' CONCLUSIONS ANS PERSPECTIVES This paper presents numerical results on classical dy- namics and quantization in two bi-parametric billiard families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ESB comprises two ellipses of minor semi- axe unitary, major semi-axe a, and a rectangular region of length 2t [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' First, we investigate the classical dynamics of these billiards where we built detailed diagrams for the chaotic fraction (ρc) of their phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' After that, we investigated the nns distributions p(s) for these systems, a measure of short correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BRB distribution [40], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6), very well fitted all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These results perfectly agree with the expected from the ergodic parameter α that signals the possibility of quantum dynamical local- ization when α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' All sets of eigenvalues used as data are in a range of energy that satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the E-C3B family, the eigenstates can be split into sin- glets and doublets subspaces due the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The first subspace presents similar results to the previous family, reinforcing the agreement with the expected energy range set with α < 1 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The doublets subspace, whose for the chaotic cases is expected a GUE distribution shows the more relevant result in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' All found statistics present intermediated results between Poisson and GUE distributions for the parameter space (ae, be) where the classical phase space is mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A BRB-like formula [27], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10), well fitted all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This formula was tested for ρc < 1 and α < 1 in just a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Particularly in the E-C3B family, the minimum value of the chaotic fraction of the classical phase space is ρc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In this perspective, a phase diagram analog to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 even more intricate is generated, possible further explorations of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The parameter β in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' How the new distribution, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10), uses the same arguments to include the parameter γ is merito- rious in a future comparison between this quantity and other localization metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Another theme meritorious of investigation is the level statistics in an energy range that α ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' ACKNOWLEDGMENTS Useful discussions with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' de Aguiar and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Terto are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 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+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. 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Chem. Inf. Model. 49(7) +(2009), 1655–1663. https://doi.org/10.1021/ci900060x. +23 + diff --git a/9tA0T4oBgHgl3EQfO_-M/content/tmp_files/load_file.txt b/9tA0T4oBgHgl3EQfO_-M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..934d79d2b319fc22deb9dbe463a136307d898df8 --- /dev/null +++ b/9tA0T4oBgHgl3EQfO_-M/content/tmp_files/load_file.txt @@ -0,0 +1,1906 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf,len=1905 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' sohan1993sharma@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='com, vijaykumarbhat2000@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='com, sahilsharma2634@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='com Abstract Carbon nanotube Y-junctions are of great interest to the next generation of innovative multi-terminal nanodevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Topological indices are graph-theoretically based parameters that describe various structural properties of a chemical molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The entropy of a graph is a topological descriptor that serves to characterize the complexity of the underlying molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The concept of entropy is a physical property of a thermodynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Graph entropies are the essential thermophysical quantities defined for various graph invariants and are applied to measure the heterogeneity and relative stabilities of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In this paper, several neighborhood degree sum-based topological indices including graph-based entropies of carbon nanotube Y-junction graphs are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Keywords: Armchair carbon nanotube, graph entropy, NM-polynomial, topological indices, Y- junction graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' MSC (2020): 05C10, 05C35, 05C90 1 Introduction Nanotechnology is currently popular because of its evolving, electron transfer property and low-cost implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Nanotubes [1], were discovered in 1985 and carbon nanotubes [2] in 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', [3, 4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions are a great option for the production of nanoscale electronic devices with better switching and reliable transport properties at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more applications of carbon nanotube Y-junctions, we refer to [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The first proposed branched carbon nanotube was of Y shape, commonly known as Y-junction or three-terminal junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions are classified as an armchair, zig-zag, or chiral depending on the chirality of connected carbon nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, they can be single-walled or multi-walled, symmetric or asymmetric, capped or uncapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A carbon nanotube is called uncapped if both ends are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A Y-junction is called symmetric if the nanotubes joining in the Y shape are identical, heptagons appeared isolated, and are distributed symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For various symmetric and asym- metric carbon nanotube Y-junctions, we refer to [12, 13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' A carbon nanotube Y-junction is formed by joining three identical carbon nanotubes in a Y- shaped pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions contain exactly six hexagons as well as heptagons at the branch- ing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The first structural model of symmetrical single-walled armchair carbon nanotube Y- junctions was proposed by Chernozatonskii [16] and Scuseria [17], independently, in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These junctions were experimentally observed [18] in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more applications and properties of carbon nanotube Y-junction graphs, we refer to [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Molecular graphs are a visual representation of a chemical molecule with vertices representing atoms and edges representing bonds between the atoms [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let G = (V (G), E(G)) be a molecular graph with vertex set V (G) and edge set E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Any edge of the graph connecting its vertices u and v, is denoted by e = uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Two vertices of graph G are said to be adjacent if there exists an edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='02169v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='mes-hall] 3 Jan 2023 vertex v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', d(v)= |{u : e = uv ∈ E(G)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', dn(v) = � u d(v): uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The connected graph G is said to be k-connected if its connectivity is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Topological indices are the numerical values calculated from molecular graphs to describe various structural properties of the chemical molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' They are frequently used to model many physico- chemical properties in various quantitative structure-property/activity relationship (QSPR/QSAR) studies [23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In 1947, the chemist Harold Wiener [26] initiated the concept of topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', without using any weight lab [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The Zagreb indices and their variations have been used to investigate molecular complexity, ZE- isomerism, and chirality [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In general, the Zagreb indices have shown applicability for deriving multilinear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more neighborhood degree sum-based topological indices, their properties, and applications, we refer to [24, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The process of computing the topological indices of a molecular graph from their definitions is complex and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In short, with the help of algebraic polynomials, one can easily compute various kinds of graph indices within a short span of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The NM-polynomial plays vital role in the computation of neighborhood degree sum-based topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let dn(v) denotes the neigh- borhood degrees sum of vertex v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, the neighborhood M-polynomial (NM-polynomial) of G is defined as [30, 32, 33] NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Recently, various neighborhood degree sum-based topological indices have been computed via the NM-polynomial technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For example, Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [30, 34] obtained some neighborhood and multiplicative neighborhood degree sum-based indices of molecular graphs by using their NM- polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Kirmani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [24] and Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [35], investigated some neighborhood degree sum-based topological indices of antiviral drugs used for the treatment of COVID-19 via the NM- polynomial technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Shanmukha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [36] computed the topological indices of porous graphene via NM-polynomial method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For more neighborhood degree sum-based topological indices via NM- polynomials, we refer to [24, 35, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Some neighborhood degree sum-based topological indices and their derivation from NM-polynomial are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Based on such a probability distribution, the entropy of a graph can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Thus, the structural information content of a graph is defined as the entropy of the underlying graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Entropy-based methods are powerful tools to investigate various problems in cybernetics, mathematical chemistry, pattern recognition, and computational physics [22, 39, 42, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 2 Table 1: Description of some topological indices and its derivation from NM-polynomial Topological index Formula Derivation from NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) Third version of Zagreb index [28]: NM1(G) � uv∈E(G) � dn(u) + dn(v) � (Dx + Dy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood second Zagreb index [29]: NM2(G) � uv∈E(G) � dn(u)dn(v) � (DxDy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood second modified Zagreb index [30]: nmM2(G) � uv∈E(G) � 1 dn(u)dn(v) � (SxSy)(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 Neighborhood harmonic index [30]: NH(G) � ab∈E(G) 2 dn(u)+dn(v) 2SxT (NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y))|x=y=1 where, Dx = x � (∂(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='x,y)) ∂x � , Dy = y � (∂(NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='x,y)) ∂y � , Sx = � x 0 NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='t,y) t dt, Sy = � y 0 NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='x,t) t dt, T (NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y)) = NM(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Entropy is a measure of randomness, uncertainty, heterogeneity, or lack of information in a sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Based on information indices, there are various approaches to deriving graph entropy from the topological structure of a given chemical molecule [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Tan and Wu [46] study network heterogeneity by using vertex-degree based entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Nadeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [52] calculated some degree-based topological indices for armchair carbon semicapped and capped nanotubes and investigated their chemical and physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Baˇca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [53] computed some degree-based topological indices of a carbon nanotube network and studied its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Azeem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [54] calculated some M-polynomials based topological indices of carbon nanotube Y-junctions and their variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Ahmad [55], studied some ve-degree based topological indices of carbon nanotube Y-junctions and discussed their proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Rahul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [57] calculated some degree-based topological indices and graph-entropies of graphene, graphyne, and graphdiyne by using Shannon’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, no work has been reported on NM-polynomial based topological indices and index-entropies of Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Therefore, the main contribution of this study includes the following: Computation of NM-polynomials of carbon nanotube Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Computation of some neighborhood degree sum-based topological indices from NM-polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Some graph index-entropies in terms of topological indices are defined and computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 3 Comparative analysis of obtained topological indices and graph index-entropies of Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 2 Aim and Methodology We use the edge partition technique, graph-theoretical tools, combinatorial computation, and the degree counting method to derive our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The degree of end vertices is used to generate the patterns of edge partitions of the Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Using such partitions, a general expression of NM-polynomials is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, several neighborhood degree sum-based topological indices are obtained from the expression of these NM-polynomials with the help of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, graph index- entropies in terms of topological indices have been defined by using edge-weight functions and computed for Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The paper is structured as follows: In Section 3, we define topological index-based graph en- tropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The Y-junction graphs and their constructions are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In Section 5, the general expression of the NM-polynomials and neighborhood degree sum-based topological indices of Y-junction graphs are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Section 6 describes the graph index-entropies of Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The numerical analysis of the findings is discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Finally, the conclusion is drawn and discussed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 3 Definitions and Preliminaries In this section, we define graph index-entropies in terms of an edge-weight function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In 2008, Dehmer [40] defined the entropy for a connected graph G as follows: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' [40] Let G = (V (G), E(G)) be a connected graph of order n and g be an arbitrary information functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (2) Since an information function defined on the vertex set of a graph is an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Hence, Dehmer’s definition shows the possibility of producing various graph entropies for a variation in the selection of information functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' For such graph entropy, we can refer to [58, 59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' for eight different edge-weight functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' the third-version of Zagreb index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood second Zagreb index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood forgotten topological index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' neighborhood second modified Zagreb index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' third NDe index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' fifth NDe index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, the third-version of Zagreb index is NM1(G) = � e=uv∈E(G) β1(e) = � e=uv∈E(G) dn(u) + dn(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let m ≥ 1 and n ≥ 4 be an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, the tube Tm(n, n) contains 2mn hexagonal faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let n, m, and l be positive integers with m ≥ 1 and n = 2l, for some l ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then J = Jm(n, n) be the Y -junction graph of Y m(n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' It has 9l2 − 3l + 2 hexagonal rings along with six heptagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graph J is of order 6l2 +18l +6+24ml and size 9l2 +21l +9+36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' It has 6l2 +12l +6+24ml vertices of degree three and 12l vertices of degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Note that graph J is a 2-conneced graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Along with 2-connected Y-junction graph J, the 1-connected Y-junction graphs have also been taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' These graphs are obtained by adding pendants to the degree 2 vertices of the 2-connected graph J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Note that, each tube of J has 2n vertices of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Therefore, the graph J has 6n vertices of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The order and size of graph J1 are 6l2 + 22l + 6 + 24ml and 9l2 + 25l + 9 + 36ml, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In J2, we have 6l2 + 26l + 6 + 24ml vertices and 9l2 + 29l + 9 + 36ml edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' It has 6l2 + 30l + 6 + 24ml vertices and 9l2 + 33l + 9 + 36ml edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The carbon nanotube Y-junction graphs J, J1, J2, and J3 are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' J1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and J3 dn(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) J-frequency J1-frequency J2-frequency J3-frequency (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) 0 4l 8l 12l (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) 6l 4l 2l 0 (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) 12l 8l 4l 0 (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) 0 2l 4l 6l (7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) 0 4l 8l 12l (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) 6l 4l 2l 0 (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) 12l 8l 4l 0 (9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J be the Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 6lx5y5 + 12lx5y8 + 6lx8y8 + 12lx8y9 + (9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes has 9l2 +21l +9+36ml number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neighborhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 7 Extension of J to JiBy means of structural analysis of J,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) = {uv ∈ E(J) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(Jm(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' n)) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J) : dn(u) = 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5)| = 6l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 12l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 6l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 12l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 9l2 − 15l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J is obtained as follows: NM(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J be the Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l + 72ml + 18 (viii) NH(J) = l2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l + 4ml + 1 (ix) NI(J) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let f(x, y) = NM(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 6lx5y5 +12lx5y8 +6lx8y8 +12lx8y9 +(9l2 −15l+9+36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then, we have Dx(f(x, y)) = 30lx5y5 + 60lx5y8 + 48lx8y8 + 96lx8y9 + 9(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Dy(f(x, y)) = 30lx5y5 + 96lx5y8 + 48lx8y8 + 108lx8y9 + 9(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' D2 x(f(x, y)) = 150lx5y5 + 300lx5y8 + 384lx8y8 + 768lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' D2 y(f(x, y)) = 150lx5y5 + 768lx5y8 + 384lx8y8 + 972lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' DxDy(f(x, y)) = 150lx5y5 + 480lx5y8 + 384lx8y8 + 864lx8y9 + 81(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (Dx + Dy)f(x, y) = 60lx5y5 + 156lx5y8 + 96lx8y8 + 204lx8y9 + 18(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' DxDy(Dx + Dy)f(x, y) = 1500lx5y5 + 6240lx5y8 + 6144lx8y8 + 14688lx8y9 + 1458(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (D2 x + D2 y)f(x, y) = 300lx5y5 + 1068lx5y8 + 768lx8y8 + 1740lx8y9 + 162(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 8 SxSy(f(x, y)) = 6l 25x5y5 + 12l 40 x5y8 + 6l 64x8y8 + 12l 72 x8y9 + (9l2−15l+9+36ml) 81 x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' SyDx + SxDy(f(x, y)) = 12lx5y5 + 267l 10 x5y8 + 12lx8y8 + 145l 6 x8y9 + 2(9l2 − 15l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 2SxT(f(x, y)) = 6l 5 x10 + 24l 13 x13 + 3l 4 x16 + 24l 17 x17 + (9l2−15l+9+36ml) 9 x18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' SxTDxDy(f(x, y)) = 15lx10 + 480l 13 x13 + 384l 16 x16 + 864l 17 x17 + 81(9l2−15l+9+36ml) 18 x18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Now, using Table 1 we have (i) NM1(J) = (Dx + Dy)f(x, y)|x=y=1 = 162l2 + 246l + 648ml + 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (ii) NM2(J) = (DxDy)f(x, y)|x=y=1 = 729l2 + 663l + 2916ml + 729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (iii) NF(J) = (D2 x + D2 y)f(x, y)|x=y=1 = 1458l2 + 1446l + 5832ml + 1458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (iv) nmM2(J) = (SxSy)f(x, y)|x=y=1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (vi) ND3(J) = DxDy(Dx + Dy)f(x, y)|x=y=1 = 13122l2 + 6702l + 52488ml + 13122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (vii) ND5(J) = SyDx + SxDy(f(x, y))|x=y=1 = 18l2 + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l + 72ml + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (viii) NH(J) = 2SxT(f(x, y))|x=y=1 = l2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l + 4ml + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (ix) NI(J) = SxTDxDy(f(x, y))|x=y=1 = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (x) S(J) = S3 xQ−2TD3 xD3 y(f(x, y))|x=y=1 = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J1 be the second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 4lx3y7+4lx5y5+8lx5y8+2lx7y7+4lx7y9+4lx8y8+8lx8y9+(9l2−9l+9+36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes has 9l2 +25l+9+36ml edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neighborhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J1) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' By means of structure analysis of J1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J1) : dn(u) = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) = {uv ∈ E(J1) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J1) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J1) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J1) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J1) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J1) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J1) : dn(u) = 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and |E(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 9l2 − 9l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J1 is obtained as follows: NM(J1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J1 be the second type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='328l + 72ml + 18 (viii) NH(J1) = l2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98l + 4ml + 1 (ix) NI(J1) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J1) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 1178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Refer to Theorem 2 for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J2 be the third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 8lx3y7+2lx5y5+4lx5y8+4lx7y7+8lx7y9+2lx8y8+4lx8y9+(9l2−3l+9+36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotubes has 9l2 + 29l + 9 + 36ml number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neigh- borhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J2) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' By means of structure analysis of J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J2) : dn(u) = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) = {uv ∈ E(J2) : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 5},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J) 2 : dn(u) = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7) = {uv ∈ E(J2) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 7},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J2) : dn(u) = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8) = {uv ∈ E(J2) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 8},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J2) : dn(u) = 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9) = {uv ∈ E(J2) : dn(u) = 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' dn(v) = 9},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' and |E(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5)| = 2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 8l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8)| = 2l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 4l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' |E(9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9)| = 9l2 − 3l + 9 + 36ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J2 is obtained as follows: NM(J2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J2 be the third type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='819l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56l + 72ml + 18 (viii) NH(J2) = l2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='57l + 4ml + 1 (ix) NI(J2) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='048l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J2) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 1643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='61l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Refer to Theorem 2 for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J3 be the fourth type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Then NM(J3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' x, y) = 12lx3y7 + 6lx7y7 + 12lx7y9 + (9l2 + 3l + 9 + 36ml)x9y9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The fourth type Y-junction graph of an uncapped symmetrical single-walled armchair car- bon nanotube has 9l2 + 33l + 9 + 36ml number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let E(i,j) be the set of all edges with neighborhood degree sum of end vertices i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', E(i,j) = {uv ∈ E(J3) : dn(u) = i, dn(v) = j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Equation (1), the NM-polynomial of J3 is obtained as follows: NM(J3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Let J3 be the fourth type Y-junction graph of an uncapped symmetrical single-walled armchair carbon nanotube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='90l + 72ml + 18 (viii) NH(J3) = l2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='090l + 4ml + 1 (ix) NI(J3) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 (x) S(J3) = 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7l2 + 2085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l + 4670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9ml + 1167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Refer to Theorem 2 for proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' We first compute index-entropies of the Y-junction graph J whose edge partition is given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Third-version of Zagreb index-entropy of J From part (i) of Theorem 2, we have NM1(J) = 162l2 + 246l + 648ml + 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (20) Using Table 2 and Equation (19) in Equation (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' we get the required third-version of Zagreb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='index-entropy of J as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='Hβ1(J) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(NM1(J)) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='NM1(J) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='e∈E(J) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='β1(e)logβ1(e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(162l2 + 246l + 648ml + 162) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162l2 + 246l + 648ml + 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6l(10)(log10) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='12l(13)(log13) + 6l(16)(log16) + 12l(17)(log17) + (9l2 − 15l + 36ml + 9)(18)(log18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(162l2 + 246l + 648ml + 162) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162l2 + 246l + 648ml + 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='60l(log10) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='156l(log13) + 96l(log16) + 204l(log17) + (162l2 − 270l + 648ml + 162)(log18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='log(162l2 + 246l + 648ml + 162) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162l2 + 246l + 648ml + 162 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='60l(1) + 156l(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1139433523) +96l(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2041199827) + 204l(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2304489214) + (162l2 − 270l + 648ml + 162)(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2552725051) � ≈ log(162l2 + 246l + 648ml + 162) − 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='78l + 810ml + 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 162l2 + 246l + 648ml + 162 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Neighborhood second Zagreb index-entropy of J From part (ii) of Theorem 2, we have NM2(J) = 729l2 + 663l + 2916ml + 729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2 + 958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27l + 5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml + 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2 + 663l + 2916ml + 729 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Similarly, we compute the remaning index-entropies of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 3 shows some calculated graph index-entropies of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In this way, the topological index-based entropies for Y-junction graphs J1, J2, and J3 are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The index-based entropies of J1 J2, and J3 are given in Tables 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 12 Table 3: Index-entropies of J Entropy Values of entropies Hβ3(J) log(1458l2 + 1446l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l2+2601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+1446l+5832ml+1458 Hβ4(J) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='828ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='62l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J) log(13122l2 + 12170l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+15202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='13l+166059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31ml+41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 13122l2+12170l+52488ml+13122 Hβ6(J) log(18l2 + 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86l+72ml+18 Hβ7(J) log(l2 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95 l2+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='69l+4ml+1 Hβ8(J) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45l2+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='82ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Table 4: Index-entropies of J1 Entropy Values of entropies Hβ1(J1) log(162l2 + 314l + 648ml + 162) − 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31l2+346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='09l+813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24ml+203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 162l2+314l+648ml+162 Hβ2(J1) log(729l2 + 957l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2+1523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='54l+5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml+1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2+957l+2916ml+729 Hβ3(J1) log(1458l2 + 2074l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46l2+3991l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+2074l+5832ml+1458 Hβ4(J1) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207l2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='065l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='897ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J1) log(13122l2 + 12170l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+32699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='53l+166059ml+41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 13122l2+12170l+52488ml+13122 Hβ6(J1) log(18l2 + 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='328l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='09l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='328l+72ml+18 Hβ7(J1) log(l2 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9l2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='21l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 l2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98l+4ml+1 Hβ8(J1) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l2+36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Table 5: Index-entropies of J2 Entropy Values of entropies Hβ1(J2) log(162l2 + 382l + 648ml + 162) − 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31l2+430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='65l+813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24ml+203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 162l2+382l+648ml+162 Hβ2(J2) log(729l2 + 1251l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2+2088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='77l+5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml+1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2+1251l+2916ml+729 Hβ3(J2) log(1458l2 + 2478l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46l2+5380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+2478l+5832ml+1458 Hβ4(J2) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='819l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='099l2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='007l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='396ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='819l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J2) log(13122l2 + 17638l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+50196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l+166059ml+50196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95 13122l2+17638l+52488ml+13122 Hβ6(J2) log(18l2 + 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56l+72ml+18 Hβ7(J2) log(l2 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='57l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9l2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='61l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 l2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='57l+4ml+1 Hβ8(J2) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='048l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l2+46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='387l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='048l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 13 Table 6: Index-entropies of J3 Entropy Values of entropies Hβ1(J3) log(162l2 + 450l + 648ml + 162) − 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31l2+515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='23l+813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24ml+203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 162l2+450l+648ml+162 Hβ2(J3) log(729l2 + 1545l + 2916ml + 729) − 1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22l2+2654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='14l+5564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88ml+1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 729l2+1545l+2916ml+729 Hβ3(J3) log(1458l2 + 3330l + 5832ml + 1458) − 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46l2+6769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l+12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84ml+3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 1458l2+3330l+5832ml+1458 Hβ4(J3) log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18l2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='35l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11l2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92l+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='11 Hβ5(J3) log(13122l2 + 23106l + 52488ml + 13122) − 41514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75l2+67694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4l+166059ml+41514 13122l2+23106l+52488ml+13122 Hβ6(J3) log(18l2 + 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='90l + 72ml + 18) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41l2+27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='79l+21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67ml+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='41 18l2+75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='90l+72ml+18 Hβ7(J3) log(l2 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='090l + 4ml + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9l2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='04l+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6ml+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 l2+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='090l+4ml+1 Hβ8(J3) log(40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2 + 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l + 162ml + 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5) − 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37l2+56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44l+105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48ml+26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='37 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5l2+106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='95l+162ml+40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, we plot line and bar graphs for comparison of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Here, we use the logarithm of the base 10 for calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The numerical values of topological indices for Y-junction graph J are given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The logarithmic values of Table 7 are plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The third NDe index has the most dominating nature compared to other topological indices, whereas neighborhood second modified Zagreb index grew slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='55 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7436 288966 467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='38 968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 25950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56 [3,3] 8190 35523 71406 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27693 623718 962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='58 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='07 2040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='63 55857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='79 [4,4] 14106 61701 123882 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='39 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81026 1089690 1637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='44 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='76 3517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='34 97442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 [5,5] 21642 95169 190938 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='96 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3436 1686882 2492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 5399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='05 150703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 [6,6] 30798 135927 272574 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='63 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8769 2415294 3527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='16 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='14 7685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='76 215642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 [7,7] 41574 183975 368790 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4103 3274926 4742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='02 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='83 10377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='47 292258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 [8,8] 53970 239313 479586 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27 317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9436 4265778 6136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88 398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='52 13474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18 380551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9] 67986 301941 604962 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24 440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4769 5387850 7711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='74 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='21 16975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='89 480522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 [10,10] 83622 371859 744918 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0103 6641142 9466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9 20882.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 592170 Figure 2: Graphical comparison among topological indices of Y-junction graph J 14 indices 1000000 NM,(J) 100000 NF(J) nmr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='(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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The logarith- mic values of these topological indices are plotted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, we see that the logarithemic values of NR−1/2 and NH for J1 are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='34052 299902 490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='656 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='96 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 26879.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='94 [3,3] 8394 36405 74190 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='22 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='51078 640122 996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='984 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='94 2088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 57251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='86 [4,4] 14378 62877 127994 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='79 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='68103 1111562 1683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='312 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 3581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 99300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98 [5,5] 21982 96639 196578 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 146.' metadata={'source': 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2448102 3595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='968 204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88 7781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 218430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [7,7] 42050 186033 378086 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1918 3313202 4822.' metadata={'source': 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+page_content='624 352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='84 13601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 384269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 [9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9] 68598 304587 618714 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='14 443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5323 5437062 7814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='952 441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='82 17119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='35 484704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 [10,10] 84302 374799 761198 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7026 6695822 9581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='28 540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 21042 596816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The log- arithmic values of these indices are plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The vertical axis of Figure 4 shows the comparison clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figure 4 shows that the logarithmic values of ND3 are extremely high when compared to other topological indices of J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 4, we see that the graph of NR−1/2 and NH are almost coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='948 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='49121 310838 509.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='71 2136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='144 58645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='93 [4,4] 14650 64053 128010 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='186 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98241 1133434 1720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24 99.' metadata={'source': 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5558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='24 155350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [6,6] 31614 139455 278766 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='824 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2192 2480910 3651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='36 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='42 7876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='788 221219 [7,7] 42526 188091 376014 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='793 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2192 3351478 4886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='92 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='99 10600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='34 298764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 [8,8] 55058 244017 487842 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='862 358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9648 4353266 6302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48 357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='56 13728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='88 387987 [9,9] 69210 307233 614250 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='031 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7104 5486274 7898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48 447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='13 17262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='43 488886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 [10,10] 84982 377739 755238 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='456 6750502 9673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 21200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='98 601463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figure 5 depicts the graphical comparison of these indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Table 10 and Figure 5 show that the values of topological indices strictly increase as the values of l and m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='15 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6419 321774 529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='18 1064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 28694 [3,3] 8802 38169 77058 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='82 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9628 672930 1055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='27 2183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='85 59973 [4,4] 14922 65229 131418 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='59 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='284 1155306 1761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='36 3708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 102929 [5,5] 22662 99579 200358 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='46 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='605 1768902 2647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='45 5637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='75 157562 [6,6] 32022 141219 283878 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='43 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='926 2513718 3713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='54 7972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2 223873 [7,7] 43002 190149 381978 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='247 3389754 4959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='63 10711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='7 301861 [8,8] 55602 246369 494658 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='67 363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='568 4397010 6385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='72 13856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 391526 [9,9] 69822 309879 621918 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='94 453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='889 5535486 7991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='1 451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='81 17405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 492868 [10,10] 85662 380679 763758 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='31 554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='209 6805182 9777 551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 6, we see that entropy measures of Hβ1, Hβ2, Hβ3, and Hβ8 almost 16 indices 1000000 100000 NF(J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' NM,(J3) NF(J3) 10000 "M,(J3) NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='363878 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='349537 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='351098 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='293045 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='469266 1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='751708 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='162725 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='567487 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='674998 [4,4] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='912369 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='901799 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='670239 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='670554 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='629107 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='696846 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='154321 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='61983 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Table 12 and Figure 7, we find that measures of graph index-entropies Hβ1, Hβ2, Hβ3, Hβ5, Hβ6, and Hβ1 are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='374116 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='362875 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='537668 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='588733 [10,10] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='676529 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='673482 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='674123 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='740586 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2 Hβ,(J) Hβ,(J) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ, (J) Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' The graphical comparison of index-entropies of Y-junction graph J2 is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Figure 8, we see that graph index-entropies of J2 increases as the values of l and m increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='388128 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='375827 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='346955 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='633105 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='336917 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='391354 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='698832 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='650775 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='672367 [4,4] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='924151 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='916793 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='9007 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='074455 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='902766 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='926068 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='876596 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='905747 [5,5] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='104655 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='098533 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='381681 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='377087 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='367462 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='473394 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='370602 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='3882828 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='331363 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='371323 [8,8] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='492905 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='488816 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='480327 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='573399 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='48337 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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+page_content='662948 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='583139 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='592399 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='540309 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='583714 [10,10] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='680065 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='676703 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='659833 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='744042 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='672602 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='680861 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='628548 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='673185 Figure 8: Graphical comparison among index-entropies of J2 In Table 14, we calculate some graph index-entropies of Y-junction graph J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figure 9 shows the graphical comparison among index-entropies of J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' From Table 14 and Figure 9, we see that index entropies Hβ1, Hβ2, Hβ3, Hβ6, and Hβ8 of J3 are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ(J,) Hβ,(J,) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Hβ,(J,) Hβ(J,) Hβ,(J) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ,(J,) Hβ,(J,) Hβ(J,) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Hβ,(J2) I Hβ,(J2) I Hβ,(J2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ,(J2) Hβ,(J2) Hβ,(J2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='401692 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} 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+page_content='530087 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='684252 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='632058 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Also, various neighborhood degree sum-based topological indices are retrieved from the expression of these polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' In addition, eight graph entropies in terms of these topologi- cal indices have been defined and calculated for Y-junction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Furthermore, some numerical values of topological indices and index-entropies of Y-junction graphs are plotted for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=', to predict thermochemical data, physicochemical properties, and structural information of carbon nanotube Y-junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Data Availability No data was used to support the findings of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Conflicts of Interest There are no conflicts of interest declared by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Funding Statement The authors received no specific funding for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' 19 Hβ,(J3) Hβ,(J3) Hβ,(J3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Hβ,(J3) Hβ,(J3) Hβ,(Js) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 Hβ,(J3) Hβ,(J3) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 Index-entropies 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Figures and Tables are prepared by Sohan Lal and Sahil Sharma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' All authors reviewed and edited the final draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Kroto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tA0T4oBgHgl3EQfO_-M/content/2301.02169v1.pdf'} +page_content=' Heath, S.' metadata={'source': 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0000000000000000000000000000000000000000..4f9e83fa9e645a6395eb22ac6c1251a2d6583fb1 --- /dev/null +++ b/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 +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. 1–18, 2022. +[2] Jianqiang Wang, Dandan Ding, Zhu Li, and Zhan Ma, +“Multiscale +point cloud geometry compression,” 2021 Data Compression Conference +(DCC), pp. 73–82, 2021. +[3] D Graziosi, O Nakagami, S Kuma, et al., “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. 9, 2020. +[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., “Emerging mpeg standards for point +cloud compression,” IEEE Journal on Emerging and Selected Topics in +Circuits and Systems, vol. 9, no. 1, pp. 133–148, 2018. +[5] Chao Cao, Marius Preda, Vladyslav Zakharchenko, Euee S Jang, and +Titus Zaharia, +“Compression of sparse and dense dynamic point +clouds—methods and standards,” Proceedings of the IEEE, vol. 109, +no. 9, pp. 1537–1558, 2021. +[6] Maurice Quach, Jiahao Pang, Dong Tian, Giuseppe Valenzise, and +Fr´ed´eric Dufaux, +“Survey on deep learning-based point cloud com- +pression,” Frontiers in Signal Processing, 2022. +[7] Gexin Liu, Jianqiang Wang, Dandan Ding, and Zhan Ma, “PCGFormer: +Lossy point cloud geometry compression via local self-attention,” in +IEEE VCIP, 2022. +[8] Ruixiang Xue, Jianqiang Wang, and Zhan Ma, “Efficient LiDAR point +cloud geometry compression through neighborhood point attention,” +ArXiv, vol. abs/2208.12573, 2022. +[9] WG7, MPEG 3D Graphics Coding, “Description of exploration exper- +iment 5.3 on AI-based dynamic pc coding,” ISO/IEC JTC 1/SC 29/WG +7 N00386, July 2022. +[10] Eduardo Peixoto, +“Intra-frame compression of point cloud geometry +using dyadic decomposition,” IEEE Signal Processing Letters, vol. 27, +pp. 246–250, 2020. +[11] Shuai Gu, Junhui Hou, Huanqiang Zeng, and Hui Yuan, “3D point cloud +attribute compression via graph prediction,” +IEEE Signal Processing +Letters, vol. 27, pp. 176–180, 2020. +[12] Evaristo Ramalho, Eduardo Peixoto, and Edil Medeiros, “Silhouette 4D +with context selection: Lossless geometry compression of dynamic point +clouds,” IEEE Signal Processing Letters, vol. 28, pp. 1660–1664, 2021. +[13] Dorina Thanou, Philip A. 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b/ANFLT4oBgHgl3EQfwzCR/content/tmp_files/load_file.txt @@ -0,0 +1,508 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf,len=507 +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'} +page_content='m-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='mth ' 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='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'} +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='m-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='mth ' 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='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'} +page_content='scale ' 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'} +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'} +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='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'} +page_content='(1-stage) ' 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='(8-stage) ' 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='(8-stage) ' 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='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='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'} +page_content='824/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'} +page_content='812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFLT4oBgHgl3EQfwzCR/content/2301.12165v1.pdf'} +page_content='445/0.' metadata={'source': 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'/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 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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 +∗ debjyoti@post.bgu.ac.il +[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 +∗ debjyoti@post.bgu.ac.il +[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). + diff --git a/BtFQT4oBgHgl3EQfNjak/content/tmp_files/load_file.txt b/BtFQT4oBgHgl3EQfNjak/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0442b25824d94a232fb9fb34a47739d7e7f48aeb --- /dev/null +++ b/BtFQT4oBgHgl3EQfNjak/content/tmp_files/load_file.txt @@ -0,0 +1,565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf,len=564 +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The impenetrable surface is modelled by restricting the DNA configurations at the z ≥ 0 plane, with attractive interactions for monomers at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Further, we consider two variants for z = 0 occupations by ds segments, where one or two surface interactions are counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This consideration has significant consequences, to the extent of changing the stability of the bound phase in the adsorbed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Interestingly, adsorption changes to first-order on coinciding with the melting transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' In vitro, the melting transition is induced by changing the temperature or pH of the DNA solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' However, the physiological condition would allow neither extremes of temperature nor pH level inside the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Therefore, the cell has to rely on other ambient factors to locally modify the stability of the ds structure of the DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', with proteins or cell membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Again, adsorption is instrumental in packaging DNA inside the virus heads [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' In all these instances, the surface-DNA interaction can be tuned by changing the nature of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This tunability calls for a detailed phase mapping arising from the interaction of the DNA with the adsorbing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The melting and the adsorption transition individu- ally, forms the subject of many theoretical and experi- mental studies in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Theoretically, lattice mod- els have been useful in extracting sensible results on par with the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The melting transition was shown to be first-order when excluded volume interactions are fully included [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' On the other hand, the polymer ad- sorption transition was shown to be continuous [2, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Self-avoidance is duly implemented among the intra- and inter-strand segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' We found that the melting vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' ad- sorption phase diagram is drastically different for the two different schemes of interaction between the ds and the adsorbing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For specific values of the coupling po- tentials, the two transitions overlap, with the continuous adsorption transition becoming first-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' One end of the DNA is grafted in the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The other end is free to wander in the z ≥ 0 direction, with the z = 0 plane impenetrable and attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For each interaction with the z = 0 surface, there is an energetic gain of −ϵs, represented by the reduced variable q = ϵs/kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Further, we consider two variants: model I and model II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The difference in the two variants is in the strength of the ds interaction with the surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' in model arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='13272v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='soft] 30 Jan 2023 2 (a) (b) Adsorbing surface (c) ds ss bubble Y-fork FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) Schematic diagram for the (a) lat- eral view of model I, and (b) planar view of model II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' In (a) representing model I, only one strand is interacting with the surface effectively in the bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' While both the strands are simultaneously in contact in model II, as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (c) Two- dimensional depiction of our lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Such consideration comes from the speculation that when interacting sidewise, like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 1(a), there would be an effective interaction of one strand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' By contrast, when both the strands touch the plane simultaneously, each strand would contribute [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' These two scenarios may arise depending on the hardness of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' While metallic surfaces (such as Gold) used during experiments are hard, biological surfaces tend to be much softer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' A schematic diagram of our model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The adsorbing surface can generally be of complex geometry with different degrees of roughness and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' How- ever, we choose a smooth and impenetrable flat surface for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For simulation, we use the pruned and en- riched Rosenbluth method (PERM) to sample the equi- librium configurations, averaging over 108 tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' We set the Boltzmann constant kB = 1 throughout our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The bound and the unbound phases are dominated by energy and entropy, respectively, depending upon whichever minimizes the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For our model, in the ab- sence of any adsorbing surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', q = 0), the melting takes place at gc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3413 with the crossover exponent φm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='94 [9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The exponent φa is expected to be universal, and the most recent improved estimate of the critical exponent from computer simulations suggest φa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='48(4) [10, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Naively, one would expect four distinct phases when melting and adsorption are considered together [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Overall, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [15], it was found that the bound state is stabilized in the presence of an adsorbing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' By contrast, on the experimental side, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [16] had demonstrated that directly adsorbed DNA hybrids are significantly less stable than if free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Therefore, fur- ther study of the melting-adsorption interplay, employing more versatile models is essential for a complete under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Model I: In this model variant, we consider equal sur- face interaction energy for both ss and ds segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This choice of interaction yields four equilibrium phases, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', bound-desorbed (BD), unbound-desorbed (UD), unbound-adsorbed (UA), and the bound-adsorbed (BA) phase [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The melting and the adsorption lines are obtained by varying g and q, respectively, while keep- ing one of them fixed [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The error bars in qc and gc are of the size of the plotting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' As the two lines (gc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3413 and qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856) approach each other, the bound state is primarily stabilized for increasing q, which is somewhat surprising [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 2(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This increased stabil- ity of the bound state persists for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='26(6) <∼ q <∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This short period of stability is followed by a steady increase in the threshold g for bound state for q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4, separating the destabilized bound and unbound state in the adsorbed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' One can understand this using the energy-entropy 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='8 1 BD UD UA BA (a) (b) (c) (d) g q melt ads 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 PnsX(2N) ns/(2N) 700 800 900 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 20 -15 -10 -5 0 5 10 15 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='04 Cs/N2Φa-1 (q-qc)NΦa 1000 900 800 700 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) (a) Model I phase diagram for melting ‘melt’ and adsorption ‘ads’ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The different phases are: bound- desorbed (BD), unbound-desorbed (UD), unbound-adsorbed (UA), and bound-adsorbed (BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The dotted lines represent the transition points for the individual cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' for melting gc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3413 and for adsorption qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (b) Scaling plots of the probability distribution (Pns) of surface contacts (ns) on the BA → UA transition line corresponding to g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 and qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='659, and for chain lengths N = 700 to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (c) A zoom in of the phase diagram in (a) showing a decrease in the threshold g for bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (d) Scaling plot of surface contact fluctuation Cs for g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5, using qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='659 and φa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' argument;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='753(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 2](d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This observation is supported by the scaling plot of the surface contact probability distribution (Pns) at a point (g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='659) above the melting phase bound- ary, using the scaling exponent φa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='99 for data col- lapse [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 2](b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For higher g values, the BA phase undergoes a continuous desorption around limg→∞ qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Summarizing the results of model I, we see, that the bound phase is stabilized only for a small range of q val- ues [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 2(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Otherwise, the bound state is mainly destabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='265(5), the two transitions re- main decoupled without affecting each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Results involving model I is in accordance with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [16], where adsorbed DNA hybrids are found to be less stable than their free counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Using this scheme of interaction, the phase plane is divided into four distinct phases viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', BD, UD, UA and the BA phase [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Intriguingly, the adsorption transition is pro- moted to first-order in this overlapping region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Adja- cent to this overlapping region, and bounded by the lines g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3413 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856 on the other two sides, is a small triangular island (denoted by a) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 3], akin to the Borromean phase found in nuclear systems [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Adsorption in this region follows the same scal- 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 BD UD UA BA a (ar1) (a) (b) (c) g q melt ads 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='7 PnsX(2N) ns/(2N) 500 600 700 800 900 1000 6 -4 -2 0 2 4 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1 Cs/N2φa-1 (q-qc)Nφa 1000 900 800 700 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) (a) Model II phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The dif- ferent phases are: bound-desorbed (BD), unbound-desorbed (UD), unbound-adsorbed (UA) and bound-adsorbed (BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Dashed lines represent, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='753 in red and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1428 in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Dotted lines represent, g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3413 and q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (b) Probability distribution of surface contacts (Pns) at gc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='25 and qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='278 (arrow ar1 in (a)), for chain lengths N = 700 to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (c) Scaling plot for fluctuation of average number of surface contacts per unit length Cs for g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='25 using φa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='98 and qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='277(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' ing exponent as of the first-order melting transition with φa = φm ∼ 1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 3(c)] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' A first-order adsorption is also evident from the probability distribution of the surface contacts (Pns) at the transition point, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', for gc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='25 and qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='278 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 3(b) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The melting transition, however, remains unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Below a, the ad- sorbed phase is destabilized for a small range of g values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For the transition from BA to UA phase the melting is two dimensional for sufficiently large q with φm ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 when the system is completely adsorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Unlike model I, the bound state in model II is stabi- lized in the presence of the adsorbing surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='753(3), the two-dimensional melting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' A similar argument also applies for the adsorption transition for which the critical adsorption strength qc decreases and saturates at limg→∞ qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1428 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Although our results from model II are in line with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [15], qualitatively, we obtain all four possible phases, instead of three, as in [15], where the UA phase was absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Biologically, adsorption-induced stability could be important to guard DNA native form against thermal fluctuation and external forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Importantly, adsorption can energetically compensate for the bending of the rigid ds segments, thereby, providing an alternative to bubble mediated bending [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Conclusion: To conclude, in this paper, we elucidate the role of adsorption in modifying the melting transi- tion and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Two separate models were consid- ered, which differs in the strength of interaction with the surface along the ds segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The two models show significant differences: model I shows that the ds structure is mostly destabilized in the presence of an attractive surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' This finding resemble the re- sult from the experiment performed with DNA hybrids in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' On the other hand, model II shows that DNA is only stabilized in the presence of an attractive surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Although this model is similar to the theoretical model of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [15], there are significant improvements, such as we consider excluded volume interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Moreover, we found the presence of all four possible phases, which is not the case in Ref [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Findings from both the models carry biological significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Our work, therefore, contributes toward completing the picture by connecting the experimental and theoretical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Acknowledgement: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' was supported by the German-Israeli Foundation through grant number I- 2485-303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Part of the simulations were carried out on the Samkhya computing facility at the Institute of Physics, Bhubaneswar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 5 ∗ debjyoti@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='il [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Cloutier and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Widom, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Cell 14, 355 (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Yan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Marko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 93, 108108 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' E 97, 022503 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Allahverdyan et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' al, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} 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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' E 79, 031903 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Schreiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 83, 4288–4295 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [18] A growing peak on either side of the distribution, and a deepening valley in between, is typical of a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The valley represent suppressed states due to the growing surface term between the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The inter-peak gap converges to a non-zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' However, for models where this surface/interface, separating two coex- isting phases, is reduced to a point, this valley is absent [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Also see SM [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Garel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Orland, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Orlandini, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' B 12, 261-268 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Therefore, the effective ad- sorbed energy per unit length (N) is doubled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [21] Double-stranded (ds) bound DNA segments are about 25 times rigid than the single-stranded (ss) unbound DNA segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' These ss segments flanked by ds segments on either side are known as bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' These bubbles can act as hinge for bends in DNA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' SUPPLEMENTARY MATERIAL I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) A typical configuration showing strand A (S1) and strand B (S2) with the adsorbing plane at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' We use the pruned and enriched Rosenbluth algorithm (PERM) [1] to simulate the configurations of the dsDNA over an attractive surface [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Two strands are grown at once, adding monomers on the top of the lastly added monomer of both the strands at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Sn = Sn(A) × Sn(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Then, a choice is made according to the importance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' At each step the local partition function is calculated as wn = � Sn exp(g×l+q×k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' If r < 1 and pruning fails, the configuration is contin- ued to grow but with Wn = Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Averages are taken over 108 tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' One of the important aspects in simulating lattice self-avoiding walks is in checking if the immediate next sites are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The straightforward way is to check if any of the last N − 1 steps occupy the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Here, the checking of self-avoidance is ≈ O(1), with no possibility of hashing collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' A possible way out is to use a linked list method e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' the AVL tree binary search [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' In AVL, the algorithm works by creating a tree like structure where each node represent an occupied lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Each entry for a new step is associated with search, insertion and rebalancing the tree branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Using 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='35 0 1 2 3 4 5 6 7 8 Pn,nsX(2N)φa ns/(2N)φa 500 600 700 800 900 1000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) Scaling plot of surface contacts proba- bility distribution (Pn,ns) for different lengths N = 500−1000, at qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='285 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='7 in model II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For data-collapse we use φa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Notice that, while N is the number of max- imum possible bps, 2N is the maximum number of possible surface contacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Sterling approximation, and for large N, this is approx- imately O(N ln N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Moreover, the AVL algorithm can be easily incorporated in the recursive structure of the PERM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' There- fore, deciding the behavior of the transition becomes dif- ficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' A first-order transition is characterised by doubly peaked distribution with growing depth of the valley in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' It grows exponentially deep in the thermodynamic limit, 7 P ∼ exp(−σLd−1), where L is the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' However, for certain models (or problems) this interface can be reduced to a point separating the two phases e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' In these situations the valley is absent and the surface free energy is no longer extensive in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Finally, Pn,ns is found using the formula, Pn,ns(q, g) = Zn,ns(q, g) �2n ns=0 Zn,ns(q, g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (S3) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S2, we show the scaling plot for Pn,ns for the adsorption transition in the unbound state corresponding to q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='285 and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Taking derivative of ln Z(q, N) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (S6) Therefore, at the critical adsorption point the quan- tity ns/N φa−1 should be N independent for N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S3(b) the estimated critical adsorp- tion point using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (S6) is qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1431(5) for g = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' For higher g’s, when the chain is completely bound, this should converge to qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' One must be careful to use the appropriate φa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' for continuous transitions we use φa = 1/2, and φa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='92 for first-order transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Further, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6840386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Here, we simulate chains of length upto N = 10, 000, to see ns/N φa−1 and γ′ 1,eff upto N = 5000 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' However, since our model has added complexities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=', two complementary monomers from different strands can occupy the same site to form a bp, we think that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (S6) to be more reliable to estimate qc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' However, for first-order adsorption the method of data collapse is used using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (S8) and (S9) but with q in place of g and, nc and Cc replaced with ns and Cs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 8 0 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='22 Cs q 1000 800 600 400 200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 100 1000 ns/N-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 N q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1428 q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1430 q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1431 q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1432 q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1433 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) (a) Fluctuation of surface contacts per unit length Cs for model II, g = 5, and lengths N = 100 to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (b) Long-length behavior of the average surface contacts per unit length (ns) scaled by N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 for different q values around the critical adsorption point for g = 5 in model II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' The adsorption transition is estimated to be qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='143 denoted by the dashed blue line in (a), and to be qc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='1431(5) from (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 100 1000 ns/NΦ-1 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2800 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content="1 γ'1,eff N-Φ 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='2900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='3000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (Color online) Scaled average surface contacts per unit length ns/N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 in (a) and γ′ 1,eff from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' (S7) in (b), using φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='5 for different q values and g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='17 in model II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' 9 ∗ debjyoti@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content='il [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Grassberger, Pruned-enriched Rosenbluth method: simulations of θ polymers of chain length up to 1, 000, 000, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' E 56, 3682 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Adelson-Velsky and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Landis, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} +page_content=' Nauk SSSR 146, 263 (1962) [Soviet Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtFQT4oBgHgl3EQfNjak/content/2301.13272v1.pdf'} 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mode 100644 index 0000000000000000000000000000000000000000..4519d0d014db538ae0fff112afeeb23a6fce09ad --- /dev/null +++ b/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/2301.03112v1.pdf.txt @@ -0,0 +1,1426 @@ +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 +[Ada56] +J. F. Adams. ‘On The Cobar Construction’. In: Proceedings of the National Academy of +Sciences 42.7 (1956), pp. 409–412. +[AMN18] +B. Antieau, A. Mathew and T. Nikolaus. ‘On the Blumberg–Mandell Künneth theorem +for TP’. In: Selecta Mathematica 24 (2018). +[Ant19] +B. Antieau. ‘Periodic cyclic homology and derived de Rham cohomology’. In: Annals of +K-Theory 4.3 (2019), pp. 505–519. +[BCN21] +L. Brantner, R. Campos and J. Nuiten. PD Operads and Explicit Partition Lie Algebras. +2021. arXiv: 2104.03870. +[Bha12] +B. Bhatt. Completions and derived de Rham cohomology. 2012. arXiv: 1207.6193. +[BL22] +B. Bhatt and J. Lurie. Absolute prismatic cohomology. 2022. arXiv: 2201.06120. +[BMS19] +B. Bhatt, M. Morrow and P. Scholze. ‘Topological Hochschild homology and integral +p-adic Hodge theory’. In: Publications mathématiques de l’IHÉS 129 (1 2019), pp. 199– +310. +[Con85] +A. Connes. ‘Non-commutative differential geometry’. In: Publications Mathématiques de +L’Institut des Hautes Scientifiques 62 (1985), pp. 41–144. + +REFERENCES +17 +[DH22] +I. Dan-Cohen and A. Horev. Koszul duality for left modules over associative algebras. +2022. arXiv: 2210.11861. +[GM95] +J. Greenlees and P. May. ‘Generalized Tate Cohomology’. In: Memoires of the American +Mathematical Society. Vol. 543. 1995. +[GS09] +J. Greenlees and B. Shipley. ‘An algebraic model for free rational G-spectra for connected +compact Lie groups G’. In: Mathematische Zeitschrift 269 (2009). +[HKR62] +G. Hochschild, B. Kostant and A. Rosenberg. ‘Differential Forms on Regular Affine +Algebras’. In: Transactions of the American Mathematical Society 102 (1962), pp. 383– +408. +[Hoy14] +M. Hoyois. ‘A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace for- +mula’. In: Algebraic & Geometric Topology 14.6 (2014), pp. 3603–3658. +[KN18] +A. Krause and T. Nikolaus. Lectures on Topological Hochschild Homology and Cyclotomic +Spectra. Available on the second authors’s website. 2018. +[LQ84] +J.-L. Loday and D. Quillen. ‘Cyclic homology and the Lie algebra homology of matrices.’ +In: Commentarii mathematici Helvetici 59 (1984), pp. 565–591. +[Lur09] +J. Lurie. Higher Topos Theory. Princeton University Press, 2009. +[Lur10] +J. Lurie. DAG V. Available on the author’s website. 2010. +[Lur15] +J. Lurie. Rotation invariance in algebraic K-theory. Available on the author’s website. +2015. +[Lur16] +J. Lurie. Higher Algebra. Available on the author’s website. 2016. +[Lur18] +J. Lurie. Elliptic Cohomology I: Spectral Abelian Varieties. Available on the author’s +website. 2018. +[MNN17] +A. Mathew, N. Naumann and J. Noel. ‘Nilpotence and descent in equivariant stable +homotopy theory’. In: Advances in Mathematics 305 (2017), pp. 994–1084. +[Qui70] +D. Quillen. ‘On the (co)homology of commutative rings’. In: Applications of Categorical +Algebra. Vol. 17. 1970. +[Rak20] +A. Raksit. Hochschild homology and the derived de Rham complex revisited. 2020. arXiv: +2007.02576. +[Toë14] +B. Toën. ‘Derived algebraic geometry’. In: EMS Surv. Math Schi. 1 2 (2014), pp. 153– +240. +[TV11] +B. Toën and G. Vezzosi. ‘S 1 -equivariant simplicial algebras, de Rham theory and +multiplicative HKR theorems’. In: Compositio Mathematica 147 (2011). +[WG91] +C. A. Weibel and S. C. Geller. ‘Étale descent for hochschild and cyclic homology’. In: +Commentarii Mathematici Helvetici 66 (1991), pp. 368–388. +WWU Münster, Mathematisches Institut, Einsteinstr. 62, 48149 Münster, Germany +Email address: konrad.bals@uni-muenster.de + diff --git a/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/load_file.txt b/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f5397b4450e3c01feed40b4c7d35491a1c243e3 --- /dev/null +++ b/DdE1T4oBgHgl3EQfWQSA/content/tmp_files/load_file.txt @@ -0,0 +1,739 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf,len=738 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='03112v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='AT] 8 Jan 2023 PERIODIC CYCLIC HOMOLOGY OVER Q KONRAD BALS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let X be a derived scheme over an animated commutative ring of characteristic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We give a complete description of the periodic cyclic homology of X in terms of the Hodge completed derived de Rham complex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular this extends earlier computations of Loday-Quillen to non-smooth algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For the purpose of this paper passing here to the Tate-construction, they prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1 ([LQ84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Assume Q ⊂ k commutative and R a smooth commutative k-algebra, then HP∗(R/k) ∼= � n∈Z H∗−2n dR (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' k) In this paper we give a generalization of this computation to the non-smooth and non-affine situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [WG91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact, similarly we get a sheaf HPk extending periodic cyclic homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We recall the details in Appendix A and write HH(X/k) := Γ(X, HHk) and HP(X/k) := Γ(X, HPk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' simplicial commutative) k- algebras CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' even in the fpqc topology via a different argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 2If P• is a simplicial resolution of the k-algebra R, it suffices to check that | HH(P•/k)| ≃ HH(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1 2 KONRAD BALS where � LΩ∗ X/k is the completion of LΩ∗ X/k with respect to the Hodge filtration LΩ≥• −/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In [Ant19] Antieau constructs the HKR-filtration on HP(X/k) with n-th associated graded � LΩ∗ X/k[2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 and we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, the HKR-filtration is exhaustive, if and only if � LΩ∗ X/k is (homologically) bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This criterion will give us a large class of examples with exhaustive HKR-filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Passing to filtered colimits we get Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In section 4 we construct the corresponding multiplicative structure on � n∈Z � LΩ∗ X/k[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' On homotopy groups the induced graded ring structure comes from LΩ≤n X/k((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is not Hausdorff because not every element is detected in some π∗LΩ≤n R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' With this we can almost completely describe the graded ring π∗ HP(X/k) in terms of π∗ � LΩ∗ X/k: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' there is a finite subset I0 ⊂ {i+j = n},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' such that for all finite subset J ⊂ {i + j = n} containing I0 we have cn − � (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='j)∈J ai · bj ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To fully understand the homotopy groups HP∗(X/k) algebraically, one, furthermore, has to analyze the lim1-terms contributing to π∗ � LΩ∗ R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We begin in section 2 with a formality statement for S1-actions in the derived cat- egory over rational algebras (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3) in order to recall a coherent version of the HKR-theorem in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This allows us to coherently compute HP for smooth algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particu- lar we will use the multiplicativity of the Tate-filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The Tate-filtration itself and its multiplic- ative structure in the rational setting will be reviewed in the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Furthermore in section 3 we will exploit the consequences for the HKR-filtration and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, the last section (4) is completely devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Throughout this note we are freely using the ∞-categorical language as developed in [Lur09] and [Lur16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, we will use the 1-category CDGAk of commutative differential graded algebras over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' functors from Zop ≤ into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a symmetric monoidal category C we equip the category Fil(C) := Fun(Zop ≤ , C) with the Day convolution tensor product ⊗Day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The n-th associated graded grnF of F is given by the cofibre of the map F n+1 → F n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, a splitting (An) of F canonical gives an identification grnF ≃ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We write Fil∧(C) ⊂ Fil(C) for the full subcategory on complete filtrations and denote by (−)∧ its left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' I would like to thank Achim Krause, Jonas McCandless and Thomas Nikolaus for helpful discussions on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, again I want to thank Thomas Nikolaus for bringing this project up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Formality over Q The explicit computations heavily rely on strong formality properties that hold if working over Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In this section we will prove a strong version of the HKR-theorem for Hochschild homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This enables us to establish a coherent versions of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1 copied from [LQ84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 4 KONRAD BALS Throughout the first section, let k be a discrete commutative Q-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The key ingredient is a formality statement of C∗(S1, k), due to [TV11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The multiplication S1 × S1 → S1 and the diagonal S1 → S1 × S1 exhibit S1 as an associative bialgebra in spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because the symmetric monoidal structure on S is cartesian, by the dual of [Lur15][Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8] the coalgebra structure given by the diagonal refines to a cocommutative coalgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Therefore C∗(S1, k) acquires the structure of a cocommutative bialgebra in D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 ([TV11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For completeness reasons we would like to include a proof here: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Both objects C∗(S1, k) and Λk(ǫ) have canonical augmentations coming from S1 → ∗ in S and ǫ �→ 0 in Ch∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will in fact show, that they even agree as augmented cocommutative algebras in D(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Using the adjunction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [Lur16] Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='174) bar: Algaug(coCAlgD(k)) coCAlgaug(D(k)) :cobar it satisfies to construct a map of (co-)augmented cocommutative coalgebras under the bar-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus � k ⊗k[x] k �∨ ≃ Λk(ǫ∨)∨ ≃ Λk(ǫ) for ǫ a dual basis to ǫ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ From now on, to shorten notation we set A := Λk(ǫ) for |ǫ| = 1 primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus the equivalence C∗(S1, k) ≃ A as cocommutative bialgebras gives a symmetric monoidal equivalence of their module categories (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' in [Rak20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The above equivalence induces the identity on underlying objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, given a complex X ∈ D(k) equipping X with an action of S1 is equivalent to providing a module structure over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Informally, this amounts to a map d: k · ǫ[1] ⊗ X ≃ X[1] → X and coherent homotopies witnessing d2 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let CDGAk denote the 1-category of commutative differential graded algebras over k as introduced in the Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Forgetting the differential, there is a functor CDGAk → CAlgCh∗(k) sending (C∗, d) ∈ CDGA to � i∈Z Ci[i] ∈ CAlgCh∗(k) with 0 differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To avoid confusion we will write U : CDGAk → CAlgBS1 k for this functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a k-algebra R the de Rham complex ΩH R/k by definition lives in CDGAk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [Qui70]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Postcomposing with the map above on homotopy groups, we get a map ΩH R/k → HH∗(R/k) → ΩH R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ 6 KONRAD BALS Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We can now get a very good understanding of the Tate-construction for such formal objects: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For (C∗, d) ∈ CDGAk we write |C∗| for the chain algebra (C−∗, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This gives a functor |−|: CDGAk → CAlgCh∗(k) of 1-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By definition we have |ΩH R/k| ≃ Ω∗ R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' With this notation set, we can make the classical computations of periodic cyclic homology in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This is also done for example in the lectures [KN18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For (C∗, d) ∈ CDGAk there is a natural map |C∗| → (UC∗)tS1 in CAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because of the lax monoidal natural transformation (−)hS1 → (−)tS1, it suffices to estab- lish a natural map |C∗| → ChS1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Under the symmetric monoidal equivalence Fun(BS1, D(k)) ≃ ModAD(k) the functor (−)hS1 corresponds to mapA(k, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now put P∗ = (A⟨t∨⟩, dP ) as the free divided power algebra on a primitive generator t∨ in degree 2 with dP t∨ = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The computation in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11 actually completely describes UChS1 ∗ and under the equivalence UCtS1 ∗ ≃ UChS1 ∗ ⊗khS1 ktS1 we already get a full identification UCtS1 ∗ ≃ (UC∗((t)), td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For C∗ = ΩH R/k for R smooth over a rational algebra k we thus could have a full understanding of HP(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, we will not directly use this, but rather proof a general statement with more structure, that generalizes to non-smooth and animated algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Main Theorem Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Unraveling, Fil• H|C∗| precisely gives the stupid or brutal filtration on the chain complex |C∗| ∈ Ch∗(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, for X ∈ Fun(BS1, Sp) let Fil• T XtS1 be the Tate filtration on XtS1, see Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is a complete commutatively multiplicative and exhaustive filtration with associated graded grnFilT XtS1 ≃ X[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The Tate-filtration also restricts to a complete (and exhaustive) filtration on Fil0 T XtS1 ≃ XhS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the concrete description of ChS1 ∗ in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11, we can identify the Tate-filtration with the t-adic filtration on (C∗�t�, td) via Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='12 and the map from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11 refines to a map of commutatively multiplicative filtrations Fil• H|C∗| → Fil• T UCtS1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' To show that we get an equivalence of complete filtrations, it is enough to check on associated graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We can finally return to our situation of interest and immediately get a description of HP(R/k) in more general situations: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' (3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' First assume that k is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We want to show that both sides commute with sifted colimits as functors to Fil∧(D(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We thus can reduce to the case that k is an ordinary Q-algebra and R smooth over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then the equivalence immediately follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 by putting C∗ = ΩH R/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the general case of an animated morphism k → R between animated Q-algebras we can give the exact same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Choose a simplicial resolution kn → Rn of polynomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' [AMN18] proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4) and the second from the facts that HH(−/Q) commutes with colimits in CAlgQ and that the colimit is sifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus also in the general case, the statement reduces to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Finally, in order to compute the periodic cyclic homology in our case, we only have to understand the left hand filtration in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 8 KONRAD BALS Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let k be an animated ring with Q ⊂ π0k and X a derived scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Then there is a natural equivalence of underlying objects in Modk HP(X/k) ≃ � n∈Z � LΩ∗ X/k[−2n] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because both sides are sheaves in the Zariski topology on X we are reduced to the case X = SpecR for R ∈ CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the above Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 there is a natural equivalence of filtrations � Fil• HLΩ∗ R/k ⊗Day Fil• T ktS1�∧ → Fil• T HP(R/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because the Tate-filtration is exhaustive on HP(R/k) it suffices to compute the underlying object of the left filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally we want to compute the underlying object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' the colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now putting everything together gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We want to use the result to investigate the exhaustiveness of the HKR-filtration constructed in [Ant19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For more details c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' or [BL22] section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4, the HKR-filtration on HP(R/k) can be identified with the filtration by partial products of � n∈Z � LΩ∗ Rn/kn[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Precisely, Fili HKR HP(R/k) ≃ � n≤−i � LΩ∗ Rn/kn[−2n] PERIODIC CYCLIC HOMOLOGY OVER Q 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By definition of the HKR-filtration we only have to construct equivalences in the case R over k a smooth algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, HP(R/k) decomposes into the product of the associated gradeds of the HKR-filtration, which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of the theorem the HKR-filtration from [Ant19] is exhaustive if and only if � LΩ∗ X/k is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This is precisely the case when � LΩ∗ X/k[2n] eventually leaves any fixed degree for n → ∞, precisely when it is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' LΩ1 R/k has Tor-amplitude in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We recover this statement in our situation via the observation that the lci-condition forces � LΩ∗ X/k to be concentrated in degrees (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, with a result in [Bha12] in the rational setting we can even prove a more drastic result: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If k is a discrete Q-algebra and X a qcqs-scheme over k, then the HKR-filtration is exhaustive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the last Corollary we want to prove that � LΩ∗ X/k is bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, it satisfies to show the claim for X = SpecR with an arbitrary k-algebra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Hence, we can further reduce the claim to the case k Noetherian and R finite type over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In this situation the result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10 in [Bha12] gives a concrete description of � LΩ∗ R/k, in particular it sits in homological degree (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Multiplicative Structure In the Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus we are able to deduce properties of the induced commutative algebra structure on � n∈Z � LΩ∗ R/k[−2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But first we will describe algebra structures on these big products more generally: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a complete and exhaustive commutative multiplicative filtration R• ∈ CAlgFil(Modk) on a commutative algebra R ∈ CAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We define R�t±1� := colim � R• ⊗Day Fil• T ktS1�∧ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the situation of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 the equivalence refines to a natural equivalence HP(X/k) ≃ � LΩ∗ X/k�t±1� in CAlgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact, in the situation of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 we can demystify the object � LΩ∗ X/k�t±1�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The object R�t±1� does not fully depend on R• as a complete filtered object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will show a very special case, of this feature: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' If F • ∈ Modk is a filtered object with F n = 0 for n but finite n, then colim(F • ⊗Day Fil• T ktS1)∧ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4, we get an equivalence � n∈Z F •−n[−2n] ∼ −→ F •⊗DayFil• T ktS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, the underlying object of F • is 0 and thus also of the complete filtration F • ⊗Day Fil• T ktS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For the last statement, we note that the construction colim(− ⊗Day Fil• T ktS1)∧ is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 we can have further identifications of commutative algebras HP(X/k) ≃ � LΩ∗ X/k�t±1� ∼ −→ limm LΩ≤m X/k((t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We start in a general setting: Given a complete multiplicative exhaustive filtration R• on a k-algebra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Set R•/Rm to be the filtration with (R•/Rm)(l) := Rl/Rm for l ≤ m and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because R• is complete we have R• ∼ −→ limm R•/Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 and Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' And now the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular this map is an equivalence in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We can finally get to the description of the homotopy groups HP∗(X/k) explained in the in- troduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Disregarding the multiplicative structure on HP∗(X/k) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Concretely, this means a neighborhood basis of 0 is given by the kernels of these maps above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, the topology cannot separate points that lie in every single such kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' lie in the kernel of the surjective map π∗ � LΩ∗ X/k → lim π∗LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In degree i this is precisely given by lim1 πi+1LΩ≤m X/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact lim π∗LΩ≤m X/k is the "Hausdorffization" of this non-Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the equation (4) the coefficient cn is a limit of the net � i+j=n ai · bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For the general case, one only has to correctly modify the degrees of elements, the arguments are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ This concludes the description of HP∗(X/k) given in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' HP of Schemes In this section, we want to carefully describe the extension of Hochschild and periodic cyclic homology to (derived) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will refer to [Lur18] [Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1], [Lur10] and [Toë14] for an introduction to derived schemes over animated commutative (aka simplicially commutative) rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will only sketch the definition: 12 KONRAD BALS Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This notion generalizes ordinary schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' quasi-separated if π0X is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let X be a derived k-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' A Zariski-sheaf with values in a category C on X is a C-valued sheaf on the topological space |X|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' ∩ Uik for S = {i0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' , ik}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The first statement is a special case of the infinity categorical comparison Lemma for Grothen- dieck sites proven in [Hoy14] Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3, and the second claim is [Lur18] Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ We now do the standard procedure of extending an algebraic functor CAlgan k/ → C to a sheaf on geometric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We proceed in steps: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' for all affine open derived subschemes U ⊂ X, the sheaves recover Hochschild homology, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' periodic cyclic homology: Γ(U, HHk) ≃ HH(OX(U)/k) Γ(U, HPk) ≃ HP(OX(U)/k) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Set Ue to be the set of affine open derived subschemes of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We want to argue, that these are already Zariski-sheaves on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because X is quasi-affine, intersections of affines are computed in a surrounding affine derived scheme, and are affine again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The collection Ue, thus, satisfies the conditions 1), 2) of Propos- ition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 6Under the appropriate notion of equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 7In fact X is affine, if and only if X = SpecR for R ∈ CAlgan k/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' PERIODIC CYCLIC HOMOLOGY OVER Q 13 functors HH(−/k) and HP(−/k) satisfy the finite limit condition of Fun′(Uop e , Modk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' As the Tate- construction commutes with finite limits, it is enough to only show the claim for Hochschild homo- logy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Indeed, it suffices to check this locally on V , so we can reduce to distinguished opens D(f) ⊂ V ⊂ U for f ∈ π0OX(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But using that HH(−/k) commutes with filtered colimits we can identify both sides with HH(OX(U)[f −1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Finally, assume that F : I → Ue is a finite diagram with colimit U as appearing in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 in the other direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given an arbitrary derived k-scheme X, we can furthermore extend Hochschild and periodic cyclic homology to sheaves HHk and HPk on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Moreover, for all open qcqs derived subschemes U ⊂ X we have Γ(U, HPk) ≃ Γ(U, HHk)tS1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let Ue now be the set of quasi-affine open derived subschemes of X, which satisfies 1) and 2) of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the last Lemma HH(−/k) and HP(−/k) extend to sheaves on Uop e and by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4 thus further extend to sheaves on entire X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Now take U ⊂ X a quasi-compact quasi-separated derived open subscheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because of quasi- compactness there exist a finite open cover of U by affine open subschemes U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Un and by the sheaf condition we get Γ(U, HPk) ≃ lim S⊂[1,n] Γ(US, HPk) in the notation of Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Each US is now quasi-affine as an open derived subscheme of an affine and quasi-compact by the quasi-separatedness of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, because the limit above is finite, it satisfies to check the claim for U quasi-compact quasi-affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' But in this case Γ(U, HPk) ≃ HP(OX(U)/k) ≃ HH(OX(U)/k)tS1 ≃ Γ(U, HHk)tS1 □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Tate Filtration In this section we want to review the construction of the classical Tate-filtration introduced in [GM95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This content is not new and also recently has been explained in [BL22] section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We would like to particular put a focus on multiplicative structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Given a representation ρ: S1 → GL(V ) of S1, the representation sphere SV is the one-point compactification of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Furthermore we define SV := Σ∞SV as the suspension spectrum of the representation sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, the S1-action really uses the representation S1 → GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 14 KONRAD BALS Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For V = C there is the standard representation given by S1 ≃ U(1) ֒→ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Its representation sphere sits in the pushout S1 ∗ ∗ SV with S1-acting freely on itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus, after adding basepoints to the top row Σ∞ gives a fibre sequence S[S1] := Σ∞ + S1 → S → SV of spectra with S1-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Construction B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Let V be a finite dimensional representation of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will see this later in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This definition would not be sensible if this would not give a filtration on XtS1 and we are bound to prove: Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For X ∈ SpBS1 the Tate filtration Fil• T XtS1 is complete with underlying object XtS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because homotopy fixed points, as a limit, preserve completeness it satisfies to prove that lim S−nV ⊗ X ≃ 0 as a spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Indeed, via Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3 we can identify the fibre as S−nV ⊗ S[S1], which is an induced S1-spectrum, such that Tate vanishes on this fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By the observation above the right hand filtration is constant at XtS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Thus together we see colim Filn T XtS1 ≃ XtS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ PERIODIC CYCLIC HOMOLOGY OVER Q 15 We are interested in possible algebra structures on the Tate filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because (−)tS1 is lax monoidal, XtS1 for an algebra X ∈ Alg(Sp) inherits an algebra structure again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' However, the question of algebra structures on FilT XtS1 with respect to the Day convolution is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' We will use the following different description of the filtered category as a modules over a graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' This insight comes from Lurie in [Lur15] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2 and in this form is in [Rak20] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a stable symmetric monoidal category C with unit 1, let 1[β] denote the underlying graded object of the unit in Fil(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' It is a commutative algebra in Gr(C) with underlying graded object � n≤0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In fact remembering this action of 1[β] recovers the full filtered object: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8 ([Rak20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' For a symmetric monoidal stable category C the above functor Fil(C) → Mod 1[β](Gr(C)) is an equivalence of symmetric monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='35 in [MNN17]: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='9 ([MNN17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular as a Q�t�- module map it is determined by the image of 1 in Q·t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because this map is not 0 as seen in the proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='6, up to a unit, it is given by multiplication by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The filtration S−•V ⊗Q can be given a commutative algebra structure in Fil(SpBS1 Q ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Under the symmetric monoidal equivalences from the cited Theorems B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='8 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular sending β to s · t gives Q�t�[s±1] the desired commutative algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 16 KONRAD BALS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' By construction FilT (−)tS1 is the composite ModBS1 k (−)⊗(S−•V ⊗k) −−−−−−−−−−→ Fil(ModBS1 k ) (−)hS1 −−−−→ Fil(Modk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' The second functor has a canonical lax structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because k is a commutative algebra over Q, also (S−•V ⊗ k) inherits a commutative algebra structure via Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' □ 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In the notation of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='11, there is an identification of the t-adic filtration on UChS1 ∗ ≃ (UC∗�t�, td) with the Tate filtration Fil≥0 T UCtS1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Using the cocommutative bialgebra A := Q[ǫ]/ǫ2 as defined in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='1 we have the symmetric monoidal equivalence of categories SpBS1 Q ∼ −→ ModASpQ (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Choosing P∗ = (A⟨t∨⟩, dP ) for t∨ primitive in degree 2 and dP (t∨) = ǫ as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content='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∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' Because again the map is non-zero, the image of t∨ has to be a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE1T4oBgHgl3EQfWQSA/content/2301.03112v1.pdf'} +page_content=' In particular, up to isomorphism the maps ChS1 ∗ [−2n − 2] → ChS1 ∗ [−2n] are given by multiplication with the dual t in (C∗�t�, td).' metadata={'source': 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a/FNE1T4oBgHgl3EQfqgXp/content/tmp_files/2301.03345v1.pdf.txt b/FNE1T4oBgHgl3EQfqgXp/content/tmp_files/2301.03345v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ffce93c891a94ee8bba78b4719748c3dcbb38358 --- /dev/null +++ b/FNE1T4oBgHgl3EQfqgXp/content/tmp_files/2301.03345v1.pdf.txt @@ -0,0 +1,1265 @@ +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 +References +H. Ahn, J. Kwak, S. Lim, H. 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Unsupervised fea- +ture learning via non-parametric instance discrimination. +In Proceedings of the IEEE conference on Computer Vi- +sion and Pattern Recognition, 2018. + diff --git a/FNE1T4oBgHgl3EQfqgXp/content/tmp_files/load_file.txt b/FNE1T4oBgHgl3EQfqgXp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..646f7a69ccbe015a3163422edcd0d6f0760f09e5 --- /dev/null +++ b/FNE1T4oBgHgl3EQfqgXp/content/tmp_files/load_file.txt @@ -0,0 +1,1091 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf,len=1090 +page_content='CaSpeR: Latent Spectral Regularization for Continual Learning Emanuele Frascaroli1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Riccardo Benaglia1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Matteo Boschini1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Luca Moschella2 Cosimo Fiorini3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Emanuele Rodol`a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Artificial Neural Networks forget catastrophi- cally whenever they face a changing training data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Rehearsal-based Continual Learn- ing (CL) approaches have been established as a versatile and reliable solution to overcome this limitation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' however, sudden input disruptions and memory constraints are known to alter the consistency of their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Hence, we propose a geometric regularizer that enforces weak requirements on the Laplacian spectrum of the latent space, promoting a par- titioning behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Fi- nally, we conduct additional analysis to provide insights into CaSpeR’s effects and applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 1 INTRODUCTION Within the natural world, intelligent creatures continually learn to adapt their behavior to changing external condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In doing so, they seamlessly blend novel notions with previous understanding into a cohesive body of knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' On the contrary, ANNs greedily fit the data they are currently trained on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Rehearsal-based CL methods struggle to separate the latent-space projections of replay data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Recent literature favors the employment of rehearsal meth- ods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Recent studies characterize this phenomenon arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='03345v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='LG] 9 Jan 2023 CaSpeR: Latent Spectral Regularization for Continual Learning in terms of abruptly divergent gradients upon introducing new classes (Caccia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022b) or deteriorating decision surface (Bonicelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' A well-known outcome is the accumulation of a predictive bias in favor of the currently seen classes (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Specifi- cally, we observe that the learner struggles to separate la- tent projections of replay examples belonging to different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Given the Riemannian nature of the latent space of DNNs (Arvanitidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018), we naturally revert to spectral geometry to model and constrain its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Spectral geometry is preferred over other geometric tools as it focuses on the latent-space structure without imposing constraints on individual coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In this work, we introduce a loss term aimed at endow- ing the model’s latent space with a cohesive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In summary, we make the following contributions: We study the interference in rehearsal CL models by investigating the geometry of their latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' To the best of our knowledge, this is the first attempt at a geometric characterization of catastrophic forgetting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We propose Continual Spectral Regularizer: a simple geometrically motivated loss term, inducing the online learner to produce well-organized latent embeddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We validate our proposal by combining it with sev- eral SOTA rehearsal-based CL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This aim can be pursued through different classes of solutions (De Lange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021): architectural meth- ods explicitly allocate separate portions of the model to separate tasks (Mallya and Lazebnik, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' regularization methods rely on a loss term to pre- vent the model from changing either its structure (Kirk- patrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018) or its response (Li and Hoiem, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Due to their versatility and effectiveness, current re- search efforts focus primarily on the latter class of ap- proaches (Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Recent trends highlight in- terest in improving several aspects of the basic ER formula, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', by introducing better-designed memory sampling strategies (Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Bang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021), combin- ing replay with other optimization techniques (Lopez-Paz and Ranzato, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Riemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021) or providing richer replay signals (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Ebrahimi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' One of the most prominent challenges for the enhancement of rehearsal methods is the imbalance between stream and replay data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' To counter this effect, researchers have come up with architec- tural modifications of the model (Hou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Douil- lard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020), purposed alterations to the learning ob- jective of the final classifier (Ahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Caccia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022) or the outright removal of it, by applying representa- tion learning instead (Cha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This is achieved through a geometrically motivated regularization term that can be easily combined with any existing replay method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (a) A quantitative evaluation measured as Label-Signal Variation (σ) within the LGG for buffer data points – lower is better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (b) TSNE embedding of the features computed by X-DER for buffered examples in later tasks (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Interference between classes is visibly reduced if CaSpeR is applied (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' All experiments are carried out on Split CIFAR-100, (a) uses buffer size 500, (b) uses 2000 (best seen in colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019), shape explo- ration and reconstruction (Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020), shape mod- eling (Moschella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022) and adversarial attacks on shapes (Rampini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Instead, we pose a weaker requirement: the gap between nearby eigen- values must be maximized, regardless of its exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3 METHOD Our approach exploits tools from spectral geometry to reg- ularize the model’s latent space to hinder forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='1 we describe the Continual Learning paradigm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='2 we present a preliminary experiment, highlighting the problem we want to address;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3 we il- lustrate our geometric regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' The parameters θ include both the weights of the feature extractor and the classifier, θf and θc respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Each task consists of a sequence of images and their correspond- ing labels τi = {(xi 1, yi 1), (xi 2, yi 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' If the model does not include mechanisms to prevent forgetting, the ac- curacy on all previous tasks will collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Rehearsal-based CL methods preserve a portion of examples from previous tasks and store them in a buffer B, with fixed size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (2) For instance, Experience Replay (ER) simply employs a cross-entropy loss over a batch of examples from B: ℓer ≜ CrossEntropy � Fθ(xb), yb� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (3) There exist different strategies for sampling the task data- points to fill the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' These will be explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 4, along with detail on the ℓb employed by each baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='T})1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In order to measure the sparsity of the 1Please refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3 for a detailed description of this pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', �b i ab,i) 5: L ← I − D−1/2AD−1/2 ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 5 6: λ ← Eigenvalues(L) 7: ℓCaSpeR ← −λg+1 + �g j=1 λj ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 6 Output ℓCaSpeR latent space w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' classes representations, we compute the Label-Signal Variation σ (Lassance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This effect can also be observed qualitatively by consider- ing a TSNE embedding of the points in B (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 2b for X-DER), which suggests that the distances between ex- amples from different classes are reduced in later tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We remark that both evaluations improve when our proposed regularizer is applied on top of the evaluated methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3 CaSpeR: Continual Spectral Regularizer Motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This structure can be enforced through loss regularizers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', in (Cosmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020), a minimum-distortion criterion is applied on the latent space of a VAE for a shape genera- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' A body of results from spectral graph theory, dating back at least to (Cheeger, 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Sinclair and Jerrum, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Shi and Malik, 2000), explain the gap occurring between neighboring Laplacian eigenvalues as a quantitative mea- sure of graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Building the LGG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We take the examples in B and for- ward them through the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' their features are used to build a k-NN graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' following (Lassance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021), we refer to it as the latent geometry graph (LGG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Spectral Regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Finally, we compute the eigenvalues λ of L and sort them by ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Let g be the number of different classes within the buffer, we calculate our regularizing loss as: ℓCaSpeR ≜ −λg+1 + g � j=1 λj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (6) The proposed loss term is weighted through the hyperpa- rameter ρ and added to the stream classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Over- all, our model optimizes the following objective: argmin θ ℓstream + ℓb + ρ ℓCaSpeR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (7) Through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Therefore, our loss indirectly en- courages the points in the buffer to be clustered without strict supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We refer the reader to Algorithm 1 for a step-by-step summary of the outlined procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Efficient Batch Operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' While seemingly straightfor- ward, the operation of CaSpeR entails the cumbersome task of constructing the entire LGG G at each forward step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' To avoid a slow training procedure with high memory requirements, we propose an efficient approximation of our initial objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' By repeating these random samplings in each forward step, we optimize a Monte Carlo approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Frascaroli, Benaglia, Boschini, Moschella, Fiorini, Rodol`a, Calderara 4 EVALUATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='1 Evaluation protocol Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' and Class Incremental Learning (Class-IL), where the model learns to make predictions in the absence of task information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' On the one hand, Class-IL is recognized as a more realistic and challenging benchmark (Farquhar and Gal, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' on the other, Task-IL is especially relevant for the quantification of forgetting, as it is unaffected by data imbalance biases (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Benchmarked models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Incremental Classifier and Representation Learn- ing (iCaRL) (Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2017): this method seeks to learn the best representation of data that fits a nearest-neighbor classifier w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' class prototypes stored in the buffer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Dark Experience Replay (DER++) (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020a): another variant of ER, which combines the standard classification replay with a distillation loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' eXtended-DER (X-DER) (Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022a): a method which improves DER++ by addressing its shortcomings and focusing on organically accommo- dating future knowledge2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Pooled Outputs Distillation Network (POD- Net) (Douillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020): the authors extend iCarl’s classification method: their model learns multiple representations for each class and adopts two additional distillation losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' ER-ACE and DER++ use reservoir sam- pling (Vitter, 1985), which might lead to uneven class rep- resentation within the stored examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021) per class, we adjust the latter sampling strategy to enforce equity, as done in (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Split CIFAR-100: CIFAR100 (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2009) contains 100 classes with 500 images per class, where each image has a dimension of 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We split the dataset into 10 subsets of 10 classes each;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Split miniImageNet: miniImageNet (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2016) is a subset of the ImageNet dataset where each image is resized to 84 × 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We use the 20 tasks per 5 classes protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We mainly quantify the performance of the com- pared models in terms of Final Average Accuracy ( ¯AF ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=',T −1} at i, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' , T − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' (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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This measure derives from the widely employed Forgetting met- ric (Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018), which tends to be more forgiv- ing of those methods that do not properly learn the current task (Buzzega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Hyperparameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' this generally indicates a pathological case where the model did not fully exploit the input stream of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' CaSpeR: Latent Spectral Regularization for Continual Learning Table 1: Class-IL results – ¯AF ( ¯F ∗ F ) – for SOTA rehearsal CL methods, with and without CaSpeR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Class-IL Split CIFAR-100 Split miniImageNet Joint (UB) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='11±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='07 (−) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='76±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='10 (−) Finetune (LB) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='38 (100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='00) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='87 (100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='00) Buffer Size 500 2000 2000 5000 ER-ACE 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='63 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='03) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='63 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='78) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='31 (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='06) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='17 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='99) + CaSpeR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='70+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='07 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='61) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='74+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='11 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='17) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='36+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='05 (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='90) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='89+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='72 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='36) iCaRL 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='94 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='24) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='95 (30.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='71) PODNet 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='61 (55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='06) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='12 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='73) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='82 (52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32) 20.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='66) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32 (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='33) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='97 (46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='74) same number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Moreover, we employ the same backbone for all experiments on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In particular, we use Resnet18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2016) for Split CIFAR-100 and EfficientNet-B2 (Tan and Le, 2019) for Split miniImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' The best hyperparameters for each model-dataset configuration are found via grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We refer the reader to the Appendix for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='2 Experimental results We report a breakdown of the results of our evaluation in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 1 (Class-IL) and 2 (Task-IL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' At first glance, CaSpeR leads to a steady improvement in ¯AF across all evaluated methods and settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' However, some interesting addi- tional trends emerge upon closer examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Firstly, we notice that the improvement in accuracy does not grow with the memory buffer size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This is in con- trast with the typical behaviour of replay regularization terms (Cha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' to coarsening (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2020), CaSpeR does not need a large pool of data to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Remarkably, the majority of the evaluated methods achieve comparable ¯AF gains for both CL settings on Split CIFAR- 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' While still improving over the baselines, we see a reduced ¯AF improvement in the Split miniImageNet benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' As a final note, PODNet appears to be an outlier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Nevertheless, CaSpeR is still capable of impacting its training positively, delivering a stabilizing effect that is especially relevant when the memory buffer is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This suggests that the additional clustering facil- itates the model’s convergence, which aligns with the ob- servations we make in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3, where we exploit CaSpeR with limited supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 5 MODEL ANALYSIS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2018) trained on top of the latent representations produced by the methods of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3, we report the results for 5-NN and 11-NN classifiers using the final buffer B as a support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Task-IL Split CIFAR-100 Split miniImageNet Joint (UB) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='84 (−) 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='46 (−) Finetune (LB) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='10 (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='84) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='05 (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='37) Buffer Size 500 2000 2000 5000 ER-ACE 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='86 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='73) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='69 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='02) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='34 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='99) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='38 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='59) + CaSpeR 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='14+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='28 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='91) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='51+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='82 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='38) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='25 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='05) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='03 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='53) iCaRL 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='38 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} 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(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='67) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='93+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='94 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='65) DER++ 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='55 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='24) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='60 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='96) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='15 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='22) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='81 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='59) + CaSpeR 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='40+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='85 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='28) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='78+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='18 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='04) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='92 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='47) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='51 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='91) X-DER 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='28 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='43) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='55 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='92) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='95) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='70 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='71) + CaSpeR 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='98 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='47) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='77+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='27) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='99+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='67 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='88) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='71+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='01 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32) PODNet 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='37 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='76) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='63 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='16) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='60 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='00) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='15 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='71) + CaSpeR 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='70 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='85) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='90+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='27 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='46 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='61) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='24+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='09 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='18) ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='85 ER-ACE ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='75 ER-ACE + CaSpeR ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='83 X-DER ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='64 X-DER +CaSpeR ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='81 iCaRL ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='79 iCaRL + CaSpeR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', τ5 after training up to τ5 and τ10 respectively (Split CIFAR-100 buffer size 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' The closer C|·| to the diagonal, the less geometric distortion between G5 and G10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We report the first 25 rows and columns of C|·|, focusing on smooth (low-frequency) correspondences (Ovsjanikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2012), and apply a C|·| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='15 threshold to increase clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Table 3: Class-IL ¯AF values of k-NN classifiers trained on top of the latent representations of replay data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Results on Split CIFAR-100 for Buffer Size 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' k-NN Clsf w/o CaSpeR w/ CaSpeR (Class-IL) 5-NN 11-NN 5-NN 11-NN ER-ACE 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='73 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='41 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='75+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='02 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='29+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='88 iCaRL 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='86 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='78 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='00+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='14 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='33+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='55 DER++ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='21 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='24 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='75+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='54 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='00+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='76 X-DER 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='44 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='62 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='47+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='03 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='49+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='87 PODNet 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='11 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='60 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='88+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='77 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='94+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', τ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In other words, the matrix C encodes the similarity between the Laplacian eigenspaces of the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In an ideal scenario where the latent space is subject to no modification between τ5 and τ10 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' previously learned classes, T is an isomorphism and C is a diagonal ma- trix (Ovsjanikov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In a practical scenario, T is only approximately isomorphic and, the better the ap- proximation, the more C is sparse and funnel-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 3, we report C|·| ≜ abs(C) for ER-ACE, DER++, iCaRL and X-DER on Split CIFAR-100, both with and without CaSpeR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' On the other hand, in line with the considerations made in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='2, the improvement is only marginal for iCaRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Its different training regime, which is less discriminative in nature, seemingly induces a limited amount of change on the structure of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2017): ODE ≜ 1 ||C||2 F � i � j̸=i c2 i,j, (12) where || · ||F indicates the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' CaSpeR pro- duces a clear decrease in ODE, signifying an increase in the diagonality of the functional matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='3 Continual Semi-supervised Learning In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='2, we shed light on some interesting properties of CaSpeR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', its ability to operate well in a low-data regime and its role in facilitating the convergence of under- performing baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Both issues naturally emerge in the Continual Semi-Supervised Learning (CSSL) setting (Bos- chini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022b), a recently-proposed CL experimental benchmark, where only a fraction of the examples on the input stream are associated with an annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 4, we report the results of an experiment on Split CIFAR-100 in the CSSL setting with only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='8% or 5% annotated labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Typical CL methods operating in this scenario are forced to discard a consistent amount of data (ER-ACE), leading to majorly reduced performance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='8% labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' This indicates that CaSpeR manages to limit the impact of the noisy labels produced by pseudo-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Table 4: Class-IL ¯AF values on Split CIFAR-100, with re- duced amount of annotations (CSSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Buffer size 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' † indicates results taken from (Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' CSSL w/o CaSpeR w/ CaSpeR Labels % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='8% 5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='8% 5% ER-ACE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='46 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='87 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='09 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='16+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='29 PsER-ACE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='31 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='35 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='69+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='38 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='42+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='07 CCIC 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='5† 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='5† 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='22+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='72 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content='82 Finally, we show that CaSpeR can be easily applied to CCIC (Boschini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=', 2022b) – a CSSL method that lever- ages both labeled and unlabeled data – to improve its ¯AF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 6 CONCLUSION In this work, we investigate how the latent space of a CL model changes throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We find that latent- space projections of past exemplars are relentlessly drawn closer together, possibly interfering and paving the way for catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' We show that our approach can be easily combined with any rehearsal-based CL approach, improving the performance of SOTA methods on standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' The appli- cability of geometric-based constraints to unsupervised or self-supervised CL scenarios is still a work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Due to the abstract nature of our setting, we do not believe that this work can have a detrimental impact on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Frascaroli, Benaglia, Boschini, Moschella, Fiorini, Rodol`a, Calderara References H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Ahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Kwak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Lim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Bang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Kim, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' SS-IL: Separated Softmax for Incremental Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In IEEE International Conference on Computer Vision, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Aljundi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Lin, B.' metadata={'source': 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with a memory of di- verse samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Bonicelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Boschini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Porrello, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Spampinato, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Calderara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' On the Effectiveness of Lipschitz-Driven Rehearsal in Continual Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Boschini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE1T4oBgHgl3EQfqgXp/content/2301.03345v1.pdf'} +page_content=' Bonicelli, P.' metadata={'source': 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Weitkamp†, and Axel Munk† ‡ ∗ +†Institute for Mathematical Stochastics, University Göttingen, Goldschmidtstraße 7, 37077 Göttingen +‡Max Planck Institute for Multidisciplinary Sciences, Am Faßberg 11, 37077 Göttingen +∗Cluster of Excellence ”Multiscale Bioimaging: from Molecular Machines to Networks of Excitable Cells”(MBExC), +University Medical Center, Robert-Koch-Straße 40, 37075 Göttingen +January 4, 2023 +Abstract +Optimal transport (OT) based data analysis is often faced with the issue that the underlying cost +function is (partially) unknown. This paper is concerned with the derivation of distributional +limits for the empirical OT value when the cost function and the measures are estimated from +data. For statistical inference purposes, but also from the viewpoint of a stability analysis, +understanding the fluctuation of such quantities is paramount. Our results find direct application +in the problem of goodness-of-fit testing for group families, in machine learning applications +where invariant transport costs arise, in the problem of estimating the distance between mixtures +of distributions, and for the analysis of empirical sliced OT quantities. +The established distributional limits assume either weak convergence of the cost process in +uniform norm or that the cost is determined by an optimization problem of the OT value over +a fixed parameter space. For the first setting we rely on careful lower and upper bounds for the +OT value in terms of the measures and the cost in conjunction with a Skorokhod representation. +The second setting is based on a functional delta method for the OT value process over the +parameter space. The proof techniques might be of independent interest. +1 +Introduction +Statistically sound methods for data analysis relying on the optimal transport (OT) theory (see e.g., +Rachev and Rüschendorf (1998), Villani (2008), and Santambrogio (2015)) have won acclaim in re- +cent years. Exemplarily, we mention fitting of generative adversarial networks (Arjovsky, Chintala, +and Bottou, 2017), novel notions of multivariate quantiles (Chernozhukov et al., 2017; Hallin et al., +2021) and dependence (Nies, Staudt, and Munk, 2021; Mordant and Segers, 2022; Wiesel, 2022) +or tools for causal inference (Torous, Gunsilius, and Rigollet, 2021). +Recall that for Polish spaces X and Y and a continuous cost function c: X × Y → R, the OT +value between two (Borel) probability measures µ ∈ P(X) and ν ∈ P(Y) is defined as +OT(µ, ν, c) ≔ +inf +π∈Π(µ,ν) +� +X×Y +c(x, y) dπ(x, y), +(1.1) +where Π(µ, ν) denotes the set of couplings of µ and ν. Under mild assumptions (1.1) also admits a +dual formulation (see, e.g., Santambrogio 2015), +OT(µ, ν, c) = sup +f∈C(X) +� +X +f cc(x) dµ(x) + +� +Y +f c(y) dν(y), +(1.2) +Key words and phrases. Optimal Transport, central limit theorem, stability analysis, curse of dimensionality, empiri- +cal process, bootstrap. +MSC 2020 subject classification. Primary: 60B12, 60F05, 60G15, 62E20, 62F40; Secondary: 90C08, 90C31. +1 + +where C(X) stands for the set of real-valued, continuous functions on X. Further, f c(y) ≔ infx∈X c(x, y)− +f(x) and f cc(x) ≔ infy∈Y c(x, y) − f c(y) denote cost-transformations of f and f c under c, respec- +tively; also often referred to as c-transformations. +If X = Y and the cost function c = dp +X is the p-th power (p ≥ 1) of a metric dX on X the OT +value gives rise to the p-Wasserstein distance +Wp(µ, ν) ≔ (OT(µ, ν, dp +X))1/p, +which defines a metric on the space of probability measures with p-th moments (Villani, 2008, +Chapter 6). This metric is particularly useful for many data analysis tasks due to its potential +awareness of the “inner geometry” of X. For instance, interpreting (normalized) images, or more +precisely the corresponding pixel locations and intensities, as probability measures, it has been +argued that the distance induced by OT corresponds to the natural expectations of what appears close +or far away for the human eye (Rubner, Tomasi, and Guibas, 2000). Meanwhile, there is a plenitude +of real world showcases where OT based distances (and their associated transport plans) prove +useful for applications e.g., in cell biology (Tameling et al., 2021), genetics (Evans and Matsen, +2012; Schiebinger et al., 2019), protein structure analysis (Gellert et al., 2019; Weitkamp et al., +2022) or fingerprint analysis (Sommerfeld and Munk, 2018), to mention but a few. In these works, +the cost function is a given known quantity which is determined by the concrete application, e.g., a +tree distance on the space of phylogenetic trees as in Evans and Matsen (2012). +However, despite the various successful applications hinted at above, there are situations in +which the underlying cost naturally depends on the measures. In certain problems, e.g., Wasser- +stein based goodness-of-fit testing under group families (Hallin, Mordant, and Segers, 2021) or +Wasserstein Procrustes analysis (Grave, Joulin, and Berthet, 2019), it is central that the underly- +ing OT problem is invariant with respect to certain transformations. This can only be realized by +measure-dependent costs. Moreover, for sliced OT (Bonneel et al., 2015), the Wasserstein distance +between multiple one-dimensional projections of measures is computed. Taking the maximum over +all directions gives rise to the max-sliced Wasserstein distance (Deshpande et al., 2019) which can +be viewed in the framework of OT with measure-dependent costs since maximizing directions are +determined by the underlying measures. Motivated by these considerations, we provide in this +work a general framework for the statistical analysis of empirical OT problems under costs that are +dependent on the underlying measures. +Adopting this statistical point of view, we assume that we do not have access to the measures +µ and ν but only to independent samples {Xi}n +i=1 ∼ µ⊗n and {Yi}m +i=1 ∼ ν⊗m with n, m ∈ N. Upon +defining the empirical measures µn ≔ 1 +n +�n +i=1 δXi and νm ≔ +1 +m +�m +i=1 δYi and given a random cost +function1 cn,m such that OT(µn, νm, cn,m) estimates the quantity OT(µ, ν, c), our main focus is on +characterizing for n, m → ∞ with m/(n + m) → λ ∈ (0, 1) the limit distribution of +� +nm +n + m +� +OT(µn, νm, cn,m) − OT(µ, ν, c) +� +. +(1.3) +This is of particular interest for asymptotic tests about the relation between µ and ν for unknown c +based on the OT value. Further, this enables the derivation of confidence intervals. As it is practi- +cally more relevant, we mainly focus on the scenario where both measures µ and ν are unknown. +However, we stress that our theory also provides distributional limits for the one-sample case, i.e., +when only µ is estimated from data while ν is assumed to be known (see Remark 2.4 and Re- +mark 2.9). Moreover, although we mostly focus on empirical measures to estimate the underlying +measures, our theory also enables the derivation of distributional limits for alternative measure esti- +mators, provided that the corresponding distributional limits can be determined. +1Here, cn,m is either a direct estimator for c or chosen via an OT-related optimization problem over a parameter class. +2 + +For a fixed cost function, i.e., for cn,m ≡ c for some c ∈ C(X × Y), already various works +derived limit distribution results for the empirical OT quantity in (1.3). A specific situation arises +for probability measures on R with cp(x, y) = |x − y|p for p ≥ 1 (Munk and Czado, 1998; del Barrio, +Giné, and Matrán, 1999; del Barrio, Giné, and Utzet, 2005; Mason, 2016; del Barrio, Gordaliza, +and Loubes, 2019) where the OT plan can be represented via a quantile coupling. For this setting, +quantile process theory (Csörgö and Horváth, 1993) in combination with integrability conditions +on the underlying densities have been exploited to derive distributional limits. +Moreover, on general Euclidean spaces Rd with d ≥ 1 and p-th power costs cp(x, y) = ∥x − y∥p +with p > 1 it has been shown by del Barrio and Loubes (2019) and del Barrio, González-Sanz, and +Loubes (2021) for probability measures µ, ν with connected support and finite 2p-th moments for +n, m → ∞ with m/(n + m) → λ ∈ (0, 1) that +� +nm +n + m +� +OT(µn, νm, cp) − E +� +OT(µn, νm, cp) +�� +⇝ N(0, σ2 +µ,ν), +(1.4) +where σ2 +µ,ν > 0 if and only if µ � ν. Here and throughout, “⇝” denotes weak convergence in +the sense of Hoffman-Jørgensen (see van der Vaart and Wellner 1996, Chapter 1.3). Their proof +is based on an L2-linearization technique of the OT value and relies on the Efron-Stein inequality. +In general, the centering quantity E[OT(µn, νm, cp)] in (1.4) cannot be replaced by its population +quantity OT(µ, ν, cp) which hinders further statistical inference purposes. Indeed, for identical ab- +solutely continuous probability measures µ = ν on Rd with sufficiently many moments it follows +for d > 2p by Fournier and Guillin (2015) and Weed and Bach (2019) that +E +� +OT(µn, νm, cp) +� +≍ min(n, m)−p/d. +Moreover, for different measures µ � ν on Rd which are absolutely continuous and sub-Weibull it +has been shown for d ≥ 5 by Manole and Niles-Weed (2021) that +E +� +OT(µn, νm, cp) +� +− OT(µ, ν, cp) ≍ min(n, m)− min(p,2)/d. +These rates are also minimax optimal (up to logarithmic factors) over appropriate collections of +identical measures µ = ν (Singh and Póczos, 2018) as well as different measures µ � ν (Manole and +Niles-Weed, 2021). In particular, this demonstrates that estimation of the OT value suffers from the +curse of dimensionality and showcases that it is in general for d ≥ 5, due to the dominance of the +bias, not possible to replace E[OT(µn, νm, cp)] with OT(µ, ν, cp) in (1.4). +Nevertheless, according to the recently discovered lower complexity adaptation principle for +empirical OT (Hundrieser, Staudt, and Munk, 2022), fast convergence rates are still achieved if one +of the population measures, µ or ν, is supported on a sufficiently low dimensional domain. Based +on this observation, Hundrieser et al. (2022) proved for compactly supported µ, ν on Rd, with µ +supported on a finite set or a smooth submanifold of dimension ˜d < 2 min(p, 2) using the functional +delta method (Römisch, 2006), +� +nm +n + m +� +OT(µn, νm, cp) − OT(µ, ν, cp) +� +⇝ +sup +f∈Scp(µ,ν) +√ +λGµ( f cpcp) + +√ +1 − λGν( f cp), +(1.5) +where Scp(µ, ν) is the set of optimizers of (1.2) and Gµ, Gν denote µ-, ν-Brownian bridges, i.e., +centered Gaussian processes with covariance structure characterized by +Cov[Gµ( f cc), Gµ(gcc)] = +� +f ccgccdµ − +� +f ccdµ +� +gccdµ +for f, g ∈ C(X) +(1.6) +and likewise for Gν. The asymptotic theory laid out in (1.5) also provides a unified framework +for distributional limits of the empirical OT value under discrete population measures (Sommerfeld +3 + +and Munk, 2018; Tameling, Sommerfeld, and Munk, 2019) and the semi-discrete setting (del Barrio, +González-Sanz, and Loubes, 2022). +The central contribution of this work is to extend such distributional limits from (1.5) to settings +where the cost function is not fixed and additionally may depend on the underlying measures. We +focus on the following two special instances. +(A) The cost estimator cn,m, centered by its population counterpart c and suitably rescaled, weakly +converges in C(X × Y) to a tight limit, i.e., √nm/(n + m)(cn,m − c) ⇝ Gc in C(X × Y). +(B) There exists a collection {cθ}θ∈Θ of costs such that for any µ ∈ P(X), ν ∈ P(Y) the corre- +sponding cost function cµ,ν ≔ cθ is selected according to an optimization problem of the OT +value over Θ, i.e., either θ ∈ arg maxθ∈Θ OT(µ, ν, cθ) or θ ∈ arg minθ∈Θ OT(µ, ν, cθ). +These two settings are natural and treat a wide spectrum of problems. Furthermore, they are +strongly related. It is noteworthy that setting (B) could be treated in the framework of (A) by +estimating the optimal θ. However, this approach requires the existence of a unique population cost +function and weak convergence of the cost process as a random element in C(X × Y). Since we are +only interested in the empirical infimal or supremal OT value it is instead more natural to rely on +an alternative approach which does not require uniqueness of the population cost function or weak +convergence of the cost process. +For setting (A) we allow the cost function to be estimated from the given data and thus capture +the asymptotic dependency between the cost estimator and the empirical measures. In particular, +this enables an analysis of the empirical OT cost when the cost estimator is parametrized by a +plug-in estimator, e.g., a maximum likelihood procedure. Notably, setting (A) also allows the cost +function to be estimated from independent data. Overall, this setting covers many scenarios with +“extrinsically estimated costs”. We refer to Sections 4.1 and 4.3 for examples. For setting (B) the +motivation slightly differs. Here, the selected cost function depends on the OT problem itself and +often brings invariance of the OT problem with respect to a class of transformation parametrized by +Θ. One could describe this as OT with “intrinsically estimated costs”. Examples of this setting are +provided in Sections 4.2 and 4.4. +Under suitable assumptions we show in Theorem 2.2 for setting (A) that +� +nm +n + m +� +OT(µn, νm, cn,m) − OT(µ, ν, c) +� +⇝ +inf +π∈Π⋆c (µ,ν) π(Gc) + sup +f∈Sc(µ,ν) +√ +λGµ( f cc)+ +√ +1−λGν( f c), +where Π⋆ +c (µ, ν) represents the set of optimizers for (1.1) for µ, ν with costs c and π(Gc) ≔ +� +Gcdπ. +For setting (B) we only state below the distributional limit for supremal costs; a similar distri- +butional limit also occurs for infimal costs (Theorem 2.6). Upon defining the set S+(Θ, µ, ν) = +arg maxθ∈Θ OT(µ, ν, cθ) of maximizers we show in Theorem 2.7 that +� +nm +n + m +� +sup +θ∈Θ +OT(µn, νm, cθ) − sup +θ∈Θ +OT(µ, ν, cθ) +� +⇝ +sup +θ∈S+(Θ,µ,ν) +sup +fθ∈Scθ(µ,ν) +√ +λGµ( f cθcθ +θ +)+ +√ +1−λGν( f cθ +θ ). +In addition to these distributional limits we show for both settings (A) and (B) consistency of a +bootstrap principle. This is of practical importance since quantiles of the respective distributional +limits are difficult to express explicitly due to their dependency on the collection of primal and dual +optimizers for population measures and cost. +Our proof technique for the distributional limit under setting (A) differs from previous ap- +proaches and might be of interest in its own right. More precisely, due to the estimation of the cost +function, we cannot rely on any of the techniques from the references mentioned above. Instead, we +derive certain lower and upper bounds on the OT value which fulfill appropriate (semi-)continuity +4 + +properties. In conjunction with a Skorokhod representation for the empirical process jointly with +the cost process, this enables us to prove that the law of the empirical OT value with estimated costs +is asymptotically stochastically dominated from above and below by the asserted limit distribution. +For the analysis of setting (B) we show under suitable assumptions on the cost family {cθ}θ∈Θ and +the underlying probability measures, that the empirical OT process √n(OT(µn, νm, cθ)−OT(µ, ν, cθ))θ∈Θ +weakly converges in C(Θ) to a tight random variable. We prove this result by invoking the func- +tional delta method in conjunction with a general result on Hadamard directional differentiability +for extremal-type functionals uniformly over a compact parameter space (see Appendix A). The +latter can be viewed as an extension of Römisch (2006, Proposition 1) or Fang and Santos (2019, +Lemma S.4.9) to processes over Θ and relies on Dini’s theorem (Toma 1997, Corollary 1). Central +for this differentiability result is a certain continuity condition among the sets of maximizing ele- +ments for varying parameter. For the OT process it is fulfilled, e.g., if for every θ ∈ Θ the set of +dual optimizer Scθ(µ, ν) is unique (up to constant shift). A similar assumption has been imposed by +Xi and Niles-Weed (2022) for weak convergence of the empirical sliced OT process, which can be +viewed as a special instance of our results for general OT processes, see Section 4.4. The distribu- +tional limits for the empirical infimal and supremal OT value over θ ∈ Θ then follow by another +application of the functional delta method. +Outline +We begin our exposition by deriving in Section 2.1 an appropriate dual formulation of +the OT value which proves useful for our subsequent considerations. We then proceed with our +main contributions, distributional limits for the empirical OT value under weakly converging costs +in Section 2.2 as well as for the empirical OT value under extremal-type costs in Section 2.3. These +asymptotic results are complemented with consistency results of bootstrap resampling schemes in +Section 2.4. We discuss our assumptions for the distributional limits and the bootstrap principles in +Section 3 and provide sufficient conditions for their validity. Statistical applications of our theory +are provided in Section 4, where we also derive a deterministic (first-order) stability result for the +OT cost under joint perturbations of measures and cost function. In Section 5 we explicitly construct +functionals which enables us to “elevate” the regularity of cost estimators to that of their population +counterparts. We employ them in the proofs of our main results which are stated in Section 6. All +remaining proofs as well as auxiliary results and lemmata are relegated to the Appendices. +Notation and probability spaces +Given a set T denote by ℓ∞(T) the Banach space of bounded +functionals on T equipped with uniform norm ∥ϕ∥ ≔ supt∈T |ϕ(t)|. Moreover, if T is equipped with +a topology τ denote by C(T) the Banach space of real valued, bounded, continuous functions on T +equipped with uniform norm. If dT denotes a metric on T, then we define by Cu(T, dT), or Cu(T) +when the metric dT is clear from context, the space of real-valued, bounded, uniformly continuous +functions on (T, d). Endowed with the uniform norm, it is a Banach space as well. A real-valued +function class F on X is always be equipped with uniform norm. This specifies the Banach space +Cu(F ) which is a closed subset of the Banach space ℓ∞(F ). Moreover, for ε > 0 the covering +number N(ε, T, d) denotes the minimal number of sets with diameter 2ε to cover T, and we write +x ≲ y when there exists a constant C > 0 with x ≤ Cy. +For a topological space X the set P(X) denotes the collection of Borel probability measures +on X. Integration +� +fdµ of a real-valued Borel measurable function f : X → R with respect to +µ ∈ P(X) is abbreviated by µ( f) or µ f. Further, we denote by f#µ the pushforward of µ under f. +We define all random variables on the same probability space (Ω, A, P). We further assume a +product structure of that space to define samples and the random weights of the bootstrap, i.e., +Ω = Ω0 × Ω1 × . . . and P = P0 ⊗ P1 × . . . so that the samples only depend on (Ω0, P0), the weights +of the first bootstrap replicate on (Ω1, P1) and so on. The law of a random variable X is denoted +by L(X). We finally assume that there exist infinite sequences of measurable maps X1, X2, . . . from +5 + +(Ω0, P0) to X, respectively, and that samples of cardinality n are obtained from the infinite sequence +by projection of the first n coordinates. Outer probability measures are denoted by P∗ (see van der +Vaart and Wellner, 1996, Chapter 1.2). Denoting by BL1 the set of real-valued functions on a metric +space (T, dT) which are bounded by one in uniform norm and such that | f(x) − f(y)| ≤ dT(x, y) for +any x, y ∈ T, we define the bounded Lipschitz metric between two probability measures µ, ν as +dBL(µ, ν) := sup f∈BL1 |µ( f) − ν( f)|. For a set A and a function f, we write f(A) ≔ { f(a) | a ∈ A}. +For two subsets A, B of a vector space, A + B := {a + b | a ∈ A, b ∈ B}. +2 +Main Results +2.1 +Preliminaries +For our theory on distributional limits for the empirical OT value under estimated cost functions we +consider throughout compact Polish spaces X and Y. Given a continuous cost function c ∈ C(X×Y) +and probability measures µ ∈ P(X), ν ∈ P(Y) there always exist optimizers to both primal and dual +problem (Villani, 2008, Theorems 4.1 and 5.10). +According to Villani (2003, Remark 1.13), dual optimizers can always be selected from the +function class +Hc := +� +h : X → R +���� ∃g: Y → [− ∥c∥∞ , ∥c∥∞], h(·) = inf +y∈Y c(·, y) − g(y) +� +, +(2.1) +which yields for any µ ∈ P(X), ν ∈ P(Y) the alternative dual representation of the OT value, +OT(µ, ν, c) = sup +h∈Hc +µ(hcc) + ν(hc). +(2.2) +The function class Hc is uniformly bounded and each element exhibits the same modulus of con- +tinuity as c, hence it is compact in C(X) by the Theorem of Arzelà-Ascoli. Formula (2.2) was +exploited by Hundrieser et al. (2022) for distributional limits of the empirical OT value under a +fixed cost function. +For our purposes, we require a dual formulation over a fixed function class which holds for more +than a single cost function and to circumvent potential measurability issues we seek a function class +which is compact in C(X) (cf. Lemma 6.4). To this end, let B > 0 and consider a concave modulus +of continuity w: R+ → R+. Then, for a continuous metric dX on X we define the compact function +class F (B, w) ⊆ C(X), +F (B, w) ≔ +� +f : X → R +���� ∥ f∥∞ ≤ 2B, | f(x) − f(x′)| ≤ w(dX(x, x′)) for all x, x′ ∈ X +� +, +(2.3) +which will be utilized for a dual representation of the OT value under suitable costs. +Lemma 2.1 (Dual formulation). Let c ∈ C(X × Y) with ∥c∥∞ ≤ B and |c(x, y) − c(x′, y)| ≤ +w (dX(x, x′)) for all x, x′ ∈ X, y ∈ Y. Then, for F ≔ F (B, w) the following inclusions hold +Hc ⊆ F cc ⊆ Hc + [−2B, 2B] +and +Hc +c ⊆ F c ⊆ Hc +c + [−2B, 2B]. +Further, for arbitrary probability measures µ ∈ P(X) and ν ∈ P(Y) it follows that +OT(µ, ν, c) = sup +f∈F +µ( f cc) + ν( f c) +(2.4) +and the set of dual optimizers Sc(µ, ν) of (2.4), referred to as Kantorovich potentials, is non-empty. +The proof of Lemma 2.1 is deferred to Section 6.1. Overall, Lemma 2.1 justifies the use of +the function class F = F (B, w) for a dual OT formulation and enables us to state conditions of +distributional limits in terms of F instead of potentially varying collections of functions. +6 + +2.2 +Distributional Limits under Weakly Converging Costs +For the distributional limits in all the statements below, we consider independent and identically +distributed random variables {Xi}n +i=1 ∼ µ⊗n and independent {Yi}m +i=1 ∼ ν⊗m defined on the probability +space put forward in the introduction. Based on these samples, we define empirical measures µn ≔ +1 +n +�n +i=1 δXi and νm ≔ +1 +m +�m +i=1 δYi. All the subsequent asymptotic results are to be understood for +n, m → ∞ with m/(n + m) → λ ∈ (0, 1), which we do not recall each time for space considerations. +Our main result on the limit law for the empirical OT value under weakly converging costs is +given as follows for the two-sample case. The one-sample case is discussed in Remark 2.4(ii). +Theorem 2.2 (OT under weakly converging costs). Let c ∈ C(X × Y) and consider an estimator +cn,m ∈ C(X×Y) for c such that cn,m(x, y) is measurable for each (x, y) ∈ X×Y. Let w: R+ → R+ be a +concave modulus of continuity for c with w(δ) > 0 for δ > 0 such that |c(x, y)−c(x′, y)| ≤ w(dX(x, x′)) +for all x, x′ ∈ X, y ∈ Y. Assume for µ ∈ P(X), ν ∈ P(Y) the following. +(JW) For the function class F = F (2 ∥c∥∞ + 1, 2w) from (2.3) joint weak convergence occurs, +� +nm +n + m + +µn − µ +νm − ν +cn,m − c + ⇝ + +√ +λ Gµ +√ +1 − λ Gν +Gc + +in ℓ∞(F cc) × ℓ∞(F c) × C(X × Y), +where (Gµ, Gν, Gc) is a tight random variable and Gµ, Gν have covariance structure as in (1.6). +Further, suppose either one of the following two assumptions. +(OP) There exists a unique OT plan π ∈ Π⋆ +c (µ, ν) between µ and ν for the cost function c. +(Sup) The empirical processes Gµ +n ≔ √n(µn − µ) and Gν +m ≔ √m(νm − ν) fulfill the convergence +sup f∈F Gµ +n( f cn,mcn,m − f cc) +P∗ +−−→ 0 and sup f∈F Gν +m( f cn,m − f c) +P∗ +−−→ 0. +Then, it follows that +� +nm +n + m +� +OT(µn, νm, cn,m) − OT(µ, ν, c) +� +⇝ +inf +π∈Π⋆c (µ,ν) π(Gc) + +sup +f∈Sc(µ,ν) +√ +λ Gµ( f cc)+ +√ +1 − λ Gν( f c). +A key insight of Theorem 2.2 is that the limit distribution for the estimated OT value can be +decomposed into two terms: the fluctuation of the cost estimators evaluated at the collection of +OT plans and the Kantorovich potentials evaluated at the limit of the empirical process. Under +uniqueness of primal and dual optimizers for the population OT problem we obtain the following. +Corollary 2.3 (OT under weakly converging costs and uniqueness). In the setting of Theorem 2.2 +assume (JW)and (OP), and suppose that the set of Kantorovich potentials Sc(µ, ν) for µ, ν with cost +function c is unique (up to a constant shift)2. Then, for π ∈ Π⋆ +c (µ, ν) and f ∈ Sc(µ, ν), it follows that +� +nm +n + m +� +OT(µn, νm, cn,m) − OT(µ, ν, c) +� +⇝ π(Gc) + +√ +λGµ( f cc) + +√ +1 − λGν( f c). +(2.5) +In particular, if (Gµ, Gν, Gc) is a jointly centered Gaussian process in ℓ∞(F cc)×ℓ∞(F c)×C(X×Y), +the weak limit in (2.5) is centered normal. +2By this we mean, for any f, g ∈ Sc(µ, ν) the difference f − g is constant on supp(µ). +7 + +The proof of Theorem 2.2 is deferred to Section 6.2.1 and relies on careful lower and upper +bounds for the empirical OT value due to the primal (1.1) and dual formulation (2.4), as well as +arguments from empirical process theory. In the course of this, a key argument is the application +of Lemma 2.1 for cn,m and c. Notably, we do not demand that the cost estimator cn,m is suitably +bounded or exhibits a similar modulus of continuity as c itself. Instead, we construct by Corol- +lary 5.4 an alternative cost estimator cn,m such that the conditions, ∥cn,m∥∞ ≤ 2 ∥c∥∞ + 1 as well as +|cn,m(x, y) − cn,m(x′, y)| ≤ 2w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y, are fulfilled deterministically and +√nm/(n + m)∥cn,m − cn,m∥∞ +P→ 0. The latter implies by Lemma 6.2 that +� +nm +n + m +� +OT(µn, νm, cn,m) − OT(µn, νm, cn,m) +� +≤ +� +nm +n + m∥cn,m − cn,m∥∞ +P→ 0. +It thus suffices to show the assertion for cn,m where the dual formulation from Lemma 2.1 involving +the function class F (2 ∥c∥∞ + 1, 2w) is available. We call cn,m a regularity elevation of cn,m; details +on different kinds of regularity elevations are given in Section 5. The notion of regularity elevations +also proves to be useful for showing the validity of condition (Sup) as outlined in Section 3.3. +Remark 2.4. We like to comment on a few aspects of the derived distributional limits. +(i) The assumptions of Theorem 2.2 and sufficient conditions for their validity are discussed in +Sections 3.1 – 3.3. Effectively, (JW) delimits the theory to settings of low dimensionality. +In such settings (Sup) is often also valid as long as the population cost is sufficiently regular. +(ii) Our proof technique for Theorem 2.2 and Corollary 2.3 also asserts distributional limits for +the one-sample setting, i.e., when µ is estimated by µn and ν is assumed to be known. For this +setting, (JW) reduces to the condition +√n +�µn − µ +cn − c +� +⇝ +�Gµ +Gc +� +in ℓ∞(F cc) × C(X × Y). +Moreover, in (Sup) we only require that sup f∈F Gµ +n( f cncn − f cc) +P∗ +−−→ 0. Then, +√n +� +OT(µn, ν, cn) − OT(µ, ν, c) +� +⇝ +inf +π∈Π⋆c (µ,ν) π(Gc) + +sup +f∈Sc(µ,ν) +Gµ( f cc). +(iii) In case of a fixed cost function, i.e., when selecting cn = c, the conditions of Theorem 2.2 +reduce to F cc being µ-Donsker and F c being ν-Donsker. Further, by Lemma 2.1 this is equiv- +alent to Hc and Hc +c being Donsker for µ and ν (van der Vaart and Wellner, 1996, Theorem +2.10.1 and Example 2.10.7), respectively, matching conditions (C) and (S2) of Theorem 2.1 +in Hundrieser et al. (2022) which imply that +� +nm +n + m +� +OT(µn, νm, c) − OT(µ, ν, c) +� +⇝ +sup +f∈Sc(µ,ν) +√ +λGµ( f cc) + +√ +1 − λGν( f c). +(iv) Our proof technique also yields distributional limits for the estimated OT value when instead +of empirical measures µn and νm one considers measurable estimators ˜µn ∈ P(X), ˜νm ∈ P(Y), +respectively, that fulfill ˜µn ⇝ µ and ˜νm ⇝ ν in probability. This would mean to replace the +empirical measures µn and νm in Assumptions (JW) and (Sup) by ˜µn and ˜νm, respectively. In +addition, instead of the scaling rate √nm/(n + m) our proof technique theory also permits a +different scaling rate an,m which diverges to infinity for n, m → ∞. +8 + +(v) In Proposition 4.9 we prove that the OT value is Gateaux differentiable in (µ, ν, c) for admis- +sible directions in (∆µ, ∆ν, ∆c) ∈ (P(X) − µ) × (P(Y) − ν) × C(X × Y) with derivative, +(∆µ, ∆ν, ∆c) �→ +inf +π∈Π⋆c (µ,ν) π(∆c) + +sup +f∈Sc(µ,ν) +∆µ( f cc) + ∆ν( f c). +Hence, the asymptotic distribution described in Theorem 2.2 may also be interpreted as a +derivative of the OT value with respect to the triple (µ, ν, c) evaluated at the limit process. +Proving Theorem 2.2 via an application of the functional delta method would amount to +showing Hadamard directional differentiability of the OT value (Römisch, 2006). However, +this turns out be a challenging issue without imposing additional assumptions on the measure +and cost estimators, see Remark 4.10. +(vi) In case of a centered normal limit in (2.5) the limit variance is given by +Var �π(Gc)� + λ VarX∼µ +� f cc(X)� + (1 − λ) VarY∼µ +� f c(Y)� ++ 2 +√ +λ Cov �π(Gc), Gµ( f cc)� + 2 +√ +1 − λ Cov �π(Gc), Gν( f c)�, +where we used that the random variables X1, . . . , Xn and Y1, . . . , Yn are independent. In partic- +ular, the limit law degenerates if both Kantorovich potentials ( f cc, f c) are (µ, ν)-almost surely +constant and cn,m converges to c with a faster rate than (nm/(n + m))−1/2, uniformly on the +support of the OT plan π. For a sharp characterization of the occurrence of almost surely +constant Kantorovich potentials we refer to Section 4 of Hundrieser et al. (2022) where the +authors showcase that for most cost functions of practical interest a.s. constancy typically +does not occur if the underlying measures are different. +2.3 +Distributional Limits under Extremal-Type Costs +As noted in the introduction, could the empirical infimal or supremal OT value over a fixed collec- +tion of cost functions also be analyzed using the previously described framework. However, as part +of this approach, we would require the existence of a single underlying population cost function +as well as weak convergence of the cost estimator. To broaden the scope of our theory, we follow +in this subsection a different route to derive limiting distributions where such conditions are not +required. More precisely, we first prove a uniform distributional limit for the empirical OT process +indexed over the collection of cost functions before relying on a delta method to characterize the +distributional limits for the respective infimal and supremal statistics. +For the subsequent assertions we again adhere to the sampling convention provided at the be- +ginning of Section 2.2. The one-sample case is discussed in Remark 2.9(iii). +Theorem 2.5 (OT process uniformly over compact Θ). Let Θ be a compact Polish space and con- +sider a continuous map c: Θ → C(X × Y), θ �→ cθ. Let w: R+ → R+ be a modulus of conti- +nuity such that supθ∈Θ |cθ(x, y) − cθ(x′, y)| ≤ w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y. Assume for +µ ∈ P(X), ν ∈ P(Y) the following. +(Don) For the function class F = F (supθ∈Θ ∥cθ∥∞ , w) from (2.3) the collection � +θ∈Θ F cθcθ is µ- +Donsker and � +θ∈Θ F cθ is ν-Donsker. +(KP) For any θ ∈ Θ, the set of Kantorovich potentials Scθ(µ, ν) ⊆ F for the OT problem between µ +and ν and cost cθ is unique (up to a constant shift). +Then, upon selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Θ, it follows that +� +nm +n + m +� +OT(µn, νm, cθ) − OT(µ, ν, cθ) +� +θ∈Θ ⇝ +� √ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ ) +� +θ∈Θ +in C(Θ). +9 + +The proof of Theorem 2.5 is based on Hadamard directional differentiability of the OT cost pro- +cess, which follows from a general sensitivity analysis for extremal-type functions uniformly over +a compact parameter space (Appendix A). The assertion for the empirical OT process then follows +by invoking the functional delta method (Römisch, 2006); the proof is deferred to Section 6.3.1. +From the above result, given any functional Φ: C(Θ) → R that is Hadamard directionally +differentiable at the function OT(µ, ν, c(·)) ∈ C(Θ), Theorem 2.5 yields by another application of +the functional delta method, the distributional limit +� +nm +n + m +� +Φ�(OT(µn, νm, cθ))θ∈Θ +� − Φ�(OT(µ, ν, cθ))θ∈Θ +�� +⇝ DH +OT(µ,ν,c(·))Φ +�� √ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ )� +θ∈Θ +� +. +Here, DH +OT(µ,ν,c(·))Φ denotes the directional Hadamard derivative of Φ. This enables the derivation +of the limit distribution for the infimal mapping using Fang and Santos (2019, Lemma S.4.9) (see +also Cárcamo, Cuevas, and Rodríguez 2020, Corollary 2.3). +Theorem 2.6 (OT infimum over compact Θ). Consider the setting of Theorem 2.5. Then, upon +selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Θ, it follows that +� +nm +n + m +� +inf +θ∈Θ OT(µn, νm, cθ) − inf +θ∈Θ OT(µ, ν, cθ) +� +⇝ +inf +θ∈S−(Θ,µ,ν) +√ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ ), +where S−(Θ, µ, ν) = arg minθ∈Θ OT(µ, ν, cθ) denotes the set of minimizers of OT(µ, ν, cθ) over Θ. +In case only (Don) holds, one can still infer the limit law for the empirical supremal OT value. +Theorem 2.7 (OT supremum over compact Θ). Consider the setting of Theorem 2.5 and only as- +sume (Don). Then, it follows that +� nm +n + m +� +sup +θ∈Θ +OT(µn, νm, cθ) − sup +θ∈Θ +OT(µ, ν, cθ) +� +⇝ +sup +θ∈S+(Θ,µ,ν) +fθ∈Scθ(µ,ν) +√ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ ), +where S+(Θ, µ, ν) = arg maxθ∈Θ OT(µ, ν, cθ) denotes the set of maximizers of OT(µ, ν, cθ) over Θ. +The proofs of Theorems 2.6 and 2.7 are documented in Sections 6.3.2 and 6.3.3, respectively. +Moreover, in some contexts the compactness assumption on Θ might be too restrictive. The follow- +ing result provides an extension to non-compact spaces Θ and focuses on the infimal statistic; an +analogue statement also holds for the supremal statistic. Its proof is deferred to Section 6.3.4. +Proposition 2.8 (OT infimum over general Θ). Let Θ be a Polish space and consider a continuous +map c: Θ → C(X × Y). Let µ ∈ P(X), ν ∈ P(Y) and suppose there is a compact set K ⊆ Θ such +that S−(Θ, µ, ν) ⊆ K, there is a sequence of minimizers θn,m ∈ S−(Θ, µn, νm) with limn,m→∞ P∗(θn,m � +K) = 0, and that the assumptions of Theorem 2.6 hold with Θ replaced by K. Then, the assertion of +Theorem 2.6 on the empirical infimal OT value over Θ remains valid. +Remark 2.9. A few comments are in order concerning the weak limits for the empirical OT cost +process as well as the respective infimal and supremal statistic. +(i) In the setting of Theorem 2.5 the parameter space Θ is compact and c: Θ → C(X × Y) +is continuous, therefore the range c(Θ) is also compact in C(X × Y). In particular, by the +Theorem of Arzelà-Ascoli, we conclude that supθ∈Θ ∥cθ∥∞ < ∞ and there exists a suitable +modulus of continuity for all cost functions uniformly on Θ. +10 + +(ii) Both assumptions of Theorem 2.5 and sufficient conditions are discussed in Sections 3.4 and +3.5. Assumption (Don) appears natural in order to control the empirical OT process uniformly +over Θ, whereas (KP) is to ensure that the limit process is supported in C(Θ) and stays tight. +Our proof technique suggests that (KP) can be slightly lifted, but not much. For instance, +one could demand that Kantorovich potentials Scθ(µ, ν) which attain the supremum in the +derivative can be approximated by Kantorovich potentials Scθ′(µ, ν) for θ′ in the immediate +vicinity of θ, as required in Lemma A.3(i). In particular, if Θ ≔ {θ1, . . . , θK} is a finite set +equipped with discrete topology, then (KP) can be omitted. +(iii) The results also extend to the one-sample setting, i.e., when µ is estimated by µn and ν is +assumed to be known. For the one-sample version of Theorem 2.5 it suffices to assume +in (Don) that the function class ∪θ∈ΘF cθcθ is µ-Donsker in conjunction with (KP). Upon +selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Θ, the limit distribution is then given for n → ∞ by +√n +� +OT(µn, ν, cθ) − OT(µ, ν, cθ) +� +θ∈Θ ⇝ +� +Gµ( f cθcθ +θ +) +� +θ∈Θ +in C(Θ). +Under identical assumptions, the one-sample analogue of Theorem 2.6 is available. For the +validity of the one-sample result in Theorem 2.7 it suffices that ∪θ∈ΘF cθcθ is µ-Donsker. +(iv) The obtained weak limits highlight an intimate dependency of limit distributions to the col- +lection of Kantorovich potentials. In Theorem 2.5 the limit process is centered Gaussian +due to Assumption (KP). For fixed θ ∈ Θ the limiting random variables degenerates to a +Dirac measure at zero if the respective Kantorovich potentials are (µ, ν)-almost surely con- +stant. Moreover, the limit distribution in Theorem 2.6 is also centered normal if Kantorovich +potentials ( f cθcθ, f cθ) for µ, ν and cθ coincide (up to a constant shift) on supp(µ) × supp(ν) for +any θ ∈ S−(Θ, µ, ν). Under analogous assumptions for θ ∈ S+(Θ, µ, ν) the limit distribution +in Theorem 2.7 is centered normal. In particular, assuming (KP), this condition is fulfilled if +S−(Θ, µ, ν) or S+(Θ, µ, ν) consist of a singleton. The resulting limit distributions degenerate if +Kantorovich potentials are (µ, ν)-almost surely constant. A sharp characterization of almost +surely constant potentials is detailed in Section 4 of Hundrieser et al. (2022). +2.4 +Bootstrap Principle for Optimal Transport Costs +Since the limit distributions in Theorems 2.2, 2.6 and 2.7 involve the set of Kantorovich potentials +(and OT plans), under non-unique optimizers there is little hope for an explicit, closed-form de- +scription of the quantiles for these distributions, which is required for further practical purposes. To +circumvent this issue we suggest the use of a k-out-of-n bootstrap procedure with k = o(n) whose +consistency is shown in this subsection. +For simplicity, we state the subsequent results for equal sample sizes, i.e., n = m as well as +bootstrap samples of equal size k = o(n). Under differing sample sizes n � m one would select +bootstrap samples of size k = o(n), l = o(m) such that l/(l + k) ≈ m/(n + m). Below, we always +consider the same bootstrap approach that we now introduce. For the two sequences of i.i.d. random +variables {Xi}n +i=1 ∼ µ⊗n, {Yi}n +i=1 ∼ ν⊗n, with respective empirical measures µn, νn, consider another +sequence of i.i.d. bootstrap random variables {Xb +i }k +i=1 ∼ µ⊗k +n , {Yb +i }k +i=1 ∼ ν⊗k +n and define the bootstrap +empirical measures µb +n,k ≔ 1 +k +�k +i=1 δXb +i and νb +n,k ≔ 1 +k +�k +i=1 δYb +i . Moreover, we write in the subsequent +statement cn for the cost estimator and cb +n,k for the bootstrap cost estimator. +Proposition 2.10 (Bootstrap for OT under weakly converging costs). In the setting of Theorem 2.2, +assume (JW) and either (OP) or (Sup). Let cb +n,k ∈ C(X × Y) be the bootstrap cost estimator such +that cb +n,k(x, y) is measurable for all (x, y) ∈ X × Y. Further, assume the following. +11 + +(JW)∗ The bootstrap empirical processes are conditionally on X1, . . . , Xn, Y1, . . . Yn consistent in the +space ℓ∞(F cc) × ℓ∞(F c) × C(X × Y) for n, k → ∞ with k = o(n), i.e., +dBL +L + +√ +k + +µb +n,k − µn +νb +n,k − νn +cb +n,k − cn + +��������� +X1, . . . , Xn, Y1, . . . Yn + , L + +√n + +µn − µ +νn − ν +cn − c + + + +P∗ +−−→ 0. +In case of setting (Sup) additionally assume the following. +(Sup)∗ The unconditional bootstrap empirical processes Gµ +n,k ≔ +√ +k(µb +n,k−µ) and Gν +n,k ≔ +√ +k(νb +n,k− ν) +fulfill sup f∈F Gµ +n,k( f cb +n,kcb +n,k − f cc) +P∗ +−−→ 0, sup f∈F Gν +n,k( f cb +n,k − f c) +P∗ +−−→ 0 for n, k → ∞, k = o(n). +Then, it follows for n, k → ∞ with k = o(n) that +dBL +� +L +� √ +k +� +OT(µb +n,k, νb +n,k, cb +n,k) − OT(µn, νn, cn) +�����X1, . . . , Xn, Y1, . . . , Yn +� +, +L +� √n (OT(µn, νn, cn) − OT(µ, ν, c)) +� � P∗ +−−→ 0. +Despite not relying on the functional delta method for the derivation of the limit distribution of +the empirical OT value under weakly converging costs, we obtain a similar bootstrap principle as +Dümbgen (1993, Proposition 2) by employing an equivalent formulation for bootstrap consistency +(Bücher and Kojadinovic, 2019) in conjunction with the use of a Skorokhod representation. The +full proof is provided in Section 6.2.2. +Remark 2.11. When employing the functional delta method, the n-out-of-n bootstrap is not consis- +tent if the Hadamard directional derivative is not linear (Dümbgen, 1993, Proposition 1). Although +Proposition 2.10 does not build on a differentiability result, we show in Section 4.5 that the OT +functional is Gateaux directional differentiability with a derivative that is non-linear if primal or +dual optimizers are non-unique. Since Gateaux directional differentiability is implied by Hadamard +directional differentiability, this suggests in the regime of non-unique optimizers the inconsistency +of the naive n-out-of-n bootstrap for the empirical OT cost under weakly converging costs. +Verification of the bootstrap consistency in the settings of Theorems 2.5–2.7 is straightforward. +It is a direct consequence of consistency of the k-out-of-n bootstrap empirical processes with k = +o(n) (van der Vaart and Wellner, 1996, Theorem 3.6.13) and the functional delta method for the +bootstrap (Dümbgen, 1993, Proposition 2). Hence, we omit the proof of the following proposition. +Proposition 2.12 (Bootstrap for OT process, supremum, and infimum). Let X, Y be compact Polish +spaces and Θ a compact topological space. Consider a continuous map c: Θ → C(X × Y), θ �→ cθ, +let µ ∈ P(X), ν ∈ P(Y) and assume (Don). +(i) (OT process in C(Θ)) Then, under (KP), it follows for n, k → ∞ with k ≤ n that +dBL +� +L +� √ +k +� +OT(µb +n,k, νb +n,k, cθ) − OT(µn, νn, cθ) +� +θ∈Θ +����X1, . . . , Xn, Y1, . . . , Yn +� +, +L +� √n (OT(µn, νn, cθ) − OT(µ, ν, cθ)) +� +θ∈Θ +� P∗ +−−→ 0. +(ii) (OT infimum over Θ) Then, under (KP), it follows for n, k → ∞ with k ≤ o(n) that +dBL +� +L +� √ +k +� +inf +θ∈Θ OT(µb +n,k, νb +n,k, cθ) − inf +θ∈Θ OT(µn, νn, cθ) +������X1, . . . , Xn, Y1, . . . , Yn +� +, +L +� √n +� +inf +θ∈Θ OT(µn, νn, cθ) − inf +θ∈Θ OT(µ, ν, cθ) +�� � P∗ +−−→ 0. +12 + +(iii) (OT supremum over Θ) Then, it follows for n, k → ∞ with k = o(n) that +dBL +� +L +� √ +k +� +sup +θ∈Θ +OT(µb +n,k, νb +n,k, cθ) − sup +θ∈Θ +OT(µn, νn, cθ) +�������X1, . . . , Xn, Y1, . . . , Yn +� +, +L +� √n +� +sup +θ∈Θ +OT(µn, νn, cθ) − sup +θ∈Θ +OT(µ, ν, cθ) +�� � P∗ +−−→ 0. +Notably, we also obtain consistency of the n-out-of-n bootstrap for setting (i) since (KP) implies +linearity of the Hadamard directional derivative. +3 +Discussion of Assumptions +In this section we discuss the assumptions on the distributional limits and the bootstrap consistency. +We also provide sufficient conditions for their validity. All the proofs are deferred to Appendix B. +3.1 +Assumptions (JW) and (JW)∗: Joint Weak Convergence +For the empirical OT value under estimated costs we demand in (JW) and (JW)∗ weak convergence +of the empirical processes in ℓ∞(F cc) and ℓ∞(F c), where F = F (2 ∥c∥∞ + 1, 2w) is selected as in +Theorem 2.2. This requires F cc and F c to be µ- and ν-Donsker, respectively. Moreover, we demand +weak convergence of the estimated cost function in C(X × Y) to ensure that any sequence of OT +plans for µn, νm and cn,m tends towards an OT plan in Π⋆ +c (µ, ν). Finally, we stress the necessity of +joint weak convergence in (JW) and (JW)∗ as the limit distribution is determined by the random +variable (Gµ, Gν, Gc) and thus characterized by their dependency. +Even though apparently unavoidable, these conditions are somewhat restrictive and delimit the +theory to low dimensional settings. This is to be expected as estimation of the OT value (under +population costs) suffers from the curse of dimensionality (Manole and Niles-Weed, 2021), leading +to slow convergence rates when both population measures µ, ν exhibit high-dimensional support. +However, in view of the recently discovered lower complexity adaptation principle (Hundrieser, +Staudt, and Munk, 2022), it suffices that one measure, µ or ν, is supported on a low dimensional +space. The following proposition provides bounds on the covering numbers (see the notation section +for a definition) of F c and F cc under uniform norm which leads to a universal Donsker property +for both function classes. +Proposition 3.1 (Universal Donsker property). Let c ∈ C(X×Y) be a continuous cost function with +∥c∥∞ ≤ 1. Assume one of the three settings. +(i) X = {x1, . . . , xN} is a finite space (and no additional assumption on c). +(ii) There exists a pseudo metric3 ˜dX on X such that N(ε, X, ˜dX) ≲ ε−β for ε > 0 sufficiently +small and some β ∈ (0, 2) and c(·, y) is 1-Lipschitz under ˜dX for all y ∈ Y. +(iii) X = �I +i=1 ζi(Ui) for I ∈ N compact, convex subsets Ui ⊆ Rdi, di ≤ 3 with non-empty interior +and maps ζi : Ui → X such that for each i ∈ {1, . . . , I} the function c(ζi(·), y) is (γi, 1)-Hölder4 +on Ui for some γi ∈ (di/2, 2] for all y ∈ Y. +3A non-negative function d: M × M → R+ on a set M is a pseudo-metric if the three conditions d(x, x) = 0, +d(x, y) = d(y, x) and d(x, y) ≤ d(x, z) + d(z, y) are fulfilled for any x, y, z ∈ M. +4 A function f : U → R on a convex set U ⊆ Rd with non-empty interior is (γ, Λ)-Hölder with modulus Λ ≥ 0 and +γ ∈ (0, 1] if ∥f ∥∞ < Λ and |f (x) − f (y)| ≤ Λ ∥x − y∥γ for any x, y ∈ U. Further, f is called (γ, Λ)-Hölder for γ ∈ (1, 2] if +every partial derivative of f is (γ − 1, Λ)-Hölder. If U is not open, we assume the existence of an extension ˜f of f onto +an open convex set containing U such that ˜f is (γ, Λ)-Hölder thereon, cf. Hundrieser, Staudt, and Munk (2022). +13 + +Let B ≥ 0 and consider a modulus of continuity w: R+ → R+ with respect to a metric dX on X. +Then, for each setting there exists some α < 2 such that for ε > 0 sufficiently small, +log N(ε, F c, ∥·∥∞) = log N(ε, F cc, ∥·∥∞) ≲ ε−α +for F = F (B, w), +where the hidden constant depends for (i) on N, for (ii) on N(ε, X, ˜dX), and for (iii) on (ζi, Ui)I +i=1. +In particular, the function classes F c and F cc are universal Donsker. +The bounds for the covering numbers stated in the above proposition are essential for the weak +convergence of the empirical processes √n(µn − µ) and √m(νm − ν) and represent an important +tool for verifying (JW). In order to clarify the assumptions of Proposition 3.1, we showcase them +in a simple example. We additionally refer to Hundrieser, Staudt, and Munk (2022, Section 3) and +Hundrieser et al. (2022, Section 5) for more illustrative examples. +Example 3.2. Suppose that X and Y are compact subsets of R3 and let c : R3 × R3 → R be twice +continuously differentiable. By enlarging X to a compact, convex set and since c can be rescaled +such that c(·, y) is (2, 1)-Hölder on X, Proposition 3.1 is applicable in this setting. +To state sufficient conditions for (JW) and (JW)∗ we assume that the population cost as well as +the empirical and bootstrap estimators are determined by the underlying measures via a Hadamard +directionally differentiable functional. For simplicity, we consider in the subsequent proposition +random variables {Xi}n +i=1 ∼ µ⊗n, {Yi}n +i=1 ∼ ν⊗n of identical sample size n with empirical measures +µn, νn, and bootstrap samples {Xb +i }k +i=1 ∼ µ⊗k +n , {Yb +i }k +i=1 ∼ ν⊗k +n of size k = k(n) = o(n) with correspond- +ing bootstrap empirical measures µb +n,k, νb +n,k. +Proposition 3.3 (Joint weak convergence). Let FX, FY be bounded function classes on X and Y, +respectively, and assume there is a functional Φc : P(X) × P(Y) ⊆ ℓ∞(FX) × ℓ∞(FY) → C(X × Y) +such that, for all n, k ∈ N, +c = Φc(µ, ν), +cn = Φc(µn, νn), +and +cb +n,k = Φc(µb +n,k, νb +n,k). +If Φc is Hadamard directionally differentiable at (µ, ν) tangentially to P(X)×P(Y), and if FX ∪F cc +is µ-Donsker while FY ∪ F c is ν-Donsker, then both (JW) and (JW)∗ are fulfilled. +Remark 3.4. We like to point out that if the functional Φc is additionally continuous with respect to +the topology induced by weak convergence on P(X)×P(Y), it follows that cn(x, y) and cb +n,k(x, y) are +measurable for each (x, y) ∈ X × Y and, due to compactness of X and Y, measurable in C(X × Y). +3.2 +Assumption (OP): Uniqueness of Optimal Transport Plans +The subject of uniqueness of OT plans between probability measures and a given cost function is of +long-standing interest and has been addressed by various authors. General conditions for continuous +settings were stated in Gangbo and McCann (1996) and Levin (1999), building on previous works. +The subject has since been covered in depth in Chapters 9 and 10 of the reference textbook by +Villani (2008); further advances have been made since. +To guarantee the uniqueness of the OT plan, many works resort to the so-called Twist condition +which demands for differentiable costs the injectivity of the map y → ∇xc(x, y) for all x ∈ X. The +following proposition formalizes a uniqueness criterion based on this condition and should fulfill +the reader’s needs for many practical applications. The result can be deduced from Theorem 10.28 +and Remark 10.33 in Villani (2008). +Proposition 3.5. Assume that X, Y are compact Polish spaces where X ⊆ Rd is a Euclidean sub- +set with non-empty interior and µ is absolutely continuous with respect to the Lebesgue measure. +Further, assume that c is locally Lipschitz on X × Y, that c(·, y) is differentiable on int(X) for each +y ∈ Y and that y �→ ∇xc(x, y) is injective for each x ∈ X. Then, the OT plan π⋆ ∈ Π(µ, ν) is unique. +14 + +Even though in certain cases weaker conditions can yield uniqueness (Ahmad, Kim, and Mc- +Cann, 2011), these more general conditions are typically considerably more difficult to verify. Nev- +ertheless, unless the cost function exhibits some kind of symmetry or is constant in some region, +uniqueness of OT plans is often to be expected. Indeed, for fixed measures there is a residual set of +cost functions such that for any such costs the OT plan is unique (McCann and Rifford, 2016). +In finite discrete settings, i.e., when both underlying measures are supported on finitely many +points, results on the uniqueness of OT plans are mostly based on the theory of finite-dimensional +linear programs and we refer to Klatt, Munk, and Zemel (2022, Section 6) for a detailed account. +Among others, they provide sufficient conditions for uniqueness of OT plans which solely depend +on the cost function and the support points but are independent of the weights of the measures. For +Euclidean based costs their condition is fulfilled for Lebesgue-almost every arrangement of support +points of µ and ν, it is however violated if the support points obey some regular or repetitive pattern. +3.3 +Assumptions (Sup) and (Sup)∗: Control of Supremum over Empirical Process +Our assumptions on the suprema of the empirical processes ensure that the fluctuation on the set +of feasible dual potentials caused by estimation of the cost function is asymptotically negligible. +Let us also point out that the suprema in (Sup) and (Sup)∗ are (Borel) measurable by Lemma +6.4 and 6.5. This implies that the convergence in outer probability occurs, in fact, in probability. +Indeed, following along the proof of Lemma 6.5 and due to measurability of cn,m it follows for fixed +f ∈ F = F (2 ∥c∥∞ + 1, 2w) that both maps ω �→ Gµ +n( f cc) and ω �→ Gµ +n( f cn,mcn,m) are measurable. In +conjunction with Gµ +n((·)cc), Gµ +n((·)cn,mcn,m) ∈ Cu(F , ∥·∥∞) by Lemma 6.4 and compactness of (F , ∥·∥∞) +the measurability of the Gµ +n((·)cc − (·)cn,mcn,m) as well as its supremum follow. +In the following, we derive sufficient conditions for the validity of Assumption (Sup) (as well +as Assumption (Sup)∗). Based on empirical process theory, in order to suitably control the suprema +sup f∈F Gµ +n( f cn,m,cn,m − f cc) +and +sup f∈F Gν +n( f cn,m − f c) +a canonical route would be to impose metric entropy bounds for F cn,m,cn,m ∪ F cc and F cn,m ∪ F c. +Such bounds, however, would impose certain regularity requirements on the cost estimator cn,m. +Hence, in order not to narrow our scope concerning cost estimators, we employ the same ideas as +in Section 2.2 and approximate the cost estimator cn,m by a more regular cost estimator ˜cn,m. The +subsequent result formalizes these considerations for our context. Its proof relies on techniques +developed by van der Vaart and Wellner (2007) for empirical processes indexed over estimated +function classes. +Proposition 3.6. Let X, Y be compact Polish spaces and consider a continuous cost function c. +(i) Assume (JW) for random elements cn,m ∈ C(X×Y) and take random elements ˜cn,m ∈ C(X×Y) +with √nm/(n + m)∥cn,m − ˜cn,m∥∞ +P→0 for n, m = m(n) → ∞ and m/(n + m) → λ ∈ (0, 1) such +that for ε > 0 sufficiently small, +log N(ε, F cc, ∥·∥∞) + sup +n∈N +log N(ε, F ˜cn,m ˜cn,m, ∥·∥∞) ≲ ε−α +with α < 2. +(3.1) +Then Assumption (Sup) is fulfilled. +(ii) Assume (JW) and (JW)∗ for random elements cb +n,k ∈ C(X × Y) and let ˜cb +n,k ∈ C(X × Y) be +random elements with +√ +k∥cb +n,k − ˜cb +n,k∥∞ +P→0 for n, k = k(n) → ∞ and k = o(n) such that for +ε > 0 sufficiently small, +log N(ε, F cc, ∥·∥∞) + sup +n∈N +log N(ε, F ˜cb +n,k ˜cb +n,k, ∥·∥∞) ≲ ε−α +with α < 2. +(3.2) +Then Assumption (Sup)∗ is fulfilled. +15 + +As a straightforward corollary of Proposition 3.6 we find that (Sup) and (Sup)∗ are fulfilled if +the cost estimators cn,m and cb +n,k fulfill certain deterministic regularity conditions once n, m, k are +sufficiently large. In the large sample regime we then choose ˜cn,m ≔ cn,m and ˜cb +n,k ≔ cb +n,k. +Corollary 3.7. Let X, Y be compact Polish spaces, consider a continuous cost function c. Assume +(JW) for cn (and (JW)∗ for cb +n,k) and that c, cn (and cb +n,k) each fulfill one of the three conditions of +Proposition 3.1 for n ≥ N, k ≥ K with random variables N, K ∈ N . Then, (Sup) (and (Sup)∗) hold. +Hence, if the population cost c and the estimators cn, cn,k are determined by some parameter +θ ∈ Θ and estimators θn, θn,k, such that the regularity properties of Proposition 3.1 are met uniformly +in an open neighborhood of θ and if the estimators are consistent, then Corollary 3.7 asserts the +validity of Assumptions (Sup) and (Sup)∗. +Moreover, under mild additional assumptions on the space X and the cost function c, we can +state a functional Ψ: C(X × Y) → C(X × Y) such that ˜cn ≔ Ψ(cn) fulfills the entropy bound +(3.1) while satisfying √n +���˜cn,m − cn,m +���∞ +P→ 0 for n → ∞. We call such a functional Ψ a regularity +elevation functional since it lifts the degree of regularity of the cost estimator. Details on regularity +elevations are deferred to Section 5. +Corollary 3.8. Let X, Y be compact Polish spaces and consider a continuous cost. Assume (JW) +(and (JW)∗). Suppose that c fulfills one of the three conditions of Proposition 3.1. Under (ii) or (iii) +further assume the subsequent condition (ii)’ or (iii)’, respectively. +(ii)’ The weak limit Gc is almost surely continuous with respect to (X, ˜dX) × Y. +(iii)’ For each i ∈ {1, . . . I} the set Ui ⊆ Rdi is convex and compact, the map ζi : Ui → ζi(Ui) is +a homeomorphism, and ci ≔ c(ζi(·), ·): Ui × Y → R is continuously differentiable in u on +Ui × Y, i.e., the derivative ∇uci : int(Ui) × Y → Rd can be continuously extended to Ui × Y. +Further, there exists a continuous partition of unity5 {ηi}I +i=1 on X with supp(ηi) ⊆ ζi(Ui). +Then, Assumption (Sup) (and (Sup)∗) is fulfilled. +3.4 +Assumption (Don): Donsker Property Uniformly over Θ +For the distributional limits by Hundrieser et al. (2022) on the empirical OT value under a fixed cost +function c, the authors effectively assume that the function classes F cc and F c are µ- and ν-Donsker, +respectively (Remark 2.4(iii)). Hence, for the uniform convergence result from Theorem 2.5 it is +natural that we demand the µ- and ν-Donsker property for the unions ∪θ∈ΘF cθcθ and ∪θ∈ΘF cθ for +F = F (supθ∈Θ ∥c∥∞ , w). The validity of this condition can be ensured under assumptions on the +complexity of the domain X in conjunction with regularity conditions imposed on the cost function. +Proposition 3.9 (Universal Donsker property over Θ). Let X, Y be compact Polish spaces and let +(Θ, dΘ) be a metric space such that log N(ε, Θ, dΘ) ≲ ε−α for α < 2. Suppose that c: (Θ, dΘ) → +C(X × Y), θ �→ cθ is 1-Lipschitz and assume supθ∈Θ ∥cθ∥∞ ≤ 1. Consider one of the three settings. +(i) X = {x1, . . . , xN} is a finite space (and no additional assumption on c). +(ii) For any θ ∈ Θ there exists a pseudo metric ˜dθ,X on X such that supθ∈Θ N(ε, X, ˜dθ,X) ≲ ε−β for +β < 2 and cθ(·, y) is 1-Lipschitz under ˜dθ,X for all y ∈ Y. +(iii) X = �I +i=1 ζi(Ui) for I ∈ N compact, convex subsets Ui ⊆ Rdi, di ≤ 3 with non-empty interior +and maps ζi : Ui → X so that for each i ∈ {1, . . . , I} the function cθ(ζi(·), y) is (γi, 1)-Hölder +on Ui (recall footnote 4) for some γi ∈ (di/2, 2] for all y ∈ Y, θ ∈ Θ. +5A collection {ηi}I +i=1 is a continuous partition of unity if ηi ∈ C(X), ηi ≥ 0 for each i and �I +i=1 ηi ≡ 1 on X. +16 + +Then, for each setting, there exists some α < 2 such that +log N(ε, ∪θ∈ΘF cθcθ, ∥·∥∞) ≲ ε−α +and +log N(ε, ∪θ∈ΘF cθ, ∥·∥∞) ≲ ε−α. +In particular, ∪θ∈ΘF cθcθ, ∪θ∈ΘF cθ are universal Donsker, and Assumption (Don) is fulfilled. +The proof of Proposition 3.9 is a simple consequence of Proposition 3.1 in combination with +the subsequent lemma whose proof is deferred to Appendix B.6. +Lemma 3.10. Let X, Y be compact Polish spaces and let (Θ, dΘ) be a metric space. Suppose +c: (Θ, dΘ) → C(X × Y) is 1-Lipschitz. Then, it follows for any ε > 0 that +max +� +N�ε, ∪θ∈ΘF cθ, ∥·∥∞ +�, N�ε, ∪θ∈ΘF cθcθ, ∥·∥∞ +�� +≤ N +�ε +4, Θ, dΘ +� +sup +θ∈Θ +N +�ε +2, F cθcθ, ∥·∥∞ +� +. +3.5 +Assumption (KP): Uniqueness of Kantorovich Potentials +The uniform weak limit of the empirical OT process from Theorem 2.5 demonstrates a close relation +to the collection of Kantorovich potentials. In particular, for the limit to be supported on C(Θ) a +certain continuity property on the Kantorovich potentials Scθ(µ, ν) with respect to θ is required. +Assumption (KP) on the uniqueness of Kantorovich potentials represents a sufficient condition to +ensure this property. +The recent work by Staudt, Hundrieser, and Munk (2022) thoroughly analyzes the topic of +uniqueness in Kantorovich potentials and highlights that it is often expected. More precisely, for +differentiable costs and assuming that one probability measure is supported on the closure of a +connected open set on a smooth manifold, Kantorovich potentials are unique. As Example 3 in their +work showcases, uniqueness also occurs under continuous costs if one measure is discrete while the +other has connected support. In case both measures have disconnected support, then uniqueness can +still be guaranteed if potentials on restricted OT sub-problems are unique and if there exists, in the +language of Staudt, Hundrieser, and Munk, a non-degenerate OT plan, meaning that all connected +components of both measures are linked via that OT plan. The existence of such OT plans can be +guaranteed under mild conditions on the underlying measures (see (3.3)) and intuitively demands +that the OT problem cannot be divided into distinct sub-problems. +The following statement is a direct consequence of Staudt, Hundrieser, and Munk (2022), which +we have included for ease of reference. +Proposition 3.11. Let c: Rd × Rd → R be a differentiable cost function. Consider probability mea- +sures µ, ν ∈ P(Rd) with compact support and suppose supp(µ) = � +i∈I Xi and supp(ν) = � +j∈J Yj +for finitely many disjoint sets. Assume each set Xi is either (i) the closure of a connected open set, +(ii) the closure of a connected open set in a smooth compact submanifold of Rd, or (iii) a single +point. Further, if min(|I|, |J|) ≥ 2, suppose for all non-empty, proper I′ ⊂ I and J′ ⊂ J that +� +i∈I′ +µ(Xi) � +� +j∈J′ +ν(Y j), +(3.3) +Then, Kantorovich potentials for µ, ν and c are unique (up to a constant shift). +It follows from Theorem 1 in Staudt, Hundrieser, and Munk (2022), and relies on continuity +of Kantorovich potentials due to the compactness assumption in conjunction with uniqueness on +subproblems (Corollary 2) and the existence of non-degenerate plans (Lemma 6). +17 + +4 +Applications +In this section we employ our theory from Section 2 to obtain novel insights about various OT +related topics. All proofs for this section are deferred to Appendix C. +4.1 +Optimal transport based One-Sample Goodness-of-Fit-Testing +Hallin, Mordant, and Segers (2021) proposed to use the Wasserstein distance between a sample +measure and a reference measure for goodness-of-fit testing under group actions. In the following, +we briefly recall the setting for compactly supported measures. Let ν0 ∈ P(Rd) be compactly +supported, define Y as the convex hull of supp(ν0), and let GΘ = {gϑ : ϑ ∈ Θ} be a group of +measurable transformations gϑ : Rd → Rd that is parametrized by ϑ ∈ Θ ⊆ Rk for k ∈ N. Further, +assume that the map x �→ gϑ(x) is continuous for every ϑ ∈ Θ and that the mappings ϑ �→ gϑ and +gϑ �→ (gϑ)#ν0 are bijective (this implies the identifiability of the model parameter). Hallin, Mordant, +and Segers, 2021 consider the subsequent testing problem: +Let GΘ be a group and define M = {gϑ#ν0 : gϑ ∈ GΘ}. Given an i.i.d. sample {Xi}n +i=1 +from some unknown µ ∈ P(X) with X ⊂ Rd compact, the aim is to test +H∗ +0 : µ ∈ M +against +H∗ +1 : µ � M. +(4.1) +Note that the parameter ϑ∗ under H0, such that (gϑ∗)#ν0 = µ, is unknown. To construct a test for +the above hypothesis, which is for instance of particular interest in the analysis of location-scale +families, the authors propose to rely on the (2-)Wasserstein distance, i.e., they propose a test based +on an empirical version of +OT +� +µ, ν0, +���g−1 +ϑ∗ (·) − · +���2� += +inf +π∈Π(µ,ν0) +� ���g−1 +ϑ∗ (x) − y +���2 dπ(x, y). +For this purpose, the unknown measure µ is replaced by µn and the cost function c(x, y) = ∥g−1 +ϑ∗ (x) − +y∥2 by cn(x, y) = ∥g−1 +ϑn (x)−y∥2, where ϑn ∈ Θ denotes a suitable estimator for ϑ∗. Thus, the proposed +test statistic is given as +OT +� +µn, ν0, +���g−1 +ϑn (·) − · +���2� += +inf +π∈Π(µn,ν0) +� ���g−1 +ϑn (x) − y +���2 dπ(x, y), +(4.2) +which amounts to solving an OT problem with an estimated cost function. Hence, we can apply our +theory to derive the limiting distribution of +√n +� +OT +� +µn, ν0, +���g−1 +ϑn (·) − · +���2� +− OT +� +µ, ν0, +���g−1 +ϑ∗ (·) − · +���2�� +(4.3) +under the null hypothesis H∗ +0 in (4.1) (see Remark 4.4 for a discussion). In addition, we are able +to extend this to testing whether H∗ +0 holds approximately, which is often preferable in practice (see, +e.g., Munk and Czado, 1998; Dette and Munk, 1998; Dette and Wu, 2019). For this purpose, we +fix an estimation procedure for ϑ∗, i.e., we choose a specific estimator ϑn (taking values in Θ) for +estimating ϑ∗ and denote its population quantity by ϑo ∈ Θ (under H∗ +0 we assume ϑ∗ = ϑo). Then, +we consider the subsequent testing problem: +Let GΘ be a group. Given an i.i.d. sample {Xi}n +i=1 from some unknown µ ∈ P(X) with +X ⊂ Rd compact, the aim is to test for some prespecified ∆ > 0 the hypothesis +H0 : W2(µ, (gϑo)#ν0) ≤ ∆ +versus +H1 : W2(µ, (gϑo)#ν0) > ∆. +(4.4) +18 + +In order to construct a test for the above problem, we have to derive the distributional limits of +(4.3) under the assumption that µ � M. To this end, we employ the theory from Sections 2 and 3. +The first step for the derivation of distributional limits of (4.3) is to establish Hölder regularity +(recall Footnote 4) for costs induced by CΘ : Θ → C(X × Y), ϑ �→ ((x, y) �→ ∥g−1 +ϑ (x) − y∥2) near ϑo. +Lemma 4.1. Let X, Y ⊆ Rd be compact and denote by C(X, Rd) the space of continuous functions +from X to Rd. Assume that KΘ : Θ ⊆ Rk → C(X, Rd), ϑ �→ (x �→ g−1 +ϑ (x)) is continuous near ϑo. +Then, there is an open (w.r.t. relative topology) neighborhood U ⊆ Θ of ϑo and some Λ ≥ 0 such +that for any x ∈ X and ϑ ∈ U the cost function CΘ(ϑ)(x, ·) ≔ ∥g−1 +ϑ (x) − ·∥2 is (2, Λ)-Hölder on Y. +Next, we verify that Hadamard differentiability of KΘ at ϑo implies Hadamard differentiability +of the cost parametrizing map CΘ : Θ → C(X × Y), ϑ �→ ((x, y) �→ ∥g−1 +ϑ (x) − y∥2) at ϑo. To this end, +we additionally impose the following assumption. +(G) For ϑo there exists mϑo > 0 such that for all ϑ′ ∈ Θ in some neighborhood of ϑo, +sup +x∈Rd +���g−1 +ϑ′ (x) − g−1 +ϑo (x) +��� +1 + +���g−1 +ϑo (x) +��� +≤ mϑo +���ϑ′ − ϑo��� . +This condition is fulfilled, e.g., for location-scale families and affine transformations. A global +version of the above assumption, i.e., where the condition is to be fulfilled for any ϑ and not only +ϑ0, has been used by Hallin, Mordant, and Segers, 2021 to ensure the consistency of their goodness- +of-fit test described above. +Lemma 4.2. Assume that the function KΘ : Θ → C(X, Rd), ϑ �→ (x �→ g−1 +ϑ (x)) is Hadamard +differentiable at ϑo tangentially to Θ, i.e., for any sequence (ϑo + tnhn)n∈N ⊆ Θ such that tn ց 0 +and hn → h ∈ Rk as n → ∞, +lim +n→∞ +����� +KΘ(ϑo + tnhn) − KΘ(ϑo) +tn +− DH +|ϑoKΘ(h) +�����∞ += 0, +where DH +|ϑoKΘ(h): X → Rd is a continuous function. Then, if Assumption (G) is satisfied, CΘ is +Hadamard differentiable at ϑo tangentially to Θ with derivative DH +|ϑoCΘ(h) ∈ C(X × Y) given by +DH +|ϑoCΘ(h): X × Y → R, +(x, y) �→ 2 +� +DH +ϑoKΘ(h)(x), g−1 +ϑo (x) − y +� +. +Moreover, if √n(ϑn − ϑo) ⇝ Gϑ for n → ∞, we obtain for cn ≔ CΘ(ϑn) and c ≔ CΘ(ϑ) that +√n(cn − c) ⇝ Gc ≔ +� +2 +� +DH +ϑoKΘ(Gϑ)(x), g−1 +ϑo (x) − y +�� +(x,y)∈X×Y +in C(X × Y). +(4.5) +Under the conditions in the proposition above, our main result from Theorem 2.2 yields a (typi- +cally) non-degenerate limiting distributions for the statistic OT (µn, ν, cn) under the assumption that +µ � M. In particular, this allows us to construct an asymptotic level α test for the null hypotheses +given in (4.4) (see Munk and Czado, 1998 for the precise construction). +Proposition 4.3. Let ν0 ∈ P(Rd) for d ≤ 3 be compactly supported and define Y as the convex +hull of supp(ν0), and let X ⊆ Rd be compact. Assume that (G) holds, and suppose that GΘ : Θ → +C(X, Rd), ϑ �→ (x �→ g−1 +ϑ (x)) is continuous near ϑo and Hadamard differentiable at ϑo. Define for +Λ ≥ 0 from Lemma 4.2 the function class +F ≔ +� +f : Y → R +���� ∥f∥∞ ≤ Λ + 1, | f(y) − f(y′)| ≤ 2Λ +���y − y′��� for all y, y′ ∈ Y +� +. +19 + +Then, the function class F CΘ(ϑo) on X is universal Donsker. Moreover, for i.i.d. random variables +{Xi}n +i=1 ∼ µ⊗n consider a measurable estimator ϑn and suppose for n → ∞ joint weak convergence, +√n +� µn − µ +ϑn − ϑo +� +⇝ +�Gµ +Gϑ +� +in ℓ∞(F CΘ(ϑo)) × Rk. +(4.6) +Then, for cn ≔ CΘ(ϑn) and c ≔ CΘ(ϑ) and by denoting the limit from (4.5) as Gc, it follows that +√n (OT (µn, ν0, cn) − OT (µ, ν0, c)) ⇝ +inf +π∈Π⋆c (µ,ν0) π(Gc) + +sup +f∈Sc(µ,ν0) +Gµ( f c), +where Sc(µ, ν0) represents the set of optimizers for sup f∈F µ( f c) + ν0( f cc). +Remark 4.4. A few comments on the distributional limits are in order. +(i) Given that the function class F CΘ(ϑo) is universal Donsker and thus µ-Donsker, and assuming +that √n(ϑn − ϑo) converges in distribution, the requirement of joint convergence as required +in (4.6) is very mild. Indeed, if √n(ϑn − ϑo) can be expressed asymptotically in terms of +a suitable linear functional of an empirical process, i.e., if it admits an asymptotic influence +function ψ ∈ L2(µ) (cf. van der Vaart 1998, p. 58), joint convergence follows since the union +F cc ∪ {ψ} is µ-Donsker. +(ii) We like to point out that Proposition 4.3 also remains valid if µ ∈ M. However, under this as- +sumption it follows that (g−1 +ϑo )#µ = ν0 which implies that the corresponding OT plan between +µ and ν0 is given by π = (Id, g−1 +ϑo (·))#µ. Hence, by (4.5) the process Gc vanishes along the sup- +port of π and the first term in the limit degenerates. Further, if the support of ν0 is connected +since then Kantorovich potentials are unique up to a constant shift (Staudt, Hundrieser, and +Munk, 2022, Corollary 2) and a.s. constant (Hundrieser et al., 2022, Corollary 4.6(i)). Con- +sequently, for this setting the corresponding limit distribution is degenerate. In contrast, if ν0 +has disconnected support, non-constant Kantorovich potentials exist (Staudt, Hundrieser, and +Munk, 2022, Lemma 11) which results in a non-degenerate limit. +(iii) The elements presented for the one-sample case can also be generalized to the case where +both empirical measures undergo a transformation, either separately or jointly. One might +think of choosing the Mahalanobis distance (x − y)⊤Σ−1(x − y) as a cost function where Σ−1 +has to be estimated and could, e.g., be a diagonal matrix. As the OT value is not invariant with +respect to affine transformations, rescaling the variables would ensure that no component has +an overwhelming impact on the cost function compared to the other components. +4.2 +Optimal Transport with Embedded Invariances +In a similar spirit to the previous section, another strand of the literature (Alvarez-Melis, Jegelka, +and Jaakkola, 2019; Grave, Joulin, and Berthet, 2019) aims at making OT invariant to a class of +transformation T , with τ: Rd → Rd continuously differentiable for each τ ∈ T , by considering +inf +τ∈T OT +� +µ, ν, ∥· − τ(·)∥2� += inf +τ∈T +inf +π∈Π(µ,ν) +� +X×Y +∥x − τ(y)∥2dπ(x, y). +(4.7) +This distance is useful in many contexts, among which the word embedding problem or protein +alignment. If the class of transformations considered is the set of rotations, analyses relying on +that distance is coined Wasserstein–Procrustes Analysis (Grave, Joulin, and Berthet, 2019; Jin, Liu, +and Xia, 2021). Theorem 2.6 provides the required tools for statistical inference for the empirical +version of the optimization problem in (4.7). +20 + +Proposition 4.5. Consider a set of transformations (T , dT ) that is a compact metric space with +log N(ε, T , dT ) ≲ ε−α for α < 2. Let X, Y ⊆ Rd be compact subsets and assume that the functional +c : (T , dT ) → C(X × Y), τ �→ cτ, with cτ(x, y) = ∥x − τ(y)∥2, is L-Lipschitz for some L ≥ 0. Further, +assume for X and {ct}t∈T any of the settings from Proposition 3.9 and take µ ∈ P(X), ν ∈ P(Y) such +that the support of µ or ν is the closure of a connected open set in Rd. Then, for {Xi}n +i=1 ∼ µ⊗n and +{Yi}m +i=1 ∼ ν⊗m, respectively, with n, m → ∞ such that m/(n + m) → λ ∈ (0, 1), it holds that +� +nm +n + m +� +inf +τ∈T OT(µn, νm, cτ) − inf +τ∈T OT(µ, ν, cτ) +� +⇝ +inf +τ∈S−(T ,µ,ν) +√ +λ Gµ( f cτcτ +τ +) + +√ +1 − λ Gν( f cτ +τ ), +where fτ ∈ Scτ(µ, ν) denotes a Kantorovich potential between µ and ν and cost cτ for τ ∈ S−(T , µ, ν). +As previously noted, one can relax the requirement that T is compact to the assumption that the +sequence of estimated optimal transformation τn is contained within a compact set with probability +tending to one (Proposition 2.8). In the setting of T consisting of diffeomorphisms we have by +Lemma 1 of Nies, Staudt, and Munk, 2021 +inf +τ∈T OT +� +µ, ν, ∥· − τ(·)∥2� += inf +τ∈T OT +� +µ, (τ−1)#ν, ∥· − ·∥2� += inf +τ∈T W2 +2 +� +µ, (τ−1)#ν +� +, +for which convergence of empirical minimizers τn can be verified for various settings using results +by Bernton et al. (2019). +Remark 4.6 (Wasserstein–Procrustes). The above proposition can be applied under mild regularity +assumptions on the measures to the special orthogonal group T ≔ SO(d) for d ≤ 3. Indeed, upon +choosing X, Y as compact, convex sets of Rd setting (iii) of Proposition 3.9 is fulfilled, asserting +(Don). Moreover, if the support of µ or ν is the closure of a connected open set in Rd, then (KP) +holds and the distributional limits of Theorem 2.6 follow. +4.3 +Sketched Wasserstein Distance for Mixture Distributions +Recently, Delon and Desolneux (2020) and Bing, Bunea, and Niles-Weed (2022) investigated a +distance between (Gaussian) mixtures distributions. These distributions are ubiquitous in statistics +and machine learning, see McLachlan, Lee, and Rathnayake (2019) and the references therein. +One way of understanding that distance is to start from the Wasserstein distance between discrete +measures but instead of using a cost function between points, one replaces the points by distributions +and one must thus choose a cost between distributions. Before formally defining that concept, recall +that, for a set of distributions A := (A1, . . . , AK) of finite cardinality K, a mixture r is a convex +combination of components from A given by a vector α ∈ ∆K, i.e., r = �K +i=1 αiAi, where ∆K is +the probability simplex in RK. Given a distance d: A × A → R+ between mixture components +of A, the aforementioned authors define the Sketched Wasserstein distance between two mixture +distributions with weights α and β as +W(α, β, d) ≔ +inf +π∈Π(α,β) +K +� +k,ℓ=1 +πk,ℓd(Ak, Aℓ), +where the infimum is taken over elements of the set of couplings +Π(α, β) = +� +π ∈ ∆K×K +������ +�K +ℓ=1 πk,ℓ = αk, +for all k ∈ {1, . . . , K} +�K +k=1 πk,ℓ = βℓ, +for all ℓ ∈ {1, . . . , K} +� +. +Understanding the fluctuations of an estimator for this distance can be achieved using the theory +developed in the present paper. This is formalized in the following proposition. +21 + +Proposition 4.7. Let (αn, βn, dn) ∈ ∆K ×∆K ×RK2 ++ be measurable estimators for α, β, d, respectively. +Further, for a positive sequence (an)n∈N with limn→∞ an = ∞, assume for n → ∞ that +an + +αn − α +βn − β +dn − d + = an + +(αn,k − αk)K +k=1 +(βn,k − βk)K +k=1 +(dn(Ak, Aℓ) − d(Ak, Aℓ))K +l,k=1 + ⇝ + +Gα +Gβ +Gd + +in R2K+K2, +(4.8) +where (Gα, Gβ, Gd) represents a tight (possibly non-Gaussian) random variable on R2K+K2. Then, +an +� +W(αn, βn, dn) − W(α, β, d) +� +⇝ +inf +π∈Π⋆ +d (α, β)⟨π, Gd⟩ + +sup +f∈Sd(α,β) +⟨ f dd, Gα⟩ + ⟨ f d, Gβ⟩. +The proof follows along the same approach as for showing Theorem 2.2 and is therefore omit- +ted, see Remark 2.4 (iv). In this context, the requirement of weak convergence for the measure +estimators (αn, βn) ⇝ (α, β) in probability follows from our assumption in (4.8) since the popula- +tion measures and its estimators are supported on finitely many points. +In Bing, Bunea, and Niles-Weed (2022), they obtain distributional limits in the case where the +asymptotic fluctuation of the cost is negligible in comparison to the estimated measures. Their +results are recovered by Proposition 4.7, which in addition covers the setting where the cost is esti- +mated on the same data and converges at the same rate. Finally, we stress that the case of Gaussian +mixtures, a particularly relevant one in applications, is also covered by our theory. Nonetheless, +the lack of distributional results for estimators of the mixture parameters in that case still hinders +further developments and would be of interest for further research. +4.4 +Sliced Optimal Transport +Our theory from Section 2.3 also enables the analysis of sliced OT quantities and complement +or extend available results from the literature (Goldfeld et al., 2022; Manole, Balakrishnan, and +Wasserman, 2022; Xi and Niles-Weed, 2022; Xu and Huang, 2022). In the following, we formalize +this statement. For two Borel probability measures µ, ν ∈ P(Rd) the average-sliced and max-sliced +Wasserstein distances of order 1 ≤ p < ∞ are defined, respectively, as +W p(µ, ν) ≔ +�� +Sd−1OT(pθ +#µ, pθ +#ν, | · − · |p) dσ(θ) +� 1 +p +and W p(µ, ν) ≔ max +θ∈Sd−1 +� +OT(pθ +#µ, pθ +#ν, | · − · |p) +� 1 +p, +where pθ : Rd → R is the projection map x �→ θT x and σ represents the uniform distribution on the +unit sphere Sd−1. Note by Lemma 1 in Nies, Staudt, and Munk (2021) for any θ ∈ Sd−1 that +OT(pθ +#µ, pθ +#ν, | · − · |p) = OT(µ, ν, |pθ(·) − pθ(·)|p), +which enables to view the sliced Wasserstein quantities in the framework of Section 2.3 and asserts +by Theorems 2.5–2.7 the following result. +Proposition 4.8. Let p ≥ 1, d ≥ 2, and define for θ ∈ Sd−1 the cost cθ : Rd × Rd → R, (x, y) �→ +|pθ(·) − pθ(·)|p. Further, take compactly supported probability measures µ, ν ∈ P(Rd) with empirical +measures µn, νm, respectively. For all assertions, we let n, m → ∞ with m/(n + m) → λ ∈ (0, 1). +(i) Assume that the set of Kantorovich potentials Scθ(µ, ν) is unique (up to a constant shift) for +any θ ∈ Sd−1. Then, it follows upon selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Sd−1 that +� +nm +n + m +� +OT(µn, ν, cθ)−OT(µ, ν, cθ) +� +θ∈Sd−1⇝ +� √ +λGµ( f cθcθ +θ +)+ +√ +1 − λGν( f cθ +θ ) +� +θ∈Sd−1 in C(Sd−1) +22 + +(ii) Assume the same as in (i). Then, it follows that +� nm +n + m +� +W p +p(µn, νm) − W p +p(µ, ν) +� +⇝ +� +Sd−1 +√ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ ) dθ. +(iii) Without imposing the assumption on uniqueness of Kantorovich potentials, it follows that +� +nm +n + m +� +W +p +p(µn, νm) − W +p +p(µ, ν) +� +⇝ +sup +θ∈S+(Sd−1, µ,ν) +fθ∈Scθ(µ,ν) +√ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ ). +Comparing Proposition 4.8 to the literature for p > 1, results in Goldfeld et al. (2022) and Xi +and Niles-Weed (2022) are recovered under slightly weaker assumptions. For the analysis of both +types of empirical sliced Wasserstein distances Goldfeld et al. (2022) require the underlying mea- +sures to have compact, convex support. Moreover, for the uniform central limit theorem by Xi and +Niles-Weed (2022) of the sliced OT process, they assume for each u ∈ Sd−1 that one of the pro- +jected measures has compact, connected support. These conditions are sufficient for the uniqueness +of Kantorovich potentials, but it can also be guaranteed for measures with disconnected support +(cf. Proposition 3.11 and more generally Staudt, Hundrieser, and Munk 2022). Proposition 4.8(ii) +also complements results by Manole, Balakrishnan, and Wasserman (2022) on the trimmed sliced +Wasserstein distance as we do not require the existence of a density but the underlying measures to +be compactly supported. +For the special case p = 1, unlike in our results, distributional limits by Goldfeld et al. (2022) +and Xu and Huang (2022) for the average- and max-sliced Wasserstein distance do not require +uniqueness of the Kantorovich potentials. Further, their theory remains valid for non-compactly +supported measures by imposing suitable moment-conditions. Crucial to their approach is the spe- +cial characterization of the 1-Wasserstein distance as an integral probability metric over Lipschitz +functions (Villani, 2008, Remark 6.5), a property which we do not exploit in our general theory. +Still, under uniqueness of Kantorovich potentials, which occurs, e.g., if one measure is discrete +while the other has connected support and is absolutely continuous (Staudt, Hundrieser, and Munk, +2022, Example 3), Proposition 4.8(i) asserts weak convergence for the sliced OT process in C(Sd−1). +4.5 +Stability analysis of Optimal Transport +In addition to statistical applications, our theory for the empirical OT value under weakly converg- +ing costs enables a deterministic stability analysis of the OT problem (1.1) under joint perturbations +of the costs and the measures, which may be of independent interest, e.g., from the viewpoint of +optimization. More precisely, we prove in the following Gateaux differentiability of the OT value +in (µ, ν, c) ∈ P(X) × P(Y) × C(X × Y) for all admissible directions. This extends well-known sta- +bility results for finite-dimensional linear programs (Gal and Greenberg, 1997, Theorem 3.1) which +covers the OT problem for probability measures supported on finitely many points. +Proposition 4.9. Let µ ∈ P(X), ν ∈ P(Y) and c ∈ C(X × Y) be fixed. Define for t > 0 sufficiently +small the quantities µt = µ + t∆µ and νt = ν + t∆ν, where ∆µ ∈ (P(X) − µ) and ∆ν ∈ (P(Y) − ν), +respectively. Further, let ct = c + t∆c for some ∆c ∈ C(X × X). Then, it follows that +lim +tց0 +1 +t (OT(µt, νt, ct) − OT(µ, ν, c)) = +inf +π∈Π⋆c (µ,ν) π(∆c) + +sup +f∈Sc(µ,ν) +∆µ( f cc) + ∆ν( f c). +Remark 4.10 (On Hadamard directional differentiability). Since the set of admissible directions +(P(X) − µ) × (P(Y) − ν) × C(X × Y) is not a normed vector space, we are in general unable to infer +23 + +Hadamard directional differentiability by additionally proving Lipschitzianity of the OT problem +with respect to the measures µ, ν and the cost function c. +Invoking the same proof strategy as in Proposition 4.9 would require us to show for any se- +quence (µn, νn, cn) = (µ + tn∆µ +n, ν + tn∆ν +n, c + tn∆c +n) ∈ P(X) × P(Y) × C(X × Y) with tn ց 0 and +(∆µ +n, ∆ν +n, ∆c +n) → (∆µ, ∆ν, ∆c) in ℓ∞(F cc) × ℓ∞(F c) × C(X × Y) that +sup +f∈F +���∆µ +n( f cncn − f cc) + ∆ν +n( f cn − f c) +��� → 0. +(4.9) +Showing this remains a challenge and would enable us to omit conditions (Sup) and (Sup)∗ in the +formulations of Theorems 2.2 and 2.10, respectively. Another challenge in such an attempt is that +any such sequence (µn, νn) does not necessarily converge weakly for n → ∞ to (µ, ν), which is +relevant for our proof, since the topology induced by ℓ∞(F cc) × ℓ∞(F c) may be too weak. +Though it is likely possible to show Hadamard directional differentiability of the OT problem +jointly in the measures and the cost by selecting a sufficiently strong norm that metrizes weak con- +vergence of measures, the functional delta method would inevitably require the empirical process to +weakly converge in this norm and impose additional conditions. A similar trade-off for the choice +of the norm is natural and known in the literature (cf. Dudley 1990, p.76; Jourdain and Tse, 2021). +5 +Regularity Elevation Functionals +In this section, we construct regularity elevation maps, i.e., continuous maps Ψ: C(X × Y) → +C(X × Y) such that for measurable estimators cn with √n(cn − c) ⇝ Gc for n → ∞, it follows that +(i) √n�cn − Ψ(cn)� P→ 0 +and +(ii) Ψ(cn) fulfills certain regularity properties. +(5.1) +Based on Lipschitzianity of the OT value with respect to the cost function (Lemma 6.2), condi- +tion (i) allows us to substitute a cost estimator with one that enjoys certain regularity properties, +effectively "elevating" its level of regularity. Such maps prove useful in our work at two particu- +lar instances. For one, it enables us to assume in the proof of Theorem 2.2 that cost estimators +are suitably bounded and exhibit the same modulus of continuity as the population cost function +(cf. Corollary 5.4). This represents an important step to rely on Lemma 2.1. Moreover, the notion +of regularity elevations also represents a useful tool to prove Corollary 3.8, for which we employ +Proposition 3.6 and set ˜cn ≔ Ψ(cn) for a suitable regularity elevation map. Insofar, these maps serve +as an effective tool for the theoretical analysis of distributional limits. +The subsequent result provides a first set of conditions to ensure that condition (i) of (5.1) is met. +Its proof as well as the proof of all subsequent results of this section are detailed in Appendix D. +Proposition 5.1. Let X, Y be compact Polish spaces and let cn ∈ C(X×Y) be a (Borel measurable) +random sequence such that an(cn − c) ⇝ L in C(X × Y) for some c ∈ C(X × Y) and (an)n∈N such +that an → ∞ for n → ∞. Let U ⊆ C(X × Y) be a linear subspace such that L is a.s. contained in +U. Then, if Ψ: C(X × Y) → C(X × Y) is continuous near c, Hadamard directionally differentiable +at f with a derivative such that DH +c Ψ|U = IdU and Ψ(c) = c, it follows for n → ∞ that +an +�cn − Ψ(cn)� P→0 +for n → ∞. +Notably, in case Ψ is Hadamard differentiable with DH +f Ψ = IdC(X×Y), one may select U = +C(X × Y) and the condition on the limit L becomes vacuous. +To conclude various types of useful regularity properties, as required in (ii) of (5.1), we thus +define in the following subsections various maps such that the conditions of Proposition 5.1 are met. +Additionally, we provide suitable metric entropy bounds for F Ψ(˜c)Ψ(˜c) independent of ˜c ∈ C(X × Y). +24 + +5.1 +Regularity Elevation to Deterministic Boundedness +Consider compact Polish spaces X, Y and let c ∈ C(X × Y) be a continuous cost function such that +∥c∥∞ ≤ 1. We define the regularity elevation functional for boundedness as +Ψbdd : C(X × Y) → C(X × Y), +˜c �→ +� +(x, y) �→ max(min(˜c(x, y), 2), −2) +� +. +Proposition 5.2. For the above setting, Ψ = Ψbdd fulfills Ψ(c) = c, is continuous, and it is +Hadamard differentiable at c with DH +|cΨ = IdC(X×Y). In particular, if X is a finite space, we ob- +tain for any uniformly bounded function class G on Y that +sup +˜c∈C(X×Y) +log N(ε, GΨ(˜c), ∥·∥∞) ≲ | log(ε)|. +Hence, for our analysis of the empirical OT value under estimated cost functions we can assume +without loss of generality that cost estimators are deterministically bounded by a constant that +depends on the population cost. In the following we prove a similar insight for the modulus of +continuity for cost estimators on compact (pseudo-)metric spaces. +5.2 +Regularity Elevation to Concave Modulus of Continuity and Lipschitzianity +Consider compact Polish spaces X, Y and let ˜dX be a continuous (pseudo-)metric on X. Denote by +˜X the space X equipped with the topology induced by ˜dX which is also compact (Lemma F.2) but +potentially does not satisfy the Hausdorff property. Let c ∈ C( ˜X × Y) be a cost function such that +∥c∥∞ ≤ 1 and consider a concave modulus w: R+ → R+ with w(δ) > 0 for δ > 0 such that +|c(x, y) − c(x′, y)| ≤ w( ˜dX(x, x′)) +for any x, x′ ∈ ˜X, y ∈ Y. +(5.2) +If c(·, y) is 1-Lipschitz under ˜dX, then select w(t) ≔ t and if c(·, y) is (γ, 1)-Hölder for γ ∈ (0, 1] +(recall footnote 4), choose w(t) ≔ tγ. The regularity elevation functional for w ◦ dX is then given by +Ψw◦ ˜dX +mod : C(X × Y) → C( ˜X × Y), +˜c �→ +� +(x, y) �→ inf +x′∈X ˜c(x′, y) + 2w( ˜dX(x, x′)) +� +Proposition 5.3. For the above setting, Ψ = Ψw◦ ˜dX +mod ◦ Ψbdd fulfills Ψ(c) = c, it is continuous near c, +and it is Hadamard directionally differentiable at c with DH +|c Ψ|C( ˜X×Y) = IdC( ˜X×Y). Further, for any +uniformly bounded function class G on Y it holds that +sup +˜c∈C(X×Y) +log N(ε, GΨ(˜c), ∥·∥∞) ≲ N(ε/8, X, w ◦ ˜dX)| log(ε)|. +An appealing consequence of the above considerations is that they allow us to construct a reg- +ularity elevated estimator ˜cn,m from cn,m such that H˜cn,m ⊆ F ˜cn,m ˜cn,m, for F = F (2 ∥c∥∞ + 1, 2w) +defined in (2.3), holds deterministically. +Corollary 5.4. Let c ∈ C(X × Y), set B ≔ ∥c∥∞ + 1/2 and let w: R+ → R+ be a concave modulus +with w(δ) > 0 for δ > 0 such that (5.2) holds for a metric dX on X. Assume for a random sequence +cn ∈ C(X × Y) that an(cn − c) ⇝ Gc in C(X × Y) with an → ∞. Then, the random sequence +cn ≔ B · Ψw◦dX/B +mod +◦ Ψbdd(cn/B) ∈ C(X × Y) +satisfies an ∥cn − cn∥∞ +P→ 0 for n → ∞ and deterministically fulfills ∥cn∥∞ ≤ 2B = 2 ∥c∥∞ + 1, +relation (5.2), and the inclusion Hcn ⊆ F cncn(2 ∥c∥∞ + 1, 2w). +25 + +5.3 +Regularity Elevation to Hölder Functions of Order γ ∈ (1, 2] +Since we are able to leverage for convergence rates of the empirical OT value (recall Proposi- +tion 3.1(i)) the regularity of the underlying cost function up to Hölder degree γ ≤ 2, we provide in +this subsection a corresponding regularity elevation map. As the setting for γ ≤ 1 can be treated +using the theory from previous subsection, we only focus on the regime of γ ∈ (1, 2]. +Consider a convex, compact set X ⊆ Rd with non-empty interior. Let c ∈ C(X × Y) be a cost +function such that ∥c∥∞ ≤ 1 and assume c is continuously differentiable in x, i.e., suppose that +∇xc: int(X)×Y → Rd can be continuously extended to X×Y. Further, suppose that c(·, y) is (γ, 1)- +Hölder for each y ∈ Y for γ ∈ (1, 2]. We define the regularity elevation map for Hölder functions +of order γ ∈ (1, 2] by +Ψc,γ +Hol : C(X × Y) → C(X × Y), ˜c �→ +� +(x, y) �→ inf +x′∈X ˜c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 +√ +d +���x − x′���γ� +Notably, it is crucial that the scalar product term involves the partial derivative of the respective +(population) cost function c. Moreover, we like to point out that the image under Ψc,γ +Hol does not +necessarily lead to (γ, 1)-Hölder functions but nonetheless ensures suitable metric entropy bounds. +Proposition 5.5. For the above setting with X ⊆ Rd convex and compact, Ψ = Ψc,γ +Hol ◦ Ψbdd fulfills +Ψ(c) = c, it is continuous near c, and it is Hadamard differentiable at c with DH +|c Ψ = IdC(X×Y). +Further, for any uniformly bounded function class G on Y we obtain that +sup +˜c∈C(X×Y) +log N(ε, GΨ(˜c), ∥·∥∞) ≲ ε−d/γ. +5.4 +Combination of Regularity Elevations +Finally, we also outline a constructive way to combine regularity elevation maps defined on different +spaces. This is important since it enables to leverage regularity properties of the population cost +function for different regions of the domain. +Hence, let X, Y be compact Polish spaces and assume existence of a collection of homeomor- +phisms ζi : Ui → ζi(Ui) for 1 ≤ i ≤ I such that X = �I +i=1 ζi(Ui). Further, assume there exists a parti- +tion on unity {ηi}I +i=1 on X with supp(ηi) ⊆ ζi(Ui). Consider a continuous cost function c: X×Y → R +and let ci : Ui ×Y → R, (u, y) �→ c(ζi(u), y). Assume there exist maps Ψi : C(Ui ×Y) → C(Ui ×Y) +such that Ψi(ci) = ci and where Ψi is continuous near ci and Hadamard differentiable at ci with +derivative DH +|ciΨi = Id. Using these maps we define the combination of regularity elevations as +Ψcom : C(X × Y) → C(X × Y), +˜c �→ +(x, y) �→ +I� +i=1 +ηi(x)Ψi +� +˜c(ζi(·), ·) +� +(ζ−1 +i (x), y) + . +Indeed, by continuity of the partition of one ηi as well as the functionals Ψi and ζi, ζ−1 +i +for each +i ∈ {1, . . . , I} it follows that the range of this functional is indeed contained in C(X × Y). +Proposition 5.6. For the above setting, Ψ = Ψcom fulfills Ψ(c) = c, it is continuous near c, and it is +Hadamard differentiable at c with DH +|cΨ = IdC(X×Y). Further, for any uniformly bounded function +class G on Y we obtain +sup +˜c∈C(X×Y) +log N(ε, GΨ(˜c), ∥·∥∞) ≤ +I� +i=1 +sup +˜c∈C(X×Y) +log N(ε, GΨi(˜c(ζi(·),·)), ∥·∥∞). +26 + +6 +Proofs of Main Results +In this section, we provide the full proofs of Lemma 2.1 for the dual representation of the OT value, +Theorem 2.2 and Proposition 2.10 for the distributional limit of the empirical OT value under weakly +converging costs, as well as Theorems 2.5–2.7 and Proposition 2.8 for empirical OT with extremal- +type costs. The proofs for all auxiliary results of this subsection are deferred to Appendix E. +6.1 +Proof of Lemma 2.1: Dual Representation of Optimal Transport Value +The subsequent auxiliary lemma establishes an important property of cost-transformations which is +essential throughout this section. +Lemma 6.1 (Lipschitz property of cost-transformation). For arbitrary functions f, ˜f : X → R and +cost functions c, ˜c: X × Y → R, it follows that +���f c − ˜f ˜c���∞ ≤ +��� f − ˜f +���∞ + ∥c − ˜c∥∞. In particular, +upon selecting the constant functions ˜f, ˜c ≡ 0, it follows that ∥f c∥∞ ≤ ∥ f∥∞ + ∥c∥∞. +Proof of Lemma 2.1. For any h ∈ Hc there exists g: Y → [− ∥c∥∞ , ∥c∥∞] with h = gc, and hence +− ∥c∥∞ − sup +y∈Y +g(y) ≤ h(x) = inf +y∈Y c(x, y) − g(y) ≤ ∥c∥∞ − sup +y∈Y +g(y). +In consequence, we find that ∥h∥∞ ≤ 2 ∥c∥∞ ≤ 2B. Further, for arbitrary x, x′ ∈ X and ε > 0, +consider y′ ∈ Y such that h(x′) ≥ c(x′, y′) − g(y′) − ε. Then, it follows that +h(x) − h(x′) = +� +inf +y∈Y c(x, y) − g(y) +� +− +� +inf +y∈Y c(x′, y) − g(y) +� +≤ c(x, y′) − g(y′) − c(x′, y′) + g(y′) + ε +≤ w(dX(x, x′)) + ε. +Since ε > 0 can be chosen arbitrarily small, we obtain that |h(x) − h(x′)| ≤ w(dX(x, x′)). This +yields Hc ⊆ F and thus Hc +c ⊆ F c. Further, by Santambrogio (2015, Proposition 1.34) we infer +Hc = Hcc +c ⊆ F cc. To show the remaining inclusions note for f ∈ F that +− ∥c∥∞ − sup +x∈X +f(x) ≤ f c ≤ ∥c∥∞ − sup +x∈X +f(x). +Hence, the function g ≔ f c + supx∈X f(x) fulfills ∥g∥∞ ≤ ∥c∥∞, and since ∥ f∥∞ ≤ 2B, we find that +f cc(x) = ( f c)c = (g)c + sup +x∈X +f(x) ∈ Hc + [−2B, 2B], +which yields F cc ⊆ Hc+[−2B, 2B] as well as F c = F ccc ⊆ Hc +c +[−2B, 2B]. To show representation +(2.4), we combine the inclusions Hc ⊆ F ⊆ C(X) with the alternative dual representations (1.2) +and (2.2). For the final claim, take a maximizing sequence { fn}n∈N for (2.4) which admits by com- +pactness of F a converging subsequence { fnk}k∈N with uniform limit f ∈ F . Then by Lemma 6.1 +it follows that { f c +nk}k∈N and { f cc +nk }k∈N also uniformly converge to f c and f cc, respectively. We thus +obtain that µ( f cc) + ν( f c) = limk→∞ µ( f cc +nk ) + ν( f c +nk) = OT(µ, ν, c) which shows that f ∈ F is a +maximizing element hence the set of optimizers Sc(µ, ν) for (2.4) is non-empty. +□ +6.2 +Proofs for Distributional Limits under Weakly Converging Costs +6.2.1 +Proof of Theorem 2.2 +For the proof of Theorem 2.2 the following auxiliary results are crucial. We start with lower and +upper bound on the difference between OT values for varying costs and probability measures which +are a consequence of the OT problem having a representation in terms of an infimum over feasible +couplings as well as a supremum over feasible potentials. +27 + +Lemma 6.2 (Lower and upper bounds). Define for B > 0 and a concave modulus of continuity +w: R+ → R+ the collection +C(B, w) ≔ �¯c ∈ C(X × Y) +��� ∥¯c∥∞ ≤ B, |¯c(x, y) − ¯c(x′, y)| ≤ w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y� . +Then, for costs c, ˜c ∈ C(B, w) and probability measures µ, ˜µ ∈ P(X), ν, ˜ν ∈ P(Y) it holds that +inf +π∈Π⋆ +˜c (˜µ,˜ν) π(˜c − c) + +sup +f∈Sc(µ,ν) +(˜µ − µ) f cc + (˜ν − ν) f c +≤ OT(˜µ, ˜ν, ˜c) − OT(µ, ν, c) +≤ min +� +inf +π∈Π⋆c (˜µ,˜ν) π(˜c − c) + +sup +f∈Sc(˜µ,˜ν) +(˜µ − µ) f cc + (˜ν − ν) f c, +inf +π∈Π⋆c (µ,ν) π(˜c − c) + +sup +f∈S˜c(˜µ,˜ν) +(˜µ − µ) f cc + (˜ν − ν) f c + sup +f∈F +(˜µ − µ)( f ˜c˜c − f cc) + (˜ν − ν)( f ˜c − f c). +� +In particular, for fixed measures or fixed costs it follows that +|OT(µ, ν, ˜c) − OT(µ, ν, c)| ≤ ∥˜c − c∥∞, |OT(˜µ, ˜ν, c) − OT(µ, ν, c)| ≤ sup +f∈F cc +���(˜µ − µ) f +��� +sup +f∈F c +���(˜ν − ν) f +���. +To employ the lower and upper bounds of Lemma 6.2 for the proof of Theorem 2.2 we addition- +ally require a number of continuity and measurability properties which are captured in the following +lemma. Notably, we equip P(X) × P(Y) with the bounded Lipschitz norm, which turns it into a +Polish metric space and metrizes weak convergence of measures. +Lemma 6.3 (Continuity and Measurability). Let µ ∈ P(X), ν ∈ P(Y), and c ∈ C(X × Y). Take a +concave modulus of continuity w: R+ → R+ for c and set C ≔ C(2 ∥c∥∞ + 1, 2w) (for the definition +of C(2 ∥c∥∞ + 1, 2w) see Lemma 6.2). Further, recall the function class F = F (2 ∥c∥∞ + 1, 2w) +introduced in (2.3) and define the functions +T1 : P(X) × P(Y) × C → R, +(µ′, ν′, c′) �→ OT(µ′, ν′, c′), +T2 : P(X) × P(Y) × C × C(X × Y) → R, +(µ′, ν′, c′, hc) �→ +inf +π∈Π⋆ +c′(µ′,ν′) π(hc), +T3 : P(X) × P(Y) × C × Cu(F )2 → R, +(µ′, ν′, c′, hµ, hν) �→ +sup +f∈Sc′(µ′,ν′) +hµ( f) + hν( f), +T4 : Cu(F )4 → R, +(hµ, ˜hµ, hν, ˜hν) �→ sup +f∈F +hµ( f) − ˜hµ( f) + hν( f) − ˜hν( f). +Then, T1 and T4 are continuous, T2 is lower semi-continuous, and T3 is upper semi-continuous. If +Π⋆ +c′(µ′, ν′) is unique, T2 is continuous at (µ′, ν′, c′, hc). Moreover, for fixed (µ′, ν′, c′) the map T2 is +continuous in hc while T3 is continuous in (hµ, hν). In particular, each function Ti for 1 ≤ i ≤ 4 is +Borel measurable. +The previous two assertions fully deal with deterministic statements on the OT functional and +related terms that arise from corresponding bounds. The following two results provide the relevant +tools to control the stochastic aspects. More precisely, for our proof of Theorem 2.2 we consider a +Skorokhod representation of the random sequence detailed in (JW) which additionally fulfills the +property that µn and νn weakly converge to µ and ν, respectively. For this purpose, we state the +following measurability assertions and joint weak convergence statements. +Lemma 6.4 (Measurability of empirical process). For a Polish space X consider a totally bounded +function class G ⊆ C(X) under uniform norm. Then, the following assertions hold. +28 + +(i) Any probability measure µ ∈ P(X) defines via evaluation a uniformly continuous functional +on G, i.e., µ ∈ Cu(G). +(ii) A map ω �→ µ(ω) ∈ P(X) ⊆ Cu(G) is Borel measurable if and only if for any g ∈ G the +evaluation map ω �→ µ(ω)(g) is Borel measurable. +(iii) The empirical process √n(µn −µ) and the bootstrap empirical process +√ +k(µb +n,k −µn) are both +Borel measurable random variables in Cu(G). +Lemma 6.5 (Joint weak convergence). For the setting of Theorem 2.2, assume (JW). Then, for +n, m → ∞, weak convergence in the Polish space Cu(F )2 ×C(X × Y) × P(X) × P(Y) to a tight limit +occurs +�� +Gµ +n( f cc), Gν +m( f c) +� +f∈F , Gc +n,m, µn, νm +� +⇝ +�� +Gµ( f cc), Gν( f c) +� +f∈F , Gc, µ, ν +� +. +(6.1) +If (Sup) of Theorem 2.2 is also valid, then, for n, m → ∞, it follows in the Polish space Cu(F )4 × +C(X × Y) × P(X) × P(Y) that +�� +Gµ +n( f cc), Gµ +n( f cn,mcn,m), Gν +m( f c), Gν +m( f cn,m) +� +f∈F , Gc +n,m, µn, νn, +� +⇝ +�� +Gµ( f cc), Gµ( f cc), Gν( f c), Gν( f c) +� +f∈F , Gc, µ, ν, +� +, +(6.2) +Each sequence element for (6.1) and (6.2) as well as the weak limit are Borel measurable. +Remark 6.6 (Skorokhod representation). When dealing with weak convergence of empirical pro- +cesses in non-separable spaces, special care is required due to potential measurability issues. How- +ever, since the different maps of interest are defined between Polish spaces and measurable, we +circumvent such issues. In particular, since the random variables from Lemma 6.5 converge weakly +to a tight limit with separable support, the conditions of Billingsley (1999, Theorem 6.7) are met +and a (measurable) Skorokhod representation exists. +With these tools at our disposal, we now proceed with the proof of Theorem 2.2. +Proof of Theorem 2.2. Invoking Corollary 5.4, as √nm/(n + m)(cn,m − c) =: Gc +n,m ⇝ Gc in the +space C(X×Y), there exists cn,m such that the inclusion Hcn,m ⊆ F cn,mcn,m (recall the function classes +from Section 2.1) holds deterministically for F = F (2 ∥c∥∞ + 1, 2w) and √n(cn,m − cn,m) +P→ 0. The +latter implies by Lemma 6.2 that +� +nm +n + m +�OT(µn, νm, cn,m) − OT(µn, νm, cn,m)� P→ 0. +Hence, to prove the assertion it suffices by Slutzky’s lemma to show that +� +nm +n + m +� +OT(µn, νm, cn,m) − OT(µ, ν, c) +� +⇝ +inf +π∈Π⋆c (µ,ν)π(Gc) + +sup +f∈Sc(µ,ν) +√ +λGµ( f cc) + +√ +1 − λGν( f c). +(6.3) +Without loss of generality, we may therefore assume cn,m = cn,m. Further, set λn ≔ m/(n+m). Then, +29 + +by Lemma 6.2, the subsequent lower and upper bounds follow, +inf +π∈Π⋆cn,m(µn,νm) π(Gc +n,m) + +sup +f∈Sc(µ,ν) +� +λn Gµ +n( f cc) + +� +1 − λn Gν +m( f c) +≤ +� nm +n + m(OT(µn, νm, cn,m) − OT(µ, ν, c)) +(6.4) +≤ min +� +inf +π∈Π⋆c (µn,νm) π(Gc +n,m) + +sup +f∈Sc(µn,νm) +� +λn Gµ +n( f cc) + +� +1 − λn Gν +m( f c), +inf +π∈Π⋆c (µ,ν) π(Gc +n,m) + +sup +f∈Scn,m(µn,νm) +� +λn Gµ +n( f cc) + +� +1 − λn Gν +m( f c) ++ sup +f∈F +� +λn +� +Gµ +n( f cn,mcn,m) − Gµ +n( f cc) +� ++ +� +1 − λn +�Gν +m( f cn,m) − Gν +m( f c)� � +. +For each setting (OP) and (Sup) we show that the upper and lower bounds asymptotically converge +in distribution to the limit in (6.3), which then asserts that the empirical OT value also tends to this +limit. To this end, we take for the random variables of Lemma 6.5 a Skorokhod representation on a +probability space (Ω, A, P) (Billingsley, 1999, p. 70) which is well-defined by Remark 6.6. More +precisely, under (OP) we take the Skorokhod representation such that +�� ˜Gµ +n( f cc), ˜Gν +m( f c) +� +f∈F , ˜Gc +n,m, ˜µn, ˜νm +� a.s. +−−→ +�� ˜Gµ( f cc), ˜Gν( f c) +� +f∈F , ˜Gc, µ, ν +� +(6.5) +in Cu(F )2 × C(X × Y) × P(X) × P(Y), whereas under (Sup) we choose it such that +�� ˜Gµ +n( f cc), ˜Gµ +n( f ˜cn,m ˜cn,m), ˜Gν +m( f c), ˜Gν +m( f ˜cn) +� +f∈F , ˜Gc +n,m, ˜µn, ˜νm +� +a.s. +−−→ +�� ˜Gµ( f cc), ˜Gµ( f cc), ˜Gν( f c), ˜Gν( f c) +� +f∈F , ˜Gc, µ, ν +� +(6.6) +in Cu(F )4 × C(X × Y) × P(X) × P(Y). We also set ˜cn,m ≔ c + ˜Gc +n,m/ √nm/(n + m) which a.s. +converges to c. +For the subsequent argument recall the functions T1, T2, T3, T4 from Lemma 6.3 and their (semi- +) continuity properties. Furthermore, note that an application of Lemma 6.1 in combination with +the arguments of the proof of Lemma 6.4 (i) yields that the maps F → R, +f �→ Gµ +n( f cc), +f �→ Gµ +n( f cn,mcn,m), +f �→ Gν +m( f c), +f �→ Gν +m( f cn,m) +are uniformly continuous, i.e., elements in Cu(F ). For both settings (OP) and (Sup) it follows by +measurability of the maps T1, T2, T3 for each n, m ∈ N that +� nm +n + m(OT(µn, νm, cn,m) − OT(µ, ν, c)) d= +� nm +n + m(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) +inf +π∈Π⋆cn,m(µn,νm) π(Gc +n,m) + +sup +f∈Sc(µ,ν) +� +λn Gµ +n( f cc) + +� +1 − λn Gν +m( f c) +d= +inf +π∈Π⋆ +˜cn,m(˜µn,˜νm) π( ˜Gc +n,m) + +sup +f∈Sc(µ,ν) +� +λn ˜Gµ +n( f cc) + +� +1 − λn ˜Gν +m( f c). +Under (OP) we also notice that +inf +π∈Π⋆c (µn,νm) π(Gc +n,m) + +sup +f∈Sc(µn,νm) +� +λn Gµ +n( f cc) + +� +1 − λn Gν +m( f c) +d= +inf +π∈Π⋆c (˜µn,˜νm) π( ˜Gc +n,m) + +sup +f∈Sc(˜µn,˜νm) +� +λn ˜Gµ +n( f cc) + +� +1 − λn ˜Gν +m( f c), +30 + +whereas under (Sup) we additionally employ measurability of T4 to infer for each n ∈ N that +inf +π∈Π⋆c (µ,ν) π(Gc +n,m) + +sup +f∈Scn,m(µn,νm) +� +λn Gµ +n( f cc) + +� +1 − λn Gν +m( f c) ++ sup +f∈F +� +λn +� +Gµ +n( f cc) − Gµ +n( f cn,mcn,m) +� ++ +� +1 − λn +�Gν +m( f c) − Gν +m( f cn,m)� +d= +inf +π∈Π⋆c (µ,ν) π( ˜Gc +n,m) + +sup +f∈S˜cn,m(˜µn,˜νm) +� +λn ˜Gµ +n( f cc) + +� +1 − λn ˜Gν +m( f c) ++ sup +f∈F +� +λn +� ˜Gµ +n( f cc) − ˜Gµ +n( f ˜cn,m ˜cn,m) +� ++ +� +1 − λn +� ˜Gν +m( f c) − ˜Gν +m( f ˜cn,m) +� +. +Hence, it suffices to work with the Skorokhod representation to obtain the weak limit for the empir- +ical OT value. Invoking Lemma 6.2, identical lower and upper bounds on the quantity of interest, +√nm/(n + m)(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)), as for (6.4) can be concluded. +To obtain a suitable bound on the limit inferior of √nm/(n + m)(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) +take for both (OP) and (Sup) a measurable set A ∈ A of full measure such that the convergence +from (6.5) and (6.6) is fulfilled thereon, respectively. Then, for each ω ∈ A it follows by lower +semi-continuity of T2 jointly with continuity of T3 under fixed (µ, ν, c) that +lim inf +n,m→∞ +inf +π∈Π⋆ +˜cn,m(˜µn,˜νm) π( ˜Gc +n,m) + +sup +f∈Sc(µ,ν) +� +λn ˜Gµ +n( f cc) + +� +1 − λn ˜Gν +m( f c) +≥ +inf +π∈Π⋆c (µ,ν) π( ˜Gc) + +sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc) + +√ +1 − λ ˜Gν( f c). +Under (OP), we find for each ω ∈ A by continuity of T2 at (µ, ν, c, Gc +n,m) as a consequence of +(OP) and upper semi-continuity of T3 that +lim sup +n,m→∞ +inf +π∈Π⋆c (˜µn,˜νm) π( ˜Gc +n,m) + +sup +f∈Sc(˜µn,˜νm) +� +λn ˜Gµ +n( f cc) + +� +1 − λn ˜Gν +m( f c) +≤ +π⋆( ˜Gc) + +sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc) + +√ +1 − λ ˜Gν( f c) += +inf +π∈Π⋆c (µ,ν) π( ˜Gc) + +sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc) + +√ +1 − λ ˜Gν( f c). +Under (Sup), we note for each ω ∈ A by continuity of T2 in hc for fixed (µ, ν, c), upper semi- +continuity of T3 and continuity of T4 that +lim sup +n,m→∞ +inf +π∈Π⋆c (µ,ν)π( ˜Gc +n,m) + +sup +f∈S˜cn,m(˜µn,˜νm) +� +λn ˜Gµ +n( f cc) + +� +1 − λn ˜Gν +m( f c) ++ sup +f∈F +� +λn +� ˜Gµ +n( f cc) − ˜Gµ +n( f cn,mcn,m) +� ++ +� +1 − λn +� ˜Gν +m( f c) − ˜Gν +m( f cn,m) +� +≤ +inf +π∈Π⋆c (µ,ν) π( ˜Gc) + +sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc) + +√ +1 − λ ˜Gν( f c) ++ sup +f∈F +√ +λ +� ˜Gµ( f cc) − ˜Gµ( f cc) +� ++ +√ +1 − λ +� ˜Gν( f c) − ˜Gν( f c) +� += +inf +π∈Π⋆c (µ,ν) π( ˜Gc) + +sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc) + +√ +1 − λ ˜Gν( f c). +As the lower bound and the upper bounds for √nm/(n + m)(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) +asymptotically match for all ω ∈ A, it follows under both (OP) and (Sup) that +lim +n,m→∞ +� +nm +n + m(OT(˜µn, ˜νm, ˜cn,m)−OT(µ, ν, c)) = +inf +π∈Π⋆c (µ,ν) π( ˜Gc)+ sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc)+ +√ +1 − λ ˜Gν( f c). +31 + +As the set A has full measure we obtain that +� +nm +n + m(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) +a.s. +−−→ +inf +π∈Π⋆c (µ,ν) π( ˜Gc) + +sup +f∈Sc(µ,ν) +√ +λ ˜Gµ( f cc) + +√ +1 − λ ˜Gν( f c), +where the limit has by measurability of T2 and T3 the same Borel law as the limit in the assertion, +which finishes the proof. +□ +6.2.2 +Proof of Proposition 2.10 +Before turning to the proof of the bootstrap consistency, i.e., Proposition 2.10, we introduce an +important result on the convergence of the bootstrap empirical measure. +Lemma 6.7. For a Polish space X let µ ∈ P(X). Consider i.i.d. random variables {Xi}n +i=1 ∼ µ⊗n +to define the empirical measure µn = n−1 �n +i=1 δXi. Further, consider k(n) i.i.d. random variables +{Xb +k}n +i=1 ∼ µ⊗k(n) +n +to define the bootstrap empirical measure µb +n,k := +1 +k(n) +�k(n) +j=1 δXb +i . Then, provided +that k(n) → ∞ as n → ∞, it follows under n → ∞ that µb +n,k weakly converges to µ, in probability. +The above lemma is a corollary of Theorem 2 in (Beran, Le Cam, and Millar, 1987) and was +added to ease further referencing. We can now prove Proposition 2.10 on bootstrap consistency +under weakly converging costs. +Proof of Proposition 2.10. By Assumptions (JW) and (JW)∗, and by measurability of the empirical +and bootstrap empirical processes (Lemma 6.4) we infer using Lemma 2.2(c) ⇒ (a) in Bücher and +Kojadinovic (2019) for two bootstrap versions (µ(1) +n,k, ν(1) +n,k, c(1) +n,k), (µ(2) +n,k, ν(2) +n,k, c(2) +n,k) based on independent +bootstrap samples {X(1) +i }k +i=1, {X(2) +i }k +i=1 ∼ µ⊗k +n and {Y(1) +i }k +i=1, {Y(2) +i }k +i=1 ∼ ν⊗k +n for n, k → ∞, k = o(n) that + +√n + +µn − µ +νn − ν +cn − c + +√ +k + +µ(i) +n,k − µn +ν(i) +n,k − νn +c(i) +n,k − cn + +i=1,2 + +⇝ + + +Gµ +Gν +Gc + + +Gµ,(i) +Gν,(i) +Gc,(i) + +i=1,2 + +in +� +Cu(F cc) × Cu(F c) × C(X × Y) +�3. Since k = o(n) we also obtain by Slutzky’s lemma that + +√n + +µn − µ +νn − ν +cn − c + +√ +k + +µ(i) +n,k − µ +ν(i) +n,k − ν +c(i) +n,k − c + +i=1,2 + +≕ + + +Gµ +n +Gν +n +Gc +n + + +Gµ,(i) +n,k +Gν,(i) +n,k +Gc,(i) +n,k + +i=1,2 + +⇝ + + +Gµ +Gν +Gc + + +Gµ,(i) +Gν,(i) +Gc,(i) + +i=1,2 + +. +Herein, the triples (Gµ, Gν, Gc), (Gµ,(1), Gν,(1), Gc,(1)), and (Gµ,(2), Gν,(2), Gc,(2)) are independent and +have identical law. Notably, invoking Corollary 5.4 we may assume without loss of generality +that the empirical and bootstrap cost function cn and c(i) +n,k for i ∈ {1, 2} deterministically satisfy the +relation F¯c ⊆ F ¯c¯c, ¯c ∈ {cn, c(i) +n,k}. Moreover, by Varadarajan (1958) we know that µn ⇝ µ, νn ⇝ ν +a.s. for n → ∞, and by Lemma 6.7 we infer for i ∈ {1, 2} that µ(i) +n,k ⇝ µ, ν(i) +n,k ⇝ ν in probability for +32 + +n, k → ∞, k = o(n). Hence, Slutzky’s lemma asserts that + +� +Gµ +n, Gν +n, Gc +n, µn, νn +�T +� +Gµ,(i) +n,k , Gν,(i) +n,k , Gc,(i) +n,k , µ(i) +n,k, ν(i) +n,k +�T +i=1,2 + ⇝ + +� +Gµ, Gν, Gc, µ, ν +�T +� +Gµ,(i), Gν,(i), Gc,(i), µ, ν +�T +i=1,2 + +(6.7) +in �Cu(F cc) × Cu(F c) × C(X × Y) × P(X) × P(Y)�3. Moreover, using an analogous argument as for +the proof of Lemma 6.5 we conclude that + +�� +Gµ +n( f cc), Gν +n( f c) +� +f∈F , Gc +n, µn, νn +�T +�� +Gµ,(i) +n,k ( f cc), Gν,(i) +n,k ( f c) +� +f∈F , Gc,(i) +n,k , µ(i) +n,k, ν(i) +n,k +�T +i=1,2 + +⇝ + +�� +Gµ( f cc), Gν( f c) +� +f∈F , Gc, µ, ν +�T +�� +Gµ,(i)( f cc), Gν,(i)( f c) +� +f∈F , Gc,(i), µ, ν +�T +i=1,2 + +(6.8) +in the Polish space �Cu(F )2 × C(X × Y) × P(X) × P(Y)�3, and we use under Assumption (OP) a +Skorokhod representation for the process in (6.8). +Under Assumptions (Sup) and (Sup)∗, by measurability of cn and cn,k as maps to C(X × Y), +Lipschitzianity under c-transformation (Lemma 6.1) and Slutzky’s lemma we conclude weak con- +vergence of the random variables + +�� +Gµ +n( f cc), Gµ +n( f cncn), Gν +n( f c), Gν +n( f cn) +� +f∈F , Gc +n, µn, νn +�T +�� +Gµ,(i) +n,k ( f cc), Gµ,(i) +n,k ( f c(i) +n,k), Gν,(i) +n,k ( f c), Gν,(i) +n,k ( f c(i) +n,k) +� +f∈F , Gc,(i) +n,k , µ(i) +n,k, ν(i) +n,k +�T +i=1,2 + +⇝ + +�� +Gµ( f cc), Gµ( f cc), Gν( f c), Gν( f c) +� +f∈F , Gc, µ, ν +�T +�� +Gµ,(i)( f cc), Gµ,(i)( f cc), Gν,(i)( f c), Gν,(i)( f c) +� +f∈F , Gc,(i), µ, ν +�T +i=1,2 + +(6.9) +in the Polish space �Cu(F )4 × C(X × Y) × P(X) × P(Y)�3. For the random variables from (6.9) we +now take a Skorokhod representation. +To denote the random elements from the Skorokhod representation, we equip to the respective +random variable with a tilde, e.g., we write ˜µn for the representation of µn. Following the same +proof technique as Theorem 2.2 we thus conclude with Lemma 6.2 and Lemma 6.3 that + +√n +� +OT(˜µn, ˜νn, ˜cn) − OT(µ, ν, c) +� +√ +k +� +OT(˜µ(i) +n,k, ˜ν(i) +n,k, ˜c(i) +n,k) − OT(µ, ν, c) +� +i=1,2 + +a.s. +−−→ + +infπ∈Π⋆c (µ,ν) π( ˜Gc) + sup f∈Sc(µ,ν) ˜Gµ( f cc) + ˜Gν( f c) +� +infπ∈Π⋆c (µ,ν) π( ˜Gc,(i)) + sup f∈Sc(µ,ν) ˜Gµ,(i)( f cc) + ˜Gν,(i)( f c) +� +i=1,2 + . +Consequently, we infer for the original random variables and using that k = o(n) that + +√n +� +OT(µn, νn, cn) − OT(µ, ν, c) +� +√ +k +� +OT(µ(i) +n,k, ν(i) +n,k, c(i) +n,k) − OT(µn, νn, cn) +� +i=1,2 + +⇝ + +infπ∈Π⋆c (µ,ν) π(Gc) + sup f∈Sc(µ,ν) Gµ( f cc) + Gν( f c) +� +infπ∈Π⋆c (µ,ν) π(Gc,(i)) + sup f∈Sc(µ,ν) Gµ,(i)( f cc)+ Gν,(i)( f c) +� +i=1,2 + . +Since the three components in the limit have identical distributions and are independent, the asser- +tion follows at once from Bücher and Kojadinovic (2019, Lemma 2.2 (a) ⇒ (c)). +□ +33 + +6.3 +Proofs for Distributional Limits under Extremal-Type Costs +Before we proceed with the proofs for the results from Section 2.3 which rely on an application of +the functional delta method, we provide a simple result on the support of the limiting processes. Its +proof is deferred to Appendix E.6. +Lemma 6.8. For a Polish space X let µ ∈ P(X) and consider a bounded, measurable function class +˜F on X. Then, the following assertions hold. +(i) The contingent cone of P(X) at µ is given by TµP(X) = Cl{ µ′−µ +t |t > 0, µ′ ∈ P(X)} ⊆ ℓ∞( ˜F ). +(ii) For any ∆ ∈ TµP(X) and f, f ′ ∈ ˜F with f − f ′ ≡ κ for some κ ∈ R it holds that ∆( f) = ∆( f ′). +(iii) If ˜F is µ-Donsker, then the tight limit Gµ of the empirical process √n(µn − µ) in ℓ∞( ˜F ) is a.s. +contained in TµP(X). +6.3.1 +Proof of Theorem 2.5 +The result follows by an application of the functional delta method (Römisch, 2006). Without +loss of generality, we assume that X = supp(µ) and Y = supp(ν). This ensures that Kantorovich +potentials are by (KP) unique on the full domains X and Y. Assumption (Don) in conjunction +with independence of the underlying random variables from µ and ν ensure by van der Vaart and +Wellner (1996, Example 1.4.6) that the joint process √nm/n + m(µn − µ, νm − ν) weakly converge +in ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ). Further, by Lemma 6.8 the limit is a.s. contained in TµP(X) × +TνP(Y). It remains to show that the map +(OT(·, ·, cθ))θ∈Θ : P(X) × P(Y) ⊆ ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ) → C(Θ), +(µ, ν) �→ +θ �→ sup +f∈F +µ( f cθ,cθ) + ν( f cθ) + +is Hadamard directionally differentiable at (µ, ν) tangentially to P(X) × P(Y). In the language of +Theorem A.2, take F and Θ as they are and set +V ≔ ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ), +U ≔ P(X) × P(Y), +E((µ, ν), f, θ) ≔ µ( f cθcθ) + ν( f cθ). +Then, Assumption (EC) follows from Lemma 6.1, while (Lip) and (Lin) are simple to verify by +definition of V and E. Moreover, by Assumption (KP) the condition of point (ii) in Lemma A.3 +holds, since the evaluations of E in f with (∆µ, ∆ν) ∈ TµP(X)×TνP(Y) are invariant under constant +shifts (Lemma 6.8), and since Kantorovich potentials are unique on X and Y up to a constant shift. +This establishes (DC), and the proof is complete. +□ +6.3.2 +Proof of Theorem 2.6 +Since Θ is a compact Polish space, it follows by Fang and Santos (2019, Lemma S.4.9) (see also +Cárcamo, Cuevas, and Rodríguez 2020, Corollary 2.3) that the infimal mapping, +I : C(Θ) → R, +h �→ inf +θ∈Θ h(θ), +is Hadamard directionally differentiable at OT(µ, ν, c(·)) ∈ C(Θ) with derivative given by +DH +OT(µ,ν,c(·))I : C(Θ) → R, +∆h �→ +inf +θ∈S−(Θ,µ,ν) ∆h(θ). +Hence, applying the functional delta method (Römisch, 2006) for the infimal mapping I onto the +uniform weak limit for the empirical OT process from Theorem 2.5 asserts the claim. +□ +34 + +6.3.3 +Proof of Theorem 2.7 +From the dual formulation (2.4) the supremal OT value over Θ is given by +sup +θ∈Θ +OT(·, ·, cθ): P(X) × P(Y) ⊆ ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ) → R, +(µ, ν) �→ sup +(f,θ)∈F ×Θ +µ( f cθcθ) + ν( f cθ). +The results of Appendix A readily apply, with the choices for V, U, and E as in the proof of Theo- +rem 2.5; the only difference being that the supremum is taken over F ×Θ instead of F . In particular, +(EC), (Lip), and (Lin) are valid, whereas (DC) is now trivially fulfilled. Overall, Theorem A.2 as- +serts that supθ∈Θ OT(·, ·, cθ) is Hadamard directionally differentiable tangentially to P(X) × P(Y) +with derivative +DH +|(µ,ν) sup +θ∈Θ +OT(·, ·, cθ): TµP(X) × TνP(Y) → R, +(∆µ, ∆ν) �→ +sup +θ∈S+(Θ,µ,ν) +fθ∈Scθ(µ,ν) +∆µ( f cθcθ +θ +) + ∆ν( f cθ +θ ). +Combined with weak convergence of √nm/n + m(µn − µ, νm − ν) in ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ) +by (Don) in conjunction with the independence of the underlying samples (van der Vaart and Well- +ner, 1996, Example 1.4.6), and the inclusion of the limit in TµP(X) × TνP(Y) by Lemma 6.8, the +functional delta method (Römisch, 2006) implies the claim. +□ +Remark 6.9. In addition to the proof presented above, it is also possible to show Theorem 2.7 with +similar arguments to those found in the proof of Fang and Santos (2019, Lemma S.4.9) or Cárcamo, +Cuevas, and Rodríguez (2020, Corollary 2.3). However, their statements only provide sufficient +conditions for Hadamard directional differentiability for tangentially to the space C(∪θ∈ΘF cθcθ) × +C(∪θ∈ΘF cθ), whereas the supremal OT value is defined only on the strict subset P(X) × P(Y). +6.3.4 +Proof of Proposition 2.8 +Define ∆(µn, νm, K) ≔ infθ∈Θ OT(µn, νm, cθ) − infθ∈K OT(µn, νm, cθ). Then, +P∗�∆(µn, νm, K) � 0� ≤ P∗��∆(µn, νm, K) � 0� ∩ �θn,m ∈ K�� + P∗(θn,m � K), +and as the first summand in the above display is null while limn→∞ P∗(θn,m � K) = 0, the right- +hand side converges to zero. Hence, invoking Slutzky’s Lemma (van der Vaart and Wellner, 1996, +Example 1.4.7) it follows from Theorem 2.6 that +� +nm +n + m +� +inf +θ∈Θ OT(µn, νm, cθ) − inf +θ∈Θ OT(µ, ν, cθ) +� += +� +nm +n + m∆(µn, νm, K) + +� +nm +n + m +� +inf +θ∈K OT(µn, νm, cθ) − inf +θ∈K OT(µ, ν, cθ) +� +⇝ 0 + +inf +θ∈S−(K,µ,ν) +√ +λGµ( f cθcθ +θ +) + +√ +1 − λGν( f cθ +θ ). +The claim now follows at once after observing that S−(K, µ, ν) = S−(Θ, µ, ν). +□ +Acknowledgements: +S. Hundrieser and C.A. Weitkamp gratefully acknowledge support from the +DFG Research Training Group 2088 Discovering structure in complex data: Statistics meets opti- +mization and inverse problems. G. Mordant gratefully acknowledges support from the DFG CRC +1456 Mathematics of the Experiment A04 and A. 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More precisely, both the result on the weak convergence of the empirical +OT process from Section 2.3 and the formulation of regularity elevation functionals from Section 5 +rely on this approach. Although, these two findings are conceptually rather unrelated, their proof +techniques are based on a more general insight which we lay out in this section. +Let (V, ∥·∥V) be a normed vector space and consider sets F and Θ. Additionally, consider a +real-valued function E : V ×F ×Θ → R which assigns each triple (v, f, θ) to a some objective value +E(v, f, θ). We are interested in sensitivity results for extremal-type functionals +Ψ(v) ≔ +sup +f∈F +E(v, f, θ) + +θ∈Θ +and +˜Ψ(v) ≔ +� +inf +f∈F E(v, f, θ) +� +θ∈Θ +. +Herein, Θ provides the collection of feasible parameters which affect the optimization problem, +while F represents the collection of feasible solutions. The space V denotes another set of param- +eters that determine the optimization problem and exhibit a vector space structure. Overall, these +optimization problems characterize the general structure of processes indexed over Θ which are +pointwise defined as the supremum or infimum over a collection F and depend on some parameter +in V with an additive structure. +For our sensitivity analysis under perturbations of v it suffices to focus only on Ψ since +inf +f∈F E(v, f, θ) = − sup +f∈F +−E(v, f, θ) +for any (v, θ) ∈ V × Θ. +In the following, we first establish sufficient conditions in terms of E for the continuity properties +of Ψ and the underlying sets of optimizers. +Lemma A.1 (Continuity). Let (V, ∥·∥V) be a normed vector space, consider compact topological +spaces F and Θ whose topologies are generated by (pseudo-)metrics dF and dΘ, respectively, and +assume that E : V × F × Θ → R satisfies the following. +(EC) For any v ∈ V the functional E(v, ·, ·): F × Θ → R is continuous +(Lip) There exists some L ≥ 0 such that for any ( f, θ) ∈ F × Θ the functional E(·, f, θ): V → R is +L-Lipschitz with respect to ∥·∥V. +Then, Range(Ψ) ⊆ C(Θ) and the functional Ψ: V → C(Θ) is L-Lipschitz. Further, for any (v, θ) ∈ +V × Θ the set of optimizers S (v, θ) ≔ { f ∈ F | sup f ′∈F E(v, f ′, θ) = E(v, f, θ)} is non-empty, and for +fixed v ∈ V the set-valued map +(θ, t) ∈ Θ × R+ �→ S (v, θ; t) ≔ + f ∈ F +���� sup +f ′∈F +E(v, f ′, θ) ≤ E(v, f, θ) + t + +is upper semi-continuous in terms of inclusion, i.e., for θn → θ and tn → t any sequence fn ∈ +S (v, θn; tn) admits a converging subsequence ( fnk)k∈N in F with limit f ∈ S (v, θ; t). +Proof of Lemma A.1. By Assumption (EC) and compactness of Θ × F it follows for any v ∈ V that +E(v, ·, ·) is uniformly continuous, hence the function +wE,v : R+ → R+, +t �→ +sup +dΘ(θ,θ′)≤t +dF (f, f ′)≤t +|E(v, f, θ) − E(v, f ′, θ′)| +40 + +is finite for all t ≥ 0 and fulfills limtց0 wE,v(t) = 0. For θ, θ′ ∈ Θ we thus find that +�������sup +f∈F +E(v, f, θ) − sup +f∈F +E(v, f, θ′) +������� ≤ sup +f∈F +|E(v, f, θ) − E(v, f, θ′)| ≤ wE,v(dΘ(θ, θ′)), +which implies for v ∈ V that Ψ(v) ∈ C(Θ) and therefore Range(Ψ) ⊆ C(Θ). For the Lipschitzianity +of Ψ, note by Assumption (Lip) for any v, v′ ∈ V that +���Ψ(v) − Ψ(v′) +���C(Θ) = sup +θ∈Θ +�������sup +f∈F +E(v, f, θ) − sup +f∈F +E(v′, f, θ) +������� +≤ sup +θ∈Θ +f∈F +|E(v, f, θ) − E(v′, f, θ)| ≤ L +���v − v′���V . +To see that S (v, θ) � ∅, note that the function E(v, ·, θ): F → R is continuous for any (v, θ) ∈ +V × Θ; hence, by compactness of F the supremum over F is attained. +It remains to prove the assertion on upper semi-continuity. Consider converging sequences +tn → t ≥ 0 and θn → θ ∈ Θ and take a sequence fn ∈ S (v, θn; tn). By compactness of F a +converging subsequence ( fnk)k∈N exists with limit f ∈ F . Hence, by Assumption (EC) and since +sup f∈F E(v, f, ·) = Ψ(v)(·) ∈ C(Θ) we obtain that f ∈ S (v, θ; t) since +E(v, f, θ) + t = lim +k→∞ E(v, fnk, θnk) + tnk ≥ lim +k→∞ sup +f∈F +E(v, f, θnk) = sup +f∈F +E(v, f, θ). +□ +With these tools at our disposal, we can state our general sensitivity result. +Theorem A.2 (Differentiability). Assume in the setting of Lemma A.1 Conditions (EC) and (Lip). +Let v ∈ V and consider a convex set U ⊂ V. Denote by TvU ≔ Cl{ v−v +t +| t > 0, v ∈ U} ⊆ V its +contingent cone at v. Further, assume the following: +(Lin) For any ( f, θ) ∈ F × Θ the function ∆|vE(·, f, θ): V → R, v �→ E(v + v, f, θ) − E(v, f, θ) +is linear. +(DC) For any h ∈ TvU the function θ ∈ Θ �→ sup f∈S (v,θ) ∆|vE(h, f, θ) is lower semi-continuous. +Then, the functional +Ψ: V → C(Θ), +v �→ +sup +f∈F +E(v, f, θ) + +θ∈Θ +is Hadamard directionally differentiable at v tangentially to U with derivative given by +DH +|vΨ: TvU → C(Θ), +h �→ + sup +f∈S (v,θ) +∆|vE(h, f, θ) + +θ∈Θ +. +Theorem A.2 can be viewed as an extension of Römisch (2006, Proposition 1) and Fang and +Santos (2019, Lemma S.4.9) to a uniform perturbation result over the parameter space Θ. Addition- +ally, our result does not require regularity properties on the full domain V but only a convex set U, +an appealing property which we exploit in the context of our analysis for the OT process (where we +choose U = P(X) × P(Y)) as well as regularity elevations (see proof of Proposition 5.3). +Assumptions (EC), (Lip), and (Lin) are fairly straightforward and often simple to verify. The +first two conditions also appear to be necessary to infer that Range(Ψ) ⊆ C(Θ) and Lipschitzianity of +Ψ: V → C(Θ). Assumption (DC) is more technical and requires some knowledge on the set of opti- +mizers S (v, θ). As the proof of Theorem A.2 reveals, is the functional θ ∈ Θ �→ sup f∈S (v,θ) E(h, f, θ) +under the assumptions of Lemma A.1 always upper semi-continuous. Hence, the sole purpose of +(DC) is to ensure Range(DH +|v Ψ) ⊆ C(Θ). Sufficient conditions for its validity are stated as follows. +41 + +Lemma A.3. Assume the setting of Lemma A.1 and Theorem A.2. Then under either of the following +conditions Assumption (DC) of Theorem A.2 is fulfilled. +(i) For any θ ∈ Θ and h ∈ TvU there exists f ∈ S (v, θ) with sup f ′∈S (v,θ) ∆|vE(h, f ′, θ) = +∆|vE(h, f, θ) such that any converging sequence θn → θ admits a sub-sequence (θnk) and +a converging sequence fnk ∈ S (v, θnk) with fnk → f in F . +(ii) For any θ ∈ Θ and h ∈ TvU it holds that ∆|vE(h, f, θ) = ∆|vE(h, f ′, θ) for any f, f ′ ∈ S (v, θ). +Proof of Lemma A.3. Let θn → θ and consider an element f ∈ S (v, θ) such that ∆|vE(h, f, θ) = +sup f ′∈S (v,θ) ∆|vE(h, f, θ). For setting (i) take an arbitrary subsequence θnk and take another subse- +quence θnkl such that fnkl ∈ S (v, θnkl) converges to f for l → ∞. Then, by (EC), +lim +l→∞ ∆|vE(h, fnkl , θnkl) = ∆|vE(h, f, θ) = +sup +f ′∈S (v,θ) +∆|vE(h, f ′, θ). +This implies by monotonicity of the limit inferior and Lemma F.1 that +lim inf +n→∞ +sup +f ′∈S (v,θ) +∆|vE(h, f ′, θn) ≥ lim inf +n→∞ ∆|vE(h, fn, θn) ≥ +sup +f ′∈S (v,θ) +∆|vE(h, f ′, θ), +which asserts the validity of Assumption (DC) of Theorem A.2. For setting (ii) take fn ∈ S (v, θn) +and consider by Lemma A.1 a converging subsequence fnk with limit f ∈ S (v, θ). Hence, it holds +that ∆|vE(h, f, θ) = sup f ′∈S (v,θ) ∆|vE(h, f ′, θ) and the assertion follows from (i). +□ +Proof of Theorem A.2. The proof strategy is inspired by Römisch (2006) who performs a sensitivity +analysis for when Θ is a singleton. To extend the claim for a compact topological space Θ we +employ the subsequent version of Dini’s theorem. +Lemma A.4 (Dini’s Theorem, Toma 1997, Corollary 1). Let Θ be a compact topological space +and consider a decreasing fn : Θ → R sequence (i.e., fn ≥ fn+1 for all n ∈ N) of upper semi- +continuous functions. Further, assume that fn pointwise converges to a (lower semi-)continuous +function f : Θ → R. Then, fn converges to f uniformly on Θ. +Take a positive null sequence tn ց 0 with tn > 0 for all n ∈ N and let h ∈ TvU. Further, take a +sequence hn ∈ V such that vn ≔ v + tnhn ∈ U for all n ∈ N and hn → h in V. For any θ ∈ Θ, we then +observe by (Lin) and (Lip) for any n ∈ N the lower bound +Ψ(vn)(θ) − Ψ(v)(θ) = sup +f∈F +E(vn, f, θ) − sup +f∈F +E(v, f, θ) +≥ +sup +f∈S (v,θ) +E(vn, f, θ) − E(v, f, θ) +≥ +sup +f∈S (v,θ) +∆|vE(tnhn, f, θ) +≥ tn +sup +f∈S (v,θ) +∆|vE(h, f, θ) − 2tnL ∥h − hn∥V . +(A.1) +Analogously, we obtain the upper bound +Ψ(vn)(θ) − Ψ(v)(θ) = sup +f∈F +E(vn, f, θ) − sup +f∈F +E(v, f, θ) +≤ +sup +f∈S (vn,θ) +E(vn, f, θ) − E(v, f, θ) +≤ tn +sup +f∈S (vn,θ) +∆|vE(h, f, θ) + 2tnL ∥h − hn∥V . +(A.2) +42 + +Note that S (vn, θ) ⊆ S (v, θ; 2L ∥vn − v∥V) since any f ∗ ∈ S (vn, θ) fulfills by (Lip) the bound +E(v, f ∗, θ) ≥ E(vn, f ∗, θ) − L ∥vn − v∥V += sup +f∈F +E(vn, f, θ) − L ∥vn − v∥V +≥ sup +f∈F +E(v, f, θ) − 2L ∥vn − v∥V . +Hence, it follows from (A.2) upon defining εn ≔ supk≥n 2L ∥vk − v∥V that +Ψ(vn)(θ) − Ψ(v)(θ) ≤ tn +sup +f∈S (v,θ;2L∥vn−v∥V) +∆|vE(h, f, θ) + 2tnL ∥h − hn∥V +≤ tn +sup +f∈S (v,θ;εn) +∆|vE(h, f, θ) + 2tnL ∥h − hn∥V . +(A.3) +Combining (A.1) and (A.3) we thus obtain for any θ ∈ Θ that +sup +f∈S (v,θ) +∆|vE(h, f, θ) − 2L ∥h − hn∥V ≤ Ψ(vn)(θ) − Ψ(v)(θ) +tn +≤ +sup +f∈S (v,θ;εn) +∆|vE(h, f, θ) + 2L ∥h − hn∥V . +To conclude the claim we show that the lower and upper bound uniformly converge on Θ for n → ∞ +to the DH +|v Ψ. Since ∥hn − h∥V → ∞, it suffices to prove for the functions +Φ ≔ DH +|v Ψ: Θ → R, θ �→ +sup +f∈S (v,θ) +∆|vE(h, f, θ), +Φn : Θ → R, θ �→ +sup +f∈S (v,θ,εn) +∆|vE(h, f, θ), +that limn→∞ ∥Φ − Φn∥C(Θ) = 0. For this purpose, we employ Dini’s theorem (Lemma A.4). +In this context note, since (εn)n∈N is a decreasing null-sequence, for all n ∈ N and any θ ∈ Θ +that S (v, θ) ⊆ S (v, θ; εn+1) ⊆ S (v, θ; εn) and consequently +Φ(θ) ≤ Φn+1(θ) ≤ Φn(θ) ≤ 2 sup +θ∈Θ +sup +f∈F +E(h, f, θ) < ∞, +(A.4) +where the upper bound is finite due to Assumption (EC) and compactness of F × Θ. +Further, let us show for any θ ∈ Θ that limn→∞ Φn(θ) = Φ(θ). Take a sequence fn ∈ S (v, θ; εn) +such that Φn(θ) ≤ ∆|vE(h, fn, θ) + 1/n. Consider a converging subsequence ( fnk)k∈N with limit +f∞ ∈ S (v, θ). Then, by (EC) it follows that +lim sup +k→∞ +Φnk(θ) ≤ lim +k→∞ ∆|vE(h, fnk, θ) + 1/nk = ∆|vE(h, f∞, θ) ≤ +sup +f∈S (v,θ) +∆|vE(v, f, θ) = Φ(θ). +Recalling (A.4), it thus follows that limn→∞ Φn(θ) = Φ(θ). +To conclude the assertion with Dini’s theorem it remains to show upper-continuity of Φn and +of Φ; recall by Assumption (DC) that Φ is already lower semi-continuous. To this end, let ε ≥ 0 and +consider a converging sequence θn → θ. Select fn ∈ S (v, θn, ε) such that sup f∈S (v,θn,ε) ∆|vE(h, f, θn) ≤ +∆|vE(h, fn, θn) + 1/n. Take a subsequence fnk and select by Lemma A.1 another converging subse- +quence fnkl with limit f∞ ∈ S (v, θ; ε). Using Assumption (EC) it thus follows that +lim +l→∞ ∆|vE(h, fnkl, θnkl) + 1/nkl = ∆|vE(h, f∞, θ) ≤ +sup +f∈S (v,θ;ε) +∆|vE(h, f, θ) +Invoking monotonicity of the limit superior and Lemma F.1 we thus obtain that +lim sup +n→∞ +sup +f∈S (v,θn;ε) +∆|vE(h, f, θ) ≤ lim sup +l→∞ +∆|vE(h, fn, θn) + 1/n ≤ +sup +f∈S (v,θ;ε) +∆|vE(h, f, θ), +Hence, by Lemma F.1 we conclude that Φn is upper semi-continuous and that Φ is continuous. +Dini’s theorem (Lemma A.4) thus implies limn→∞ ∥Φ − Φn∥∞ = 0, asserting the Hadamard direc- +tional differentiability of Ψ at v tangentially to U. Finally, note that the range of DH +v Ψ is indeed +contained in C(Θ). +□ +43 + +B +Proofs for Section 3: Sufficient Criteria for Assumptions +B.1 +Proof of Proposition 3.1 +By Lemma 2.1 it follows that F c ⊆ Hc +c + [−2B, 2B] and F cc ⊆ Hc + [−2B, 2B] with Hc defined +in (2.1). Invoking Hundrieser, Staudt, and Munk (2022, Lemma 2.1) and Santambrogio (2015, +Proposition 1.34) we obtain for any ε > 0 that +N(ε, F c, ∥·∥∞) = N(ε, F cc, ∥·∥∞) ≤ +�2B +ε +� +N(ε/2, Hc +c, ∥·∥∞) = +�2B +ε +� +N(ε/2, Hc, ∥·∥∞). +For the function class Hc, the asserted uniform metric entropy bounds are available in Section 3.1 +and Appendix A of Hundrieser, Staudt, and Munk (2022). Note by uniform boundedness of the cost +function that Hc and Hc +c are uniformly bounded. The assertion on the universal Donsker property +then follows from van der Vaart and Wellner (1996, Theorem 2.5.6). +□ +B.2 +Proof of Proposition 3.3 +By assumption the functional +Φc : P(X) × P(Y) → ℓ∞(F cc) × ℓ∞(F c) × C(X × Y) +(µ, ν) �→ (µ, ν, Φc(µ, ν)), +where the domain is viewed as a subset of ℓ∞(FX ∪F cc)×ℓ∞(FY ∪F c), is Hadamard differentiable +at (µ, ν) tangentially to P(X) × P(Y). Moreover, since FX ∪ F cc is µ-Donsker it follows that +√n/2(µn − µ) ⇝ Gµ in ℓ∞(FX ∪ F cc). Likewise, since FY ∪ F c is ν-Donsker it follows that +√n/2(νn − ν) ⇝ Gν in ℓ∞(FY ∪ F c). Further, by independence of the random variables {Xi}n +i=1 +and {Yi}n +i=1 it follows from van der Vaart and Wellner, 1996, Theorem 1.4.6 that the joint empirical +processes √n/2(µn − µ, νn − ν) weakly converge in ℓ∞(FX ∪ F cc) × ℓ∞(FY ∪ F c) to (Gµ, Gν), +contained in TµP(X) × TνP(Y) by Lemma 6.8. We thus conclude by the functional delta method +(Römisch, 2006) for Φc that (JW) is fulfilled. +Moreover, by the Donsker property and independence of the random variables, we also infer by +van der Vaart and Wellner (1996, Theorem 3.6.13) in the space ℓ∞(FX ∪ F cc) × ℓ∞(FY ∪ F c) that +dBL +� +L +� √ +k +�µb +n,k − µn +νb +n,k − νn +�������X1, . . . , Xn, Y1, . . . Yn +� +, L +�Gµ +Gν +�� +P∗ +−−→ 0. +Hence, by the functional delta method for conditionally weakly converging random variables Düm- +bgen (1993) for Ψc we infer that +dBL +L + +√ +k + +µb +n,k − µn +νb +n,k − νn +cb +n,k − cn + +��������� +X1, . . . , Xn, Y1, . . . Yn + , L + +√n + +µn − µ +νn − ν +cn − c + + + +P∗ +−−→ 0. +□ +B.3 +Proof of Proposition 3.6 +Before, we start to prove Proposition 3.6, we establish an auxiliary lemma. +Lemma B.1. Let X and Y be compact Polish spaces and consider c ∈ C(X × Y). +(i) For any function g : X → R and any constant κ, it holds that (g + κ)c = gc − κ. +(ii) Let B > 0. Then, for any g : X → R and ∆c ∈ C(X × Y) with ∥g∥∞ + 2 ∥c + ∆c∥∞ ≤ B it holds +that g(c+∆c)(c+∆c)cc ∈ Hc + [−B, B]. +44 + +The proof of the above lemma can be found in Appendix E.7. +Proof of Proposition 3.6. The proof is strongly inspired by van der Vaart and Wellner, 2007, Theo- +rem 2.3 and employs standard empirical process arguments. In order to simplify the notation, we +only consider the case n = m and write cn instead of cn,n. Note that the claim for n � m follows by +the analogous arguments. +To show (i) first note by triangle inequality and using Lemma 6.1 that +sup +f∈F +|Gµ +n( f cncn − f cc)| ≤ sup +f∈F +|Gµ +n( f cncn − f ˜cn ˜cn)| + sup +f∈F +|Gµ +n( f ˜cn ˜cn − f cc)| +≤ 4 √n ∥cn − ˜cn∥∞ + sup +f∈F +|Gµ +n( f ˜cn˜cn − f cc)|. +(B.1) +The first term converges by assumption for n → ∞ in probability to zero. For the latter term note +by (JW) and the assumption on ˜cn that √n/2(˜cn − c) ⇝ Gc. By tightness of the law of Gc there +exists for any ε > 0 a compact set K ⊆ C(X × Y) such that P(Gc ∈ K) > 1 − ε; thus for any δ > 0 +the set Kδ of elements in C(X × Y) with distance less than δ > 0 to K fulfills +lim inf +n→∞ P +� � +n/2(˜cn − c) ∈ Kδ� +≥ P(Gc ∈ Kδ) > 1 − ε. +(B.2) +By compactness of K there exists a finite δ/2-covering {h1, . . . , hp} which implies that Kδ/2 ⊆ +�p +i=1 B(hi, δ), where B(h, δ) denotes the open ball of radius δ around h in the space C(X × Y). We +thus obtain +� � +n/2(˜cn − c) ∈ Kδ/2� +⊂ +p +� +i=1 +� +˜cn ∈ B(c + 2n−1/2hi, δ) +� +. +Moreover, by Santambrogio (2015, Proposition 1.34) it follows for any f ∈ F and ¯c ∈ C(X × Y) +that f ¯c¯c = f ¯c¯c¯c¯c. Therefore, by triangle inequality, +sup +f∈F +|Gµ +n( f ˜cn ˜cn − f cc)| = sup +f∈F +|Gµ +n( f ˜cn ˜cn˜cn ˜cn − f cccc)| +≤ sup +f∈F +|Gµ +n( f ˜cn ˜cn˜cn ˜cn − f ˜cn ˜cncc)| + sup +f∈F +|Gµ +n( f ˜cn ˜cncc − f cccc)| +≤ +sup +f∈F ˜cn ˜cn +|Gµ +n( f ˜cn˜cn − f cc)| + sup +f∈F +|Gµ +n( f ˜cn˜cncc − f cccc)|. +(B.3) +Assuming √n/2(˜cn − c) ∈ Kδ/2, it follows for the first term in (B.3) that +sup +f∈F ˜cn˜cn +|Gµ +n( f ˜cn ˜cn − f cc)| ≤ +sup +f∈F ˜cn ˜cn +max +i=1,...,p +sup +∥h−hi∥∞<δ +|Gµ +n( f (c+2h/ √n)(c+2h/ √n) − f cc)| +≤ +sup +f∈F ˜cn ˜cn +max +i=1,...,p +sup +∥h−hi∥∞<δ +|Gµ +n( f (c+2h/ √n)(c+2h/ √n) − f (c+2hi/ √n)(c+2hi/ √n))| ++ +sup +f∈F ˜cn ˜cn +max +i=1,...,p |Gµ +n( f (c+2hi/ √n)(c+2hi/ √n) − f cc)| +≤ 8δ + +sup +f∈F ˜cn ˜cn +max +i=1,...,p |Gµ +n( f (c+2hi/ √n)(c+2hi/ √n) − f cc)|. +(B.4) +Here, we used in the last inequality Lemma 6.1 to infer +���� f (c+2h/ √n)(c+2h/ √n) − f (c+2hi/ √n)(c+2hi/ √n)����∞ ≤ 4 ∥hi − h∥∞ / √n ≤ 4δ/ √n +45 + +in conjunction with Gµ +n(g) = √n(µn − µ)(g) ≤ 2 √n ∥g∥∞ for any measurable function g on X. Now, +define for 1≤ i ≤ p the function class +˜Gi +n ≔ ˜Gi +n(hi) ≔ +� +f (c+2hi/ √n)(c+2hi/ √n) − f cc��� f ∈ F ˜cn ˜cn� +. +For each 1 ≤ i ≤ p and any ε > 0 we then observe that +log N(ε, ˜Gi +n, ∥·∥∞) ≤ log N(ε, F ˜cn ˜cn(c+2hi/ √n)(c+2hi/ √n), ∥·∥∞) + log N(ε, ˜F ˜cn ˜cncc, ∥·∥∞) +≤2 log N(ε, F ˜cn˜cn, ∥·∥∞), +where the last step follows by Lemma 2.1 in Hundrieser, Staudt, and Munk, 2022. In consequence, +it follows by Dudley’s entropy integral (see, e.g.,Wainwright, 2019, Chapter 5) that +E + sup +f∈F ˜cn ˜cn +max +i=1,...,p +����Gµ +n( f (c+2hi/ √n)(c+2hi/ √n) − f cc) +���� + ≤ +p +� +i=1 +E +sup +g∈ ˜Gin +���Gµ +n(g) +��� + +≲ +p +� +i=1 +� 4∥hi∥∞/ √n +0 +� +log �N(ε, F ˜cn˜cn, ∥·∥∞)�dε +≲ +p +� +i=1 +� 4∥hi∥∞/ √n +0 +ε−α/2dε +≲ +p +� +i=1 +(∥hi∥∞ / √n)1−α/2, +where by assumption the hidden constants do not depend on n. We thus infer conditionally on the +event √n/2(˜cn − c) ∈ Kδ/2 for n → ∞ that +sup +f∈F ˜cn ˜cn +max +i=1,...,p Gµ +n( f (c+2hi/ √n)(c+2hi/ √n) − f cc) +P→ 0. +(B.5) +For the second term in (B.3) we assume √n/2(˜cn − c) ∈ Kδ/2 and obtain by similar arguments, +sup +f∈F +|Gµ +n( f ˜cn ˜cncc − f cccc)| ≤ 8δ + sup +f∈F +max +i=1,...,p |Gµ +n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc)|. +(B.6) +Upon defining the function class +˜Gn ≔ ˜Gn(h1, . . . , hp) ≔ +� +f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc��� f ∈ F +� +(B.7) +we note again by Lemma 6.1 that any g ∈ Gn fulfills ∥g∥∞ ≤ maxi=1,...,p 4 ∥hi∥∞ / √n. Further, for +n sufficiently large there exists a constant B > 0 such that Lemma B.1 is applicable for any f ∈ F , +and we obtain +f (c+2hi/ √n)(c+2hi/ √n)cc ∈ Hc + [−B, B]. +Hence, for sufficiently large n, it follows by Lemma 2.1 for any ε > 0 that +N(ε, ˜Gn(h1, . . . , hp), ∥·∥∞) ≤ �N(ε, F cc + [−B, B], ∥·∥∞)�2 . +46 + +Again invoking, Dudley’s entropy integral asserts such for n that +E +� +sup +f∈F +max +i=1,...,p +�����Gµ +n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc) +����� +� += E +sup +f∈ ˜Gn +���Gµ +n( ˜f) +��� + +≲ +� maxi=1,...,p 4∥hi∥∞/ √n +0 +� +log +� +N(ε, ˜Gn, ∥·∥∞) +� +dε +≤ +� maxi=1,...,p 4∥hi∥∞/ √n +0 +� +log (N(ε, F cc + [−B, B], ∥·∥∞))dε +≲ +� maxi=1,...,p 4∥hi∥∞/ √n +0 +ε−α/2dε +≲ +� +max +i=1,...,p ∥hi∥∞ / √n +�1−α/2 +. +This implies conditionally on the event √n/2(˜cn − c) ∈ Kδ/2 for n → ∞ that +sup +f∈F +max +i=1,...,p |Gµ +n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc)| +P→ 0. +(B.8) +Concluding, for any ε > 0 it follows for δ ≔ ε/32 > 0 from (B.2)–(B.8) that +lim sup +n→∞ +P +sup +f∈F +|Gµ +n( f ˜cn˜cn − f cc)| > ε + +≤ lim sup +n→∞ +P +sup +f∈F +|Gµ +n( f ˜cn ˜cn − f cc)| > ε, +� +n/2(˜cn − c) ∈ Kδ/2 + + P +� � +n/2(˜cn − c) � Kδ/2� +≤ lim sup +n→∞ +P +sup +f∈F +|Gµ +n( f ˜cn˜cn − f cc)| > ε, +� +n/2(˜cn − c) ∈ Kδ/2 + + ε +≤ lim sup +n→∞ +P + sup +f∈F ˜cn ˜cn +|Gµ +n( f ˜cn ˜cn − f cc)| > ε/2, +� +n/2(˜cn − c) ∈ Kδ/2 + ++ lim sup +n→∞ +P +sup +f∈F +|Gµ +n( f ˜cn˜cncc − f cccc)| > ε/2, +� +n/2(˜cn − c) ∈ Kδ/2 + + ε +≤ lim sup +n→∞ +P + sup +f∈F ˜cn ˜cn +max +i=1,...,p |Gµ +n( f (c+2hi/ √n)(c+2hi/ √n) − f cc)| > ε/4, +� +n/2(˜cn − c) ∈ Kδ/2 + ++ lim sup +n→∞ +P +sup +f∈F +max +i=1,...,p |Gµ +n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc)| > ε/4, +� +n/2(˜cn − c) ∈ Kδ/2 + + ε = ε, +which shows the convergence in probability of sup f∈F |Gµ +n( f ˜cn˜cn− f cc)| to zero. We thus conclude the +convergence in probability for both terms of (B.1). An analogous argument yields the convergence +sup f∈F |Gν +n( f cn − f c)| +P→ 0 for n → ∞, where we apply Lemma 2.1 of Hundrieser, Staudt, and +Munk, 2022 to obtain +sup +n∈N +log N(ε, F ˜cn ∪ F c, ∥·∥∞) ≤ sup +n∈N +� +log N(ε, F ˜cn, ∥·∥∞) + log N(ε, F c, ∥·∥∞) +� += sup +n∈N +� +log N(ε, F ˜cn ˜cn, ∥·∥∞) + log N(ε, F cc, ∥·∥∞) +� +≤ sup +n∈N +2 log N(ε, F ˜cn ˜cn ∪ F cc, ∥·∥∞) ≲ ε−α +for α < 2, +47 + +which overall verifies (Sup) of Theorem 2.2. +For (ii) note by Bücher and Kojadinovic (2019) and since k = k(n) = o(n) for n → ∞ that +√ +k(˜cb +n,k − c) = +√ +k(˜cb +n,k − cb +n,k) + +√ +k(cb +n,k − cn) + +� +k +n +√n(cn − c) ⇝ Gc. +Likewise, it follows for n → ∞ that +√ +k(µb +n,k − µ) ⇝ Gµ in ℓ∞(F cc), +√ +k(νb +n,k − ν) ⇝ Gν in ℓ∞(F c). +This means that we can pursue a similar proof strategy as for (i). Define Gµ +n,k ≔ +√ +k(µb +n,k − µ) and +Gν +n,k ≔ +√ +k(νb +n,k − ν). Then, we infer from Lemma 6.1 that +sup +f∈F +|Gµ +n,k( f cb +n,kcb +n,k − f cc)| ≤ sup +f∈F +|Gµ +n,k( f cb +n,kcb +n,k − f ˜cb +n,k ˜cb +n,k)| + sup +f∈F +|Gµ +n,k( f ˜cb +n,k˜cb +n,k − f cc)| +≤ 4 +√ +k +���cb +n,k − ˜cb +n,k +���∞ + sup +f∈F +|Gµ +n,k( f ˜cb +n,k ˜cb +n,k − f cc)|, +(B.9) +where the first term converges for n → ∞ in probability to zero. By Santambrogio (2015, Proposi- +tion 1.34) we obtain that +sup +f∈F +|Gµ +n,k( f ˜cb +n,k ˜cb +n,k − f cc)| ≤ +sup +f∈F +˜cb +n,k ˜cb +n,k +|Gµ +n,k( f ˜cb +n,k ˜cb +n,k − f cc)| + sup +f∈F +|Gµ +n,k( f ˜cb +n,k ˜cb +n,kcc − f cccc)|. +Moreover, by analogous arguments to those for (i) we obtain with probability at least 1 − ε for n +sufficiently large that +sup +f∈F +˜cb +n,k ˜cb +n,k +|Gµ +n,k( f ˜cb +n,k ˜cb +n,k − f cc)| ≤ 8δ + +sup +f∈F +˜cb +n,k ˜cb +n,k +max +i=1,...,p |Gµ +n,k( f (c+2hi/ +√ +k)(c+2hi/ +√ +k) − f cc)| +(B.10) +as well as +sup +f∈F +|Gµ +n,k( f ˜cb +n,k ˜cb +n,kcc − f cccc)| ≤ 8δ + sup +f∈F +max +i=1,...,p |Gµ +n,k( f (c+2hi/ +√ +k)(c+2hi/ +√ +k)cc − f cccc)|. +(B.11) +Next, we verify that the suprema on the right-hand sides of (B.10) and (B.11) converge (uncondi- +tionally with respect to the µn but conditionally on the set with probability at least 1 − ε) to zero. +We note by Dudley’s entropy integral for the bootstrap empirical process +√ +k(µb +n,k − µn) and the +empirical process √n(µn − µ) as well as our previous considerations that +≲E + +sup +f∈F +˜cb +n,k ˜cb +n,k +max +i=1,...,p |Gµ +n,k( f (c+2hi/ +√ +k)(c+2hi/ +√ +k) − f cc)| + += +p +� +i=1 +Eµn +Eµb +n,k + +sup +f∈F +˜cb +n,k ˜cb +n,k +| +√ +k(µb +n,k − µn)( f (c+2hi/ +√ +k)(c+2hi/ +√ +k) − f cc)| +������ µn + + ++ +� +k +nE + +sup +f∈F +˜cb +n,k ˜cb +n,k +|Gµ +n( f (c+2hi/ +√ +k)(c+2hi/ +√ +k) − f cc)| + +≲ +p +� +i=1 +Eµn +� 4∥hi∥∞/ +√ +k +0 +� +log +� +N(ε, F ˜cb +n,k ˜cb +n,k, ∥·∥∞) +� +dε + +� +k +n +� 4∥hi∥∞/ +√ +k +0 +� +log +� +N(ε, F ˜cb +n,k ˜cb +n,k, ∥·∥∞) +� +dε +≲ +p +� +i=1 +1 + +� +k +n + +� 4∥hi∥∞/ +√ +k +0 +ε−α/2dε ≲ +p +� +i=1 +1 + +� +k +n + +� +∥hi∥∞ / +√ +k +�1−α/2 , +48 + +which tends to zero for n → ∞ with k = k(n) = o(n) since the hidden constants do not depend on +n, k. Recalling the definition of the function class ˜Gk in (B.7) with n replaced by k, we obtain +≲E +sup +f∈F +max +i=1,...,p |Gµ +n,k( f (c+2hi/ +√ +k)(c+2hi/ +√ +k)cc − f cccc)| + = E +sup +f∈ ˜Gk +����Gµ +n,k( f) +���� + . +Hence, Dudley’s entropy integral in combination with our previous considerations yields +E +sup +f∈ ˜Gk +����Gµ +n,k( f) +���� + ≤ Eµn +Eµb +n,k +sup +f∈ ˜Gk +| +√ +k(µb +n,k − µn)( f)| +������ µn + + + +� +k +nE +sup +f∈ ˜Gk +|Gµ +n( f)| + +≲ +1 + +� +k +n + +� maxi=1,...,p 4∥hi∥∞/ +√ +k +0 +� +log (N(ε, F cc + [−B, B], ∥·∥∞))dε +≲ +1 + +� +k +n + +� +max +i=1,...,p ∥hi∥∞ / +√ +k +�1−α/2 +, +which goes to zero for n, k(n) → ∞ with k(n) = o(n) (the hidden constants are independent of n, k). +Using the same arguments as in (i), we conclude that +sup +f∈F +|Gµ +n,k( f ˜cb +n,k ˜cb +n,k − f cc)| +P→ 0. +Finally, analogous arguments yield that sup f∈F |Gν +n,k( f cb +n,k − f c)| +P→ 0, thus showing (ii). +□ +B.4 +Proof of Corollary 3.7 +Define the random variables +˜cn ≔ + +c +if n < N, +cn +if n ≥ N, +and +˜cb +n,k ≔ + +c +if n < N or k < K, +cb +n,k +if n ≥ N and k ≥ K. +By Proposition 3.1 the cost estimators ˜cn and ˜cb +n,k satisfy the entropy bounds in (3.1) and (3.2). +Tightness of N and K implies that √n ∥˜cn − cn∥∞ +P→ 0 and +√ +k∥˜cb +n,k − cb +n,k∥∞ +P→ 0 for n, k → ∞, +which asserts the claim by Proposition 3.6. +□ +B.5 +Proof of Corollary 3.8 +By Assumption (JW) it follows that √nm/(n + m)(cn,m − c) ⇝ Gc, whereas under (JW)∗ we infer +from Bücher and Kojadinovic (2019) and k = o(n) that +√ +k(cb +n,k − c) ⇝ Gc unconditionally. In what +follows we state the arguments for (Sup); for (Sup)∗ a similar proof strategy applies by replacing +the empirical costs process by the bootstrap cost process. +First, assume without loss of generality that the population cost function fulfills ∥c∥∞ ≤ 1. Then, +for all three settings of Proposition 3.1 it follows that log N(ε, F cc, ∥·∥∞) ≲ ε−α with α < 2. +For setting (i) we set ˜cn,m ≔ Ψbdd(cn,m), for Ψbdd defined in Section 5.1. Since ∥˜cn,m∥∞ ≤ 2 and +F = F (2 ∥c∥∞ + 1, 2w) is uniformly bounded by 6, we obtain that F ˜cn,m is uniformly bounded by 8. +By Proposition 5.1 and 5.2 both conditions of Proposition 3.6(i) are met, asserting (Sup). +For setting (ii) we take ˜cn,m ≔ Ψ +˜dX +mod ◦ Ψbdd(cn,m) for Ψ +˜dX +mod from Section 5.2. Then, ∥˜cn,m∥∞ ≤ 2 +and F ˜cn,m is uniformly bounded by 8. Moreover, by Assumption (ii)’ it follows with Proposition 5.1 +and 5.3 that √nm/(n + m)∥˜cn,m − cn,m∥∞ +P→ 0 and that +sup +n∈N +log N(ε, F ˜cn,m ˜cn,m, ∥·∥∞) ≲ N(ε/8, X, ˜dX)| log(ε)| ≲ ε−β| log(ε)| ≲ ε−2+(2−β)/2, +49 + +where we used the covering number assumption on X. (Sup) then follows from Proposition 3.6(i). +For setting (iii) define ci ∈ C(Ui × Y) as ci(u, y) ≔ c(ζi(u), y). We consider ˜cn,m ≔ Ψcom(cn,m) +where Ψcom denotes the combination (Section 5.4) of regularity elevation functionals Ψi : C(Ui × +Y) → C(Ui × Y) defined by Ψi = Ψ∥·∥γi +mod ◦ Ψbdd from Section 5.2 if γi ∈ (0, 1], and Ψi = Ψci,γi +Hol ◦ Ψbdd +from Section 5.3 if γi ∈ (1, 2], where we replace X by Ui. Then, by Propositions 5.3, 5.5, and 5.6 the +functional Ψ fulfills the assumptions of Proposition 5.1 and therefore √nm/(n + m)∥˜cn,m−cn,m∥∞ +P→ +0. Moreover, since for any ˜c ∈ C(X × Y) it holds that ∥Ψ(˜c)∥∞ < C for a deterministic constant +C ≥ 0 that only depends on the functions ci and the spaces Ui, it follows that F Ψ(˜c) is uniformly +bounded by C + 6 and therefore +sup +n∈N +log N(ε, F ˜cn,m ˜cn,m, ∥·∥∞) ≲ +I� +i=1 +sup +˜ci∈C(Ui×Y) +log N(ε, F ˜cn,mΨi(˜ci), ∥·∥∞) ≲ max +i=1,...,I ε−di/γi, +where we use for the first inequality Proposition 5.6, and for the second we employ the bounds from +Proposition 5.3 with N(ε, Ui, ∥·∥γi) ≲ ε−di/γi for 0 < γi ≤ 1 and Proposition 5.5 for 1 < γi ≤ 2. The +assertion then follows by an application of Proposition 3.6(i). +□ +B.6 +Proof of Lemma 3.10 +For ε > 0 suppose that the right-hand side is finite since otherwise the claim is vacuous. Set k = +N(ε/4, Θ, dΘ) and let {θ1, . . . , θk} be a minimal ε/4-covering of Θ. Further, for each i = 1, . . . , k let +{ f i +1, . . . , f i +ki} be a minimal ε/2-covering of F cθicθi, i.e., ki = N (ε/2, F cθicθi, ∥·∥∞). Once we show that +FX(ε) ≔ �k +i=1{ f i +1, . . . , f i +ki} is an ε-covering for � +θ∈Θ F cθcθ and that FY(ε) ≔ �k +i=1{( f i +1)cθi, . . . , ( f i +ki)cθi} +is an ε-covering for � +θ∈Θ F cθ the claim follows, since +|FY(ε)| ≤ |FX(ε)| = +k +� +i=1 +N +�ε +2, F cθicθi, ∥·∥∞ +� +≤ N +�ε +4, Θ, dΘ +� +sup +θ∈Θ +N +�ε +2, F cθcθ, ∥·∥∞ +� +. +Hence, let θ ∈ Θ and f ∈ F cθcθ, and choose ˜f ∈ F with f = ˜f cθcθ. Select θi with dΘ(θ, θi) ≤ ε/4 +and choose f i +li ∈ FX(ε) such that +���� f i +li − ˜f cθicθi +����∞ ≤ ε/2. Now, by Lipschitzianity of the cost in θ and +Lemma 6.1 we infer +��� ˜f cθicθi − ˜f cθcθ���∞ ≤ 2dΘ(θ, θi) ≤ ε/2, and it follows that +��� f i +li − f +���∞ = +���f i +li − ˜f cθcθ���∞ ≤ +���f i +li − ˜f cθicθi���∞ + +��� ˜f cθicθi − ˜f cθcθ���∞ ≤ ε, +which verifies that FX(ε) is an ε-covering of ∪θ∈ΘF cθcθ. +Moreover, for any f ∈ F cθ there exists ˜f ∈ F with f = +˜f cθ and by Santambrogio (2015, +Proposition 1.34) it follows that ˜f cθ = ˜f cθcθcθ. Hence, upon selecting f i +li ∈ FX(ε) as above, we find +by Lemma 6.1 that +���( f i +li)cθi − ˜f cθi���∞ = +���( f i +li)cθi − ˜f cθicθicθi���∞ ≤ +��� f i +li − ˜f cθicθi���∞ ≤ ε/2. +Again invoking Lemma 6.1 yields +��� ˜f cθi − ˜f cθ���∞ ≤ d(θ, θi) ≤ ε/4. Consequently, we find that +���( f i +li)cθi − f +���∞ = +���( f i +li)cθi − ˜f cθ���∞ ≤ +���( f i +li)cθi − ˜f cθi���∞ + +��� ˜f cθi − ˜f cθ���∞ ≤ 3ε +4 ≤ ε, +which proves that FY(ε) is an ε-covering of ∪θ∈ΘF cθ and finishes the proof. +□ +50 + +C +Proofs for Section 4: Applications +C.1 +Proof of Lemma 4.1 +Select U as the pre-image of {˜g ∈ C(X, Rd): ∥˜g − g−1 +ϑo ∥∞ < 1} under KΘ, which is open (rela- +tive) in Θ due to continuity. Hence, by compactness of X, the collection {g−1 +ϑ }ϑ∈U is uniformly +bounded on X. Invoking the Cauchy-Schwarz inequality and, due to compactness of Y, we infer that +{CΘ(ϑ)(x, ·)}ϑ∈U,x∈X is also uniformly bounded on Y. Further, since ∇yCΘ(ϑ)(x, y) = 2�g−1 +ϑ (x) − y� +for y ∈ int(Y) the collection {∇yCΘ(ϑ)(x, ·)} is bounded on Y uniformly over ϑ ∈ U, x ∈ X. Fi- +nally, note that HessyCΘ(ϑ)(x, y) = −2Id, independent of ϑ ∈ U, x ∈ X. Thus, by combining these +observations, we conclude the existence of Λ ≥ 0 such that the (2, Λ)-Hölder regularity is met. +□ +C.2 +Proof of Lemma 4.2 +To establish the Hadamard differentiability of CΘ at ϑo note that +����� +CΘ(ϑo + tnhn) − CΘ(ϑo) +tn +− DH +|ϑoCΘ(h) +�����∞ += +sup +(x,y)∈X×Y +����� +1 +tn +� +g−1 +ϑo+tnhn(x) − g−1 +ϑo (x), g−1 +ϑo+tnhn(x) + g−1 +ϑo (x) − 2y +� +− 2 +� +DH +ϑoKΘ(h)(x), g−1 +ϑo (x) − y +������ +≤ +sup +(x,y)∈X×Y +�������2 +� 1 +tn +� +g−1 +ϑo+tnhn(x) − g−1 +ϑo (x) +� +− DH +ϑoKΘ(h)(x), g−1 +ϑo (x) − y +������� + 1 +tn +���g−1 +ϑo+tnhn(x) − g−1 +ϑo (x) +���2� +. +For n → ∞, the first term tends to zero by Hadamard differentiability of KΘ whereas the second +term tends to zero by (G). Hence, CΘ is Hadamard differentiable at ϑo. The second assertion follows +from the functional delta method for Hadamard differentiable functionals (Römisch, 2006). +□ +C.3 +Proof of Proposition 4.3 +First note that, in comparison to Sections 2 and 3, the roles of X and Y are interchanged. The +universal Donsker property of F CΘ(ϑo) follows from Proposition 3.1(iii) since d ≤ 3 and CΘ(ϑo)(x, ·) +is (2, Λ)-Hölder for some Λ ≥ 0 uniformly in x ∈ X (Lemma 4.1). Moreover, note by measurability +of ϑn and continuity of CΘ near ϑo that cn is also measurable. By joint weak convergence (4.6) we +infer from Hadamard differentiability of CΘ at ϑo (Lemma 4.2) using the functional delta method +that the one-sample version of (JW) (recall Remark 2.4(ii)) is fulfilled. Further, since ϑn +P→ ϑo, as +n tends to infinity, we infer from Corollary 3.8 and Lemma 4.1 that the one-sample version of (Sup) +is also met. The assertion now follows at once from Theorem 2.2. +□ +C.4 +Proof of Proposition 4.5 +Note that by assumption, (T , dT ) and c fulfill the requirements of Theorem 2.6. Furthermore, note +that Assumption (Don) can be established via Proposition 3.9 and Assumption (KP) is implied +by the assumptions on the support of µ and ν (Staudt, Hundrieser, and Munk, 2022, Corollary 2). +Hence, the statement follows from Theorem 2.6. +□ +C.5 +Proof of Proposition 4.8 +Select X ⊆ Rd as a compact set which contains the supports of µ and ν. Note that Sd−1 is a compact +Polish space and consider the Lipschitz map cSd−1 : (Sd−1, ∥·∥) → C(X × X), θ �→ cθ|X×X whose +modulus depends on X and p. By compactness of X and Sd−1 it thus follows from the Theorem of +Arzelà-Ascoli that {cθ|X×X}θ∈Sd−1 is uniformly bounded and equicontinuous with a uniform modulus. +51 + +Therefore, upon choosing the function class F as in Theorem 2.5, Assertion (i) follows by Theo- +rem 2.5 once we verify that Assumption (Don) is fulfilled. To this end, note that log N(ε, Sd−1, ∥·∥) ≲ +| log(ε)|. Moreover, define for θ ∈ Sd−1 the pseudo metric ˜dθ,X(x, y) = |θT x−θTy| on X which fulfills +supθ∈Sd−1 N(ε, X, ˜dθ,X) ≲ ε−1 and for any x, x′, y ∈ X, +���cθ(x, y) − cθ(x′, y) +��� ≤ p diam(pθ(X))p−1|θT x − θT x′| ≤ p diam(X)p−1 ˜dθ,X(x, x′). +Since the upper bound for the Lipschitz modulus does not depend on θ, Proposition 3.9(ii) is appli- +cable and we conclude that � +θ∈Sd−1 F cθcθ and � +θ∈Sd−1 F cθ are universal Donsker. By applying the +continuous mapping theorem (van der Vaart and Wellner, 1996, Theorem 1.11.1) for the integration +operator over Sd−1 we obtain Assertion (ii). Finally, Assertion (iii) follows from Theorem 2.7. +□ +C.6 +Proof of Proposition 4.9 +Since X × Y is compact and by continuity of c and ∆c there exists a common modulus of continuity +w for {ct(·, y)}y∈Y,t∈[0,1]. Hence, for any t ∈ [0, 1] we have c, ct ∈ C(∥c∥∞ + ∥∆c∥∞ + 1, w) (see +Lemma 6.2 for the definition of C(·, ·)). Consequently, we infer by Lemma 6.2 the inequalities, +1 +t + +inf +π∈Π⋆ct(µt,νt) π(t∆c) + +sup +f∈Sc(µ,ν) +t∆µ( f cc) + t∆ν( f c) + +≤1 +t (OT(µt, νt, ct) − OT(µ, ν, c)) +≤1 +t + +inf +π∈Π⋆c (µ,ν) π(t∆c) + +sup +f∈Sct(µt,νt) +t∆µ( f cc) + t∆ν( f c) + sup +f∈F +t∆µ( f ctct − f cc) + t∆ν( f ct − f c) + . +Next, we observe that ∆µ = ˜µ − µ for some ˜µ ∈ P(X). This yields using Lipschitzianity under cost +transformations with respect to the cost function (Lemma 6.1) that +sup +f∈F +���∆µ( f ctct − f cc) +��� = sup +f∈F +���(˜µ − µ)( f ctct − f cc) +��� ≤ 4 ∥ct − c∥∞ = 4t +���∆c���∞ +t→0 +−−−→ 0. +Likewise, it follows that | sup f∈F ∆ν( f ct − f c)| → 0 for t → 0. Finally, since the pair (µt, νt) weakly +converges for t ց 0 to (µ, ν) it follows by Lemma 6.3 that +lim inf +tց0 +inf +π∈Π⋆ct(µt,νt) π(∆c) + +sup +f∈Sc(µ,ν) +∆µ( f cc) + ∆ν( f c) +≥ +inf +π∈Π⋆c (µ,ν) π(∆c) + +sup +f∈Sc(µ,ν) +∆µ( f cc) + ∆ν( f c) +as well as +lim sup +tց0 +inf +π∈Π⋆c (µ,ν) π(∆c) + +sup +f∈Sct(µt,νt) +∆µ( f cc) + ∆ν( f c) +≤ +inf +π∈Π⋆c (µ,ν) π(∆c) + +sup +f∈Sc(µ,ν) +∆µ( f cc) + ∆ν( f c), +which yields the claim. +□ +52 + +D +Proofs for Section 5: Regularity Elevation Functionals +D.1 +Proof of Proposition 5.1 +By the functional delta method (Römisch, 2006) and the assumptions on Ψ and L it follows that +an +� ( fn − f) +(Ψ( fn) − f) +� +⇝ +� +L +DH +f Ψ(L) +� +d= +�L +L +� +for n → ∞. +The continuous mapping theorem (van der Vaart and Wellner, 1996, Theorem 1.11.1) in combi- +nation with measurability of the random elements fn and Ψ( fn) (due to continuity Ψ near f) thus +asserts +an +�Ψ( fn) − fn +� P→ 0 +for n → ∞. +□ +D.2 +Proof of Proposition 5.2 +First note that Ψ(˜c) ∈ C(X×Y) for any ˜c ∈ C(X×Y) as a concatenation of continuous functions and +under ∥˜c∥∞ < 2 that Ψ(˜c) = ˜c, which yields Ψ(c) = c. In particular, this shows that Ψ: C(X × Y) → +C(X × Y) is continuous near c. For Hadamard differentiability at c consider a positive sequence +tn ց 0 and take a converging sequence (hn)n∈N ⊆ C(X × Y) with limit h. Since h is bounded and +∥c∥∞ ≤ 1, for n sufficiently large we have ∥c + tnhn∥∞ < 2 and therefore Ψ(c + tnhn) = c + tnhn. We +then obtain +����� +Ψ(c + tnhn) − Ψ(c) +tn +− h +�����∞ += ∥hn − h∥∞ → 0. +Finally, since for any ˜c ∈ C(X × Y) it holds that ∥gΨ(˜c)∥∞ ≤ B + 2 where B ≔ supg∈G ∥g∥∞ we find +for a finite space X that +sup +˜c∈C(X×Y) +log N(ε, GΨ(˜c), ∥·∥∞) ≤ |X|(log(B + 2) + | log(ε)|) ≲ | log(ε)|. +□ +D.3 +Proof of Proposition 5.3 +By condition (5.2) it follows for x, x′ ∈ X with ˜dX(x, x′) = 0 that c(x, y) = c(x′, y), whereas under +˜dX(x, x′) > 0 we have by w(δ) > 0 for δ > 0 that +c(x, y) ≤ c(x′, y) + w( ˜dX(x, x′)) < c(x′, y) + 2w( ˜dX(x, x′)). +This asserts for any (x, y) ∈ X × Y that +S (c, (x, y)) ≔ arg min +x′∈X +c(x′, y) + 2w +� ˜dX(x, x′) +� += {x′′ ∈ X | ˜d(x, x′′) = 0}, +and overall yields by ∥c∥∞ ≤ 1 that Ψ(c) = c. +For the second and third claim, recall from Proposition 5.2 that Ψbdd : C(X × Y) → C(X × Y) +is continuous near c and Hadamard differentiable at c with derivative IdC(X×Y). Hence, it suffices to +verify that Ψw◦ ˜dX +mod is continuous near c and Hadamard directionally differentiable with DH +|c Ψ|C( ˜X×Y) = +IdC( ˜X×Y) for which we rely on Lemma A.1 and Theorem A.2. Define the spaces V ≔ C(X × Y), +F = X, Θ = ˜X × Y and the functional +Ew◦ ˜dX : V × F × Θ = C(X × Y) × X × ( ˜X × Y) �→ R, +(˜c, x′, (x, y)) �→ −˜c(x′, y) − 2w( ˜dX(x, x′)). +53 + +For any ˜c ∈ C(X×Y) the function Ew◦ ˜dX(˜c, ·, ·): X×( ˜X×Y) → R is continuous as a sum of continu- +ous functions. Further, for any (x′, (x, y)) ∈ X×( ˜X×Y) note that the function Ew◦ ˜dX(·, x′, (x, y)): C(X× +Y) → R is 1-Lipschitz under uniform norm and that +∆cEw◦ ˜dX(˜c, x′, (x, y)) ≔ Ew◦ ˜dX(˜c + c, x′, (x, y)) − Ew◦ ˜dX(c, x′, (x, y)) = −˜c(x′, y) +is linear in ˜c ∈ C(X × Y). Hence, by Lemma A.1 we obtain continuity of the functional +Ψw◦ ˜dX +mod : C(X × Y) → C( ˜X × Y), +˜c �→ +� +(x, y) �→ inf +x′∈X ˜c(x′, y) + 2w( ˜dX(x, x′)) = − sup +x′∈X +Ew◦ ˜dX(˜c, x′, (x, y)) +� +. +Consider the closed sub-vector space U ≔ C( ˜X × Y) ⊆ C(X × Y), cf. Lemma F.2. It remains to +show Assumption (DC) of Theorem A.2. To this end, note for h ∈ C( ˜X × Y) that +h(x, y) + 2w( ˜dX(x, x′)) = h(x′, y) + 2w( ˜dX(x′, x′)) +for any x, x′ ∈ S (c, (x, y)) +since ˜dX(x, x′) = 0. This implies by Lemma A.3 that (DC) is fulfilled. Theorem A.2 thus asserts +that Ψw◦ ˜dX +mod is Hadamard directionally differentiable at c with derivative given by +DH +|c Ψw◦ ˜dX +mod : C(X × Y) → C( ˜X × Y), +h �→ +(x, y) �→ +inf +x′ : ˜dX(x′,x)=0 +h(x′, y) = − sup +x′ : ˜dX(x′,x)=0 +−∆cEw◦ ˜dX(h, x′, (x, y)) + . +Hence, if h ∈ C( ˜X × Y), then DH +|cΨw◦ ˜dX +mod (h) = h, which yields DH +|c Ψw◦ ˜dX +mod |C( ˜X×Y) = IdC( ˜X×Y). +For the last claim note that any ˜c ∈ C(X × Y) fulfills for (x, y) ∈ X × Y that +− ∥˜c∥∞ ≤ Ψw◦ ˜dX +mod (˜c)(x, y) ≤ ˜c(x, y) ≤ ∥˜c∥∞ , +and hence ∥Ψ(˜c)∥∞ = ∥Ψw◦ ˜dX +mod ◦ Ψbdd(˜c)∥∞ ≤ 2. Further, for any x, x′ ∈ X, y ∈ Y we have +Ψw◦ ˜dX +mod (˜c)(x, y) − Ψw◦ ˜dX +mod (˜c)(x′, y) ≤ inf +x′′∈X c(x′′, y) + 2w( ˜dX(x′′, x)) − c(x′′, y) − 2w( ˜dX(x′′, x′)) +≤ 2w( ˜dX(x, x′)), +(D.1) +where we used the reverse triangle inequality since w ◦ ˜dX defines a (pseudo-)metric on X. We +thus conclude for any ˜c ∈ C(X × Y) and a bounded function class G with B ≔ supg∈G ∥g∥∞ < ∞ +from ∥Ψ(˜c)∥∞ ≤ 2 and (D.1) that the elements of GΨ(˜c) are bounded by B + 2 and 2-Lipschitz under +w ◦ ˜dX as an infimum over such 2-Lipschitz functions. Hence, GΨ(˜c) ⊆ BL(B+2),2(X, w ◦ ˜dX) where +for the latter class uniform metric entropy bounds are available by Kolmogorov and Tikhomirov +(1961, Section 9), asserting for any ε > 0 +N(ε, BL(B+2),2(X, w ◦ ˜dX), ∥·∥∞) = N(ε/2, BL(B+2)/2,1(X, w ◦ ˜dX), ∥·∥∞) +≲ N(ε/8, X, w ◦ ˜dX)| log(ε)|. +□ +D.4 +Proof of Corollary 5.4 +We infer from Proposition 5.3 that Ψ ≔ B · Ψw◦dX/B +mod +◦ Ψbdd(·/B) is continuous near c and Hadamard +differentiable at c with derivative DH +|cΨ = IdC(X×Y). Hence, invoking Proposition 5.1 it follows +that an(cn − cn) +P→ 0 for n → ∞. Moreover, by definition of Ψbdd and Ψw/B,dX +mod +it follows that +∥cn∥∞ ≤ 2B and that cn fulfills (5.2) with w replaced by 2w. The inclusion now follows at once from +Lemma 2.1. +□ +54 + +D.5 +Proof of Proposition 5.5 +Since c is (γ, 1)-Hölder it follows for any x, x′ ∈ X and y ∈ Y as in Lemma A.4 of Hundrieser, +Staudt, and Munk (2022) by convexity of X that +c(x, y) = c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + Rx′(x) +with +|Rx′(x)| ≤ +√ +d +���x − x′���γ , +and consequently, for x � x′ we obtain +c(x, y) < c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 +√ +d +���x − x′���γ . +This asserts for any (x, y) ∈ X × Y that +S (c, (x, y)) ≔ arg min +x′∈X +c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 +√ +d +���x − x′���γ = {x} +and yields by ∥c∥∞ ≤ 1 that Ψ(c) = c. +To show the claim on continuity and Hadamard differentiability it suffices to verify that ΨHol is +continuous near c and that is Hadamard differentiable at c with derivative DH +|c ΨHol = IdC(X×Y) for +which we rely on Lemma A.1 and Theorem A.2. Set V ≔ C(X × Y), F ≔ X and Θ ≔ X × Y and +define the functional +EHol : V × F × Θ = C(X × Y) × X × (X × Y) → R, +(˜c, x′, (x, y)) �→ − +� +˜c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 +√ +d +���x − x′���γ� +. +For any ˜c ∈ C(X × Y) the functional EHol(˜c, ·, ·): X × (X × Y) → R is continuous by continuity of +∇xc(·, ·) and for any (x′, (x, y)) ∈ X × (X × Y) the functional EHol(·, x′, (x, y)): C(X × Y) → R is +1-Lipschitz under uniform norm while +∆cEHol(˜c, x′, (x, y)) = EHol(˜c + c, x′, (x, y)) − EHol(c, x′, (x, y)) = −˜c(x′, y) +is linear in ˜c ∈ C(X × Y). Finally, condition (DC) follows by Lemma A.3 since S (c, (x, y)) = {x} is +a singleton. Hence, by Lemma A.1 and Theorem A.2 the functional +ΨHol : C(X × Y) → C(X × Y), ˜c �→ +� +(x, y) �→ − sup +x′∈X +EHol(˜c, x′, (x, y)) +� +is continuous near c and Hadamard differentiable at c with derivative +DH +|cΨHol : C(X × Y) → C( ˜X × Y), +h �→ +� +(x, y) �→ h(x, y) = −∆cEHol(h, x′, (x, y)) +� +. +For the claim on the uniform metric entropy bound let ˜c ∈ C(X × Y), and assume (after applica- +tion of Ψbdd) that ∥˜c∥∞ ≤ 2. Define the collection of functions ( ˜Ex′,y)x′∈X,y∈Y with +˜Ex′,y : X → R, +x �→ ˜c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 +√ +d +���x − x′���γ , +which is (γ, 2)-Hölder on X. Hence, by Hundrieser, Staudt, and Munk (2022, Lemma A.5) there +exists another collection ( ˜Eσ +x′,y)x′∈X,y∈Y,σ∈(0,1] of smooth functions on X such that +sup +x′∈X +y∈Y +���� ˜Ex′,y − ˜Eσ +x′,y +����∞ ≤ Kσγ +and +sup +x′∈X +y∈Y +���� ˜Eσ +x′,y +����C2(X) ≤ Kσγ−2, +(D.2) +for all σ > 0 and some independent K > 0. Here, the C2(X)-norm of a twice continuously differen- +tiable function g : X ⊂ Rd → R is defined as +∥g∥C2(X) ≔ max +|β|≤2 +���Dβg +���∞ , +where Dβg = ∂|β|g/∂xβ1 +1 · · · xβd +d +for β ∈ Nd +0. +55 + +Note that a function with ∥g∥C2(X) ≤ Γ for Γ > 0 is absolutely bounded by Γ, it is Γ-Lipschitz, and +dΓ-semi-concave (for a formal definition see Albano, 2002 or Hundrieser, Staudt, and Munk, 2022), +since the Eigenvalues of its Hessian are bounded by d · Γ. Upon defining c(x, y) ≔ Ψ(˜c)(x, y) = +infx′∈X ˜Ex′,y(x) and cσ(x, y) ≔ infx′∈X ˜Eσ +x′,y(x) we thus obtain from (D.2) that +���c − cσ���∞ ≤ Kσγ and +that cσ is semi-concave of order Γ(σ) ≔ dKσγ−2. Hence, following along the lines the proofs of +Lemma A.4 in Hundrieser, Staudt, and Munk, 2022 we obtain for any ε > 0 and σ(ε) ≔ (ε/4K)1/γ +that +N(ε, Gc, ∥·∥∞) ≤ N(ε/2, Gcσ(ε), ∥·∥∞) = N + +ε +2Γ(σ(ε)), +Gcσ(ε) +2Γ(σ(ε)), ∥·∥∞ + ≲ +� +ε +2Γ(σ(ε)) +�−d/2 +≲ ε−d/γ. +Here, we used in the second inequality that Gcσ(ε)/2Γ(σ(ε)) is contained in the collection of functions +on X which are absolutely bounded by B ≥ 0, Lipschitz with modulus L ≥ 0 and 1-semi-concave, +where B depends on G and L depends on X, in conjunction with uniform metric entropy bounds by +Bronshtein (1976) and Guntuboyina and Sen (2013) for convex functions. In particular, since the +hidden constants do not depend on ˜c, the claim follows. +□ +D.6 +Proof of Proposition 5.6 +For the first claim note that Ψi(ci) = ci for each i ∈ {1, . . . , I} and consequently, it follows for +x ∈ ζi(Ui) that Ψi(ci)(ζ−1 +i (x), y) = c(x, y). Hence, since �I +i=1 ηi(x) ≡ 1 it follows that Ψ(c) = c. +The claim on continuity of Ψ near c follows by continuity of the functionals Ψi : C(Ui × Y) → +C(Ui × Y) near ci for each 1 ≤ i ≤ I. +For the claim on Hadamard differentiability of Ψ define for each i ∈ {1, . . . , I} the functionals +Ψ1 +com,i: C(X × Y) → C(Ui × Y), +˜c �→ ((u, y) �→ ˜c(ζi(u), y)) , +Ψ2 +com,i: C(Ui × Y) → C(ζi(Ui) × Y), +˜c �→ +� +(x, y) �→ ˜c(ζ−1 +i (x), y) +� +, +where both maps assign to the respective spaces of continuous functions since ζ−1 +i +and ζi are both +continuous. Further, note for any ˜c ∈ C(X × Y) that Ψ(˜c) = �I +i=1 ηi · Ψ2 +com,i ◦ Ψi ◦ Ψ1 +com,i(˜c). Both +functionals Ψ1 +com,i and Ψ2 +com,i are Hadamard differentiable at ci with derivative +DH +|ciΨ1 +com,i: C(X × Y) → C(Ui × Y), +h �→ ((u, y) �→ h(ζi(u), y)) , +DH +|ciΨ2 +com,i: C(Ui × Y) → C(ζi(Ui) × Y), +h �→ +� +(x, y) �→ h(ζ−1 +i (x), y) +� +. +By assumption on Ψi and chain rule we infer that Ψ is Hadamard differentiable at c with derivative +DH +c Ψ: C(X × Y) → C(X × Y), +h �→ +(x, y) �→ +I� +i=1 +ηi(x)h(ζ−1 +i (ζi(x)), y) = +I� +i=1 +ηi(x)h(x, y) = h(x, y) + +and conclude that DH +|c Ψ = IdC(X×Y). +Finally, the bound on the covering numbers is a consequence of Lemma 3.1 and Lemma A.1 in +Hundrieser, Staudt, and Munk, 2022 as they assert for arbitrary ˜c ∈ C(X × Y) that +log N(ε, GΨ(˜c), ∥·∥∞) ≤ +I� +i=1 +log N(ε, GΨ(˜c)|ζi(Ui), ∥·∥∞) +≤ +I� +i=1 +log N(ε, GΨi(˜c) ◦ ζi, ∥·∥∞) += +I� +i=1 +log N(ε, GΨi(˜c(ζi(·),·)), ∥·∥∞). +□ +56 + +E +Proofs for Section 6: Lemmata of Distributional Limits +E.1 +Proof of Lemma 6.1 +Assume ∥f − ˜f∥∞ + ∥c − ˜c∥∞ < ∞ since otherwise the claim is vacuous. For ˜f and ˜c there exists for +y ∈ Y and ε > 0 some x′ ∈ X such that ˜f ˜c(y) ≥ ˜c(x′, y) − ˜f(x′) − ε. Hence, +f c(y) − ˜f ˜c(y) = +� +inf +x∈X c(x, y) − f(x) +� +− +� +inf +x∈X ˜c(x, y) − ˜f (x) +� +≤ c(x′, y) − f(x′) − ˜c(x′, y) + ˜f(x′) + ε +≤ +��� f − ˜f +���∞ + ∥c − ˜c∥∞ + ε. +As ε > 0 can be chosen arbitrarily small, we obtain for any y ∈ Y the inequality +f c(y) − ˜f ˜c(y) ≤ +���f − ˜f +���∞ + ∥c − ˜c∥∞ . +Repeating the argument for f and c asserts the converse inequality and proves the claim. +□ +E.2 +Proof of Lemma 6.2 +Let us start by splitting the problem in two different ways, +OT(˜µ, ˜ν, ˜c) − OT(µ, ν, c) = (OT(˜µ, ˜ν, ˜c) − OT(˜µ, ˜ν, c)) + (OT(˜µ, ˜ν, c) − OT(µ, ν, c)) += (OT(˜µ, ˜ν, ˜c) − OT(µ, ν, ˜c)) + (OT(µ, ν, ˜c) − OT(µ, ν, c)). +Since c, ˜c ∈ C(2 ∥c∥∞ + 1, 2w), we can employ the dual representation of the OT value from +Lemma 2.1 with F = F (2 ∥c∥∞ + 1, 2w). Hence, for each bracket in the display above, one can +choose to plug-in a feasible plan in the primal formulation or a potential from F in the dual formu- +lation to obtain upper and lower bounds. Doing so, we obtain +inf +π∈Π⋆ +˜c (˜µ,˜ν) π(˜c − c) ≤ OT(˜µ, ˜ν, ˜c) − OT(˜µ, ˜ν, c) ≤ +inf +π∈Π⋆c (˜µ,˜ν) π(˜c − c), +sup +f∈Sc(µ,ν) +(˜µ − µ) f cc + (˜ν − ν) f c ≤ OT(˜µ, ˜ν, c) − OT(µ, ν, c) ≤ +sup +f∈Sc(˜µ,˜ν) +(˜µ − µ) f cc + (˜ν − ν) f c, +OT(µ, ν, ˜c) − OT(µ, ν, c) ≤ +inf +π∈Π⋆c (µ,ν) π(˜c − c), +OT(˜µ, ˜ν, ˜c) − OT(µ, ν, ˜c) ≤ +sup +f∈S˜c(˜µ,˜ν) +(˜µ − µ) f ˜c˜c + (˜ν − ν) f ˜c. +In particular, for the last upper bound we further note that +sup +f∈S˜c(˜µ,˜ν) +(˜µ−µ) f ˜c˜c +(˜ν−ν) f ˜c ≤ +sup +f∈S˜c(˜µ,˜ν) +(˜µ−µ) f cc +(˜ν−ν) f c + sup +f∈F +(˜µ−µ)( f ˜c˜c − f cc)+(˜ν−ν)( f ˜c − f c), +which overall yields the lower and upper bounds for the OT cost under varying measures and costs. +Finally, the bound under fixed measures µ, ν it follows by Hölder’s inequality for any π ∈ Π(µ, ν) +that |π(˜c − c)| ≤ ∥˜c − c∥∞, whereas under a fixed cost function c we have +sup +f∈Sc(˜µ,˜ν)∪Sc(µ,ν) +���(˜µ − µ) f cc + (˜ν − ν) f c��� ≤ sup +f∈F +���(˜µ − µ) f cc��� + sup +f∈F +���(˜ν − ν) f c��� += sup +f∈F cc +���(˜µ − µ) f +��� + sup +f∈F c +���(˜ν − ν) f +���. +□ +57 + +E.3 +Proof of Lemma 6.3 +The continuity of T1 is a consequence of Villani (2008, Theorem 5.20). Indeed, any converging +sequence (µn, νn, cn) with limit (µ∞, ν∞, c∞) admits a sequence of OT plans πn ∈ Π⋆ +cn(µn, νn) which +converges weakly along a subsequence, say (πnk)k∈N, to an OT plan π∞ ∈ Π⋆ +c∞(µ∞, ν∞). Hence, +lim sup +k→∞ +|T1(µnk, νnk, cnk) − T1(µ∞, ν∞, c∞)| = lim sup +k→∞ +|OT(µnk, νnk, cnk) − OT(µ∞, ν∞, c∞)| += lim sup +k→∞ +|πnk(cnk) − π∞(c∞)| +≤ lim sup +k→∞ +|(πnk − π∞)(c∞)| + +���cnk − c∞ +���∞ = 0. +Since this holds for any sequence of converging OT plans, continuity of T1 follows from Lemma F.1. +For the lower semi-continuity of T2 take a sequence (hc,n)n∈N with limit hc,∞ and consider OT +plans πn ∈ Π⋆ +cn(µn, νn) such that +inf +π∈Π⋆cn(µn,νn) π(hc,n) ≥ πn(hc,n) − 1/n. +Then, by Villani (2008, Theorem 5.20) a converging subsequence (πnk)k∈N with limit π∞ ∈ Π⋆ +c∞(µ∞, ν∞) +exists and it follows that +lim inf +k→∞ T2(µnk, νnk, cnk, hc,nk) = lim inf +k→∞ +inf +π∈Π⋆cnk (µnk,νnk) π(hc,nk) +≥ lim inf +k→∞ πnk(hc,nk) − 1/nk +≥ lim inf +k→∞ πnk(hc,∞) − +���hc,∞ − hc,nk +���∞ − 1/nk += π∞(hc,∞) ≥ T2(µ∞, ν∞, c∞, hc,∞). +Consequently, by Lemma F.1, lower semi-continuity of T2 follows. To infer upper semi-continuity +of T2, and thus continuity, at (µ∞, ν∞, c∞, hc,∞) under the assumption of a unique OT plan π⋆ ∈ +Π⋆ +c∞(µ∞, ν∞) note by Villani (2008, Theorem 5.20) that for any sequence of OT plans πn ∈ Π⋆ +cn(µn, νn) +there exists a weakly converging subsequence πnk which tends to π⋆ for k → ∞. Hence, we con- +clude that +lim sup +k→∞ +T2(µnk, νnk, cnk, hc,nk) = lim sup +k→∞ +inf +π∈Π⋆cnk (µnk,νnk) π(hc,nk) +≤ lim sup +k→∞ +πnk(hc,nk) +≤ lim sup +k→∞ +πnk(hc,∞) − +���hc,∞ − hc,nk +���∞ += π∞(hc,∞) = T2(µ∞, ν∞, c∞, hc,∞). +This implies by Lemma F.1 the upper semi-continuity of T2. Moreover, for fixed (µ′, ν′, c′) the map +T2 is continuous in hc since for any ˜hc it holds that +|T2(µ, ν, c, hc) − T2(µ, ν, c, ˜hc)| ≤ +���hc − ˜hc +���∞ . +To show upper semi-continuity of T3 take a sequence (hµ,n, hν,n)n∈N with limit (hµ,∞, hν,∞). Fur- +ther, by definition of C, it follows from Lemma 2.1 that any cn ∈ C fulfills Hcn ⊆ F cncn ⊆ F . Take +a sequence fn ∈ Scn(µn, νn) ⊆ F such that +T3(µn, νn, cn, hµ,n, hν,n) ≤ hµ( fn) + hν( fn) + 1/n. +58 + +By compactness of F there exists a uniformly converging subsequence, say ( fnk)k∈N, with limit +f∞ ∈ F . Next, we demonstrate that f∞ ∈ Sc∞(µ∞, ν∞). To this end, we note +OT(µ∞, ν∞, c∞) ≥ µ∞( f c∞c∞ +∞ +) + ν∞( f c∞ +∞ ) += lim +k→∞ µnk( f c∞c∞ +∞ +) + νnk( f c∞ +∞ ) +≥ lim +k→∞ µnk( f +cnkcnk +nk +) + νnk( f +cnk +nk ) − +���� f c∞c∞ +∞ +− f +cnkcnk +nk +����∞ − +���� f c∞ +∞ − f +cnk +nk +����∞ += lim +k→∞ OT(µnk, νnk, cnk) − +����f c∞c∞ +∞ +− f +cnkcnk +nk +����∞ − +���� f c∞ +∞ − f +cnk +nk +����∞ += OT(µ∞, ν∞, c∞), +where the last equality follows by continuity of T1. Hence, we get f∞ ∈ Sc∞(µ∞, ν∞). +By continuity of hµ and hν on F and upon denoting the norm on Cu(F ) by ∥·∥F , we infer that +lim sup +k→∞ +T3(µnk, νnk,cnk, hµ,nk, hν,nk) = lim sup +k→∞ +sup +f∈Scnk (µnk,νnk) +hµ,nk( f) + hν,nk( f) +≤ lim sup +k→∞ +hµ,nk( fnk) + hν,nk( fnk) + 1/nk +≤ lim sup +k→∞ +hµ,∞( fnk) + hν,∞( fnk) + +���hµ,∞ − hµ,nk +���F + +���hν,∞ − hν,nk +���F + 1/nk += hµ,∞( f∞) + hν,∞( f∞) ≤ T3(µ∞, ν∞, c∞, hµ,∞, hν,∞) +and consequently, by Lemma F.1, upper semi-continuity of T3 follows. Further, for fixed (µ′, ν′, c′) +the map T3 is continuous in (hµ, hν) since for another (˜hµ, ˜hν) it holds that +|T3(µ, ν, c, hµ, hν) − T3(µ, ν, c, ˜hµ, ˜hν)| ≤ +���˜hµ − ˜hµ +���F cc + +���˜hν − ˜hν +���F c . +Finally, for T4 take (h1,µ, ˜h1,µ, h1,ν, ˜h1,ν), (h2,µ, ˜h2,µ, h2,ν, ˜h2,ν) ∈ Cu(F )4 and note that +|T4(h1,µ, ˜h1,µ, h1,ν, ˜h1,ν) − T4(h2,µ, ˜h2,µ, h2,ν, ˜h2,ν)| +≤ +���h1,µ − h2,µ +���F + +���˜h1,µ − ˜h2,µ +���F + +���h1,ν − h2,ν +���F + +���˜h1,ν − ˜h2,ν +���F , +which asserts continuity. +□ +E.4 +Proof of Lemma 6.4 +For (i) take f, g ∈ G, then |µ( f) − µ(g)| ≤ ∥f − g∥∞ and hence µ: G → R defines a Lipschitz map +under uniform norm which asserts µ ∈ Cu(G). Assertion (ii) follows from Giné and Nickl (2016, p. +17). Finally, (iii) follows from (ii) since for any g ∈ G the evaluations µn(g) = n−1 �n +i=1 g(Xi) and +µb +n,k(g) = k−1 �k +i=1 g(Xb +i ) are Borel measurable. +□ +E.5 +Proof of Lemma 6.5 +We first prove that Assumption (JW) implies for n, m → ∞ with m/(n + m) → λ ∈ (0, 1) that + +√n +� +(µn − µ)( f cc) +� +f∈F +√m +� +(νm − ν)( f c) +� +f∈F +� +nm +n+m(cn,m − c) + += + +� +Gµ +n( f cc) +� +f∈F +� +Gν +m( f c) +� +f∈F +Gc +n,m + +⇝ + +� +Gµ( f cc) +� +f∈F +� +Gν( f c) +� +f∈F +Gc + +(E.1) +59 + +in the Polish space Cu(F ) × Cu(F ) × C(X × Y). To this end, consider the map +Ψ: Cu(F cc) × Cu(F c) × C(X × Y) → Cu(F ) × Cu(F ) × C(X × Y), +(α, β, γ) �→ +��α( f cc)� +f∈F , �β( f c)� +f∈F , γ +� +. +This map is well-defined (i.e., its range is correct) since for any (α, β) ∈ Cu(F cc) × Cu(F c) there +exist moduli of continuity wα, wβ: R+ → R+ such that for f, ˜f ∈ F it follows by Lemma 6.1 that +|α( f cc) − α( ˜f cc)| ≤ wα( +��� f cc − ˜f cc���∞) ≤ wα( +��� f − ˜f +���∞), +|β( f c) − β( ˜f c)| ≤ wβ( +��� f c − ˜f c���∞) ≤ wβ( +��� f − ˜f +���∞), +which assert that ((α( f cc))f∈F , (β( f c))f∈F , γ) ∈ Cu(F ) × Cu(F ) × C(X × Y). Moreover, for any +(α, β), (˜α, ˜β) ∈ Cu(F cc) × Cu(F c) we have +sup +f∈F +|α( f cc) − ˜α( f cc)| = sup +˜f ∈F cc +|α( ˜f ) − ˜α( ˜f )| +and +sup +f∈F +|β( f c) − ˜β( f c)| = sup +˜f∈F c +|β( ˜f ) − ˜β( ˜f )|, +hence the map Ψ is continuous. Consequently, Assumption (JW) and the continuous mapping +theorem (van der Vaart and Wellner, 1996, Theorem 1.11.1) assert weak convergence (E.1). +Moreover, by Varadarajan (1958) the empirical measures (µn, νn) weakly converge a.s. in P(X)× +P(Y) to (µ, ν). Note that P(X) × P(Y) is by compactness of X and Y a separable, complete metric +space (Bolley, 2008). Invoking Slutzky’s lemma (van der Vaart and Wellner, 1996, Example 1.4.7) +in conjunction with (E.1) we thus obtain the first claim. In particular, by measurability of cn,m and +Lemma 6.4, all involved quantities are Borel measurable. +For the second claim note by Lemma 6.1 that any realization of µn, νm and cn,m leads the pro- +cesses Gµ +n( f cn,mcn,m) and Gν +m( f cn,m) to be 2√n-Lipschitz and 2√m-Lipschitz in f, respectively. Thus, +they are uniformly continuous in f. Moreover, for fixed f ∈ F we can show that the function +˜Gµ +n : P(X) × C(X × Y) → R, +(˜µ, ˜c) �→ √n(˜µ − µ)( f ˜c˜c) +is upper semi-continuous (i.e., in particular measurable). Indeed, for ˜µk ⇝ ˜µ in P(X) and ˜ck → ˜c +in C(X × Y) it follows by Lemma 6.1, upper semi-continuity of f ˜c˜c and the Portmanteau Theorem +(van der Vaart and Wellner, 1996, Theorem 1.3.4) that +lim sup +k→∞ +√n(˜µk − µ)( f ˜ck˜ck) ≤ lim sup +k→∞ +√n(˜µk − µ)( f ˜c˜c) + 2 √n +��� f ˜ck˜ck − f ˜c˜c���∞ ≤ √n(˜µ − µ)( f ˜c˜c). +Hence, by Lemma 6.4(ii) we conclude that (Gµ +n( f cn,mcn,m))f∈F is Borel measurable. Likewise, we +conclude (Gν +m( f cn,m))f∈F is Borel measurable. +Consequently, by (Sup) we infer, for n, m → ∞, that +� +Gµ +n( f cc) − Gµ +n( f cn,mcn,m), Gν +m( f c) − Gν +m( f cn) +� +f∈F +P→ (0, 0) +in Cu(F )2. +The claim now follows by a combination of Slutzky’s lemma and the continuous mapping theorem +(van der Vaart and Wellner, 1996, Example 1.4.7, Theorem 1.11.1). +□ +E.6 +Proof of Lemma 6.8 +The first claim follows by an observation in Römisch (2006) since the set of probability measures +P(X) is convex. For additional insights see Aubin and Frankowska (1990, Proposition 4.2.1) +60 + +For the second claim consider a sequence ∆n = (˜µn − µ)/tn with tn > 0 and ˜µn ∈ P(X) such that +∥∆n − ∆∥ ˜F = sup f∈ ˜F |∆n( f) − ∆( f)| → 0. Then, it follows from triangle inequality that +���∆( f) − ∆( f ′) +��� = +���∆n( f) − ∆n( f ′) + (∆ − ∆n)( f) + (∆ − ∆n)( f ′) +��� +≤ +���∆n( f − f ′) +��� + 2 ∥∆ − ∆n∥ ˜F +Herein, the first term vanishes since ∆n( f − f ′) = (˜µn − µ)(κ)/tn = 0, whereas the second term +converges for n → ∞ to zero. Hence, ∆( f) = ∆( f ′). +The third claim relies on Portemanteau’s theorem (van der Vaart and Wellner, 1996, Lemma +1.3.4) which asserts using the notion of outer probabilities P∗ that +P +� +Gµ ∈ TµP(X) +� +≥ lim sup +n→∞ +P∗ � √n(µn − µ) ∈ TµP(X) +� += 1. +□ +E.7 +Proof of Lemma B.1 +We start by proving (i). Note for κ ∈ R that +( f + κ)c(y) = inf +x∈X c(x, y) − f(x) − κ = f c(y) − κ, +which yields the claim. To show assertion (ii), observe by Lemma 6.1 that +���g(c+∆c)(c+∆c)���∞ ≤ ∥g∥∞ + 2 +���c + ∆c���∞ ≤ B. +(E.2) +Further, we find that +− ∥c∥∞ − sup +x∈X +g(c+∆c)(c+∆c)(x) ≤ g(c+∆c)(c+∆c)c(y) ≤ ∥c∥∞ − sup +x∈X +g(c+∆c)(c+∆c)(x). +(E.3) +Using part (i) of this lemma, we obtain +g(c+∆c)(c+∆c)cc = +�� +g(c+∆c)(c+∆c)�c + sup +x∈X +g(c+∆c)(c+∆c)(x) +�c ++ sup +x∈X +g(c+∆c)(c+∆c)(x). +Combining (E.2) and (E.3) with the above equation demonstrates that g(c+∆c)(c+∆c)cc ∈ Hc + [−B, B] +and hence yields the claim. +□ +F +Elementary Analytical Results +Lemma F.1. Consider a real-valued sequence (an)n∈N and let K ∈ R. +(i) If for any subsequence (ank)k∈N there exists a subsequence (ankl)l∈N with lim supl→∞ ankl ≤ K, +then it follows that lim supn→∞ an ≤ K. +(ii) If for any subsequence (ank)k∈N there exists a subsequence (ankl )l∈N with lim infl→∞ ankl ≥ K, +then it follows that lim infn→∞ an ≥ K. +(iii) If for any subsequence (ank)k∈N there exists a subsequence (ankl )l∈N with liml→∞ ankl = K, +then it follows that limn→∞ an = K. +Proof. We only prove (i) and note that (ii) and (iii) can be shown analogously. +Assume that +lim supn→∞ an = infn∈N(supm≥n am) ≥ K + ε for some ε > 0. Since (supm≥n am)n∈N is decreas- +ing in n, this would imply that supm≥n am ≥ K + ε for all n ∈ N. Hence, there would exist a +subsequence of (an)n∈N, say (anl)l∈N, with anl ≥ K + ε/2 for all l ∈ N. However, this would assert +lim infl→∞ anl ≥ K + ε/2 > K, contradicting the assumption. Thus, lim supn→∞ an ≤ K. +□ +61 + +Lemma F.2. Let (X, dX) be a compact metric space and consider a continuous (pseudo-)metric +˜dX on X. Then, (X, ˜dX) is a compact (pseudo-)metric space. Moreover, given a Polish space Y it +follows that C((X, ˜dX) × Y) ⊆ C((X, dX) × Y). +Proof. The (pseudo-)metric properties are clearly fulfilled for (X, ˜dX). By continuity of ˜dX under dX +the canonical inclusion ι: (X, dX) → (X, ˜dX), x �→ x is continuous. As the image of a compactum +under a continuous map is again compact the first claim follows. For the second claim, take h ∈ +C((X, ˜dX) × Y). Then, the composition map X × Y → R, (x, y) �→ h(ι(x), y) is continuous and +therefore the canonical embedding h ◦ (ι, IdY) of h is included in C((X, dX) × Y). +□ +62 + diff --git a/KtAzT4oBgHgl3EQfVfzp/content/tmp_files/load_file.txt b/KtAzT4oBgHgl3EQfVfzp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a64cacb6b568c8cdfd351074b894a3a4500c61a1 --- /dev/null +++ b/KtAzT4oBgHgl3EQfVfzp/content/tmp_files/load_file.txt @@ -0,0 +1,2636 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf,len=2635 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='01287v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ST] 3 Jan 2023 Empirical Optimal Transport under Estimated Costs: Distributional Limits and Statistical Applications Shayan Hundrieser†, Gilles Mordant†, Christoph A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Weitkamp†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' and Axel Munk† ‡ ∗ †Institute for Mathematical Stochastics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' University Göttingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Goldschmidtstraße 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 37077 Göttingen ‡Max Planck Institute for Multidisciplinary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Am Faßberg 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 37077 Göttingen ∗Cluster of Excellence ”Multiscale Bioimaging: from Molecular Machines to Networks of Excitable Cells”(MBExC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' University Medical Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Robert-Koch-Straße 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 37075 Göttingen January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2023 Abstract Optimal transport (OT) based data analysis is often faced with the issue that the underlying cost function is (partially) unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This paper is concerned with the derivation of distributional limits for the empirical OT value when the cost function and the measures are estimated from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For statistical inference purposes, but also from the viewpoint of a stability analysis, understanding the fluctuation of such quantities is paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Our results find direct application in the problem of goodness-of-fit testing for group families, in machine learning applications where invariant transport costs arise, in the problem of estimating the distance between mixtures of distributions, and for the analysis of empirical sliced OT quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The established distributional limits assume either weak convergence of the cost process in uniform norm or that the cost is determined by an optimization problem of the OT value over a fixed parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the first setting we rely on careful lower and upper bounds for the OT value in terms of the measures and the cost in conjunction with a Skorokhod representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The second setting is based on a functional delta method for the OT value process over the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proof techniques might be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1 Introduction Statistically sound methods for data analysis relying on the optimal transport (OT) theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', Rachev and Rüschendorf (1998), Villani (2008), and Santambrogio (2015)) have won acclaim in re- cent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Exemplarily, we mention fitting of generative adversarial networks (Arjovsky, Chintala, and Bottou, 2017), novel notions of multivariate quantiles (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hallin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2021) and dependence (Nies, Staudt, and Munk, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Mordant and Segers, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Wiesel, 2022) or tools for causal inference (Torous, Gunsilius, and Rigollet, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Recall that for Polish spaces X and Y and a continuous cost function c: X × Y → R, the OT value between two (Borel) probability measures µ ∈ P(X) and ν ∈ P(Y) is defined as OT(µ, ν, c) ≔ inf π∈Π(µ,ν) � X×Y c(x, y) dπ(x, y), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) where Π(µ, ν) denotes the set of couplings of µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under mild assumptions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) also admits a dual formulation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', Santambrogio 2015), OT(µ, ν, c) = sup f∈C(X) � X f cc(x) dµ(x) + � Y f c(y) dν(y), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Optimal Transport, central limit theorem, stability analysis, curse of dimensionality, empiri- cal process, bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' MSC 2020 subject classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Primary: 60B12, 60F05, 60G15, 62E20, 62F40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Secondary: 90C08, 90C31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1 where C(X) stands for the set of real-valued, continuous functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, f c(y) ≔ infx∈X c(x, y)− f(x) and f cc(x) ≔ infy∈Y c(x, y) − f c(y) denote cost-transformations of f and f c under c, respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' also often referred to as c-transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' If X = Y and the cost function c = dp X is the p-th power (p ≥ 1) of a metric dX on X the OT value gives rise to the p-Wasserstein distance Wp(µ, ν) ≔ (OT(µ, ν, dp X))1/p, which defines a metric on the space of probability measures with p-th moments (Villani, 2008, Chapter 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This metric is particularly useful for many data analysis tasks due to its potential awareness of the “inner geometry” of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For instance, interpreting (normalized) images, or more precisely the corresponding pixel locations and intensities, as probability measures, it has been argued that the distance induced by OT corresponds to the natural expectations of what appears close or far away for the human eye (Rubner, Tomasi, and Guibas, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Meanwhile, there is a plenitude of real world showcases where OT based distances (and their associated transport plans) prove useful for applications e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', in cell biology (Tameling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2021), genetics (Evans and Matsen, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Schiebinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2019), protein structure analysis (Gellert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Weitkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2022) or fingerprint analysis (Sommerfeld and Munk, 2018), to mention but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In these works, the cost function is a given known quantity which is determined by the concrete application, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', a tree distance on the space of phylogenetic trees as in Evans and Matsen (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, despite the various successful applications hinted at above, there are situations in which the underlying cost naturally depends on the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In certain problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', Wasser- stein based goodness-of-fit testing under group families (Hallin, Mordant, and Segers, 2021) or Wasserstein Procrustes analysis (Grave, Joulin, and Berthet, 2019), it is central that the underly- ing OT problem is invariant with respect to certain transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This can only be realized by measure-dependent costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for sliced OT (Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2015), the Wasserstein distance between multiple one-dimensional projections of measures is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Taking the maximum over all directions gives rise to the max-sliced Wasserstein distance (Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2019) which can be viewed in the framework of OT with measure-dependent costs since maximizing directions are determined by the underlying measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Motivated by these considerations, we provide in this work a general framework for the statistical analysis of empirical OT problems under costs that are dependent on the underlying measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Adopting this statistical point of view, we assume that we do not have access to the measures µ and ν but only to independent samples {Xi}n i=1 ∼ µ⊗n and {Yi}m i=1 ∼ ν⊗m with n, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Upon defining the empirical measures µn ≔ 1 n �n i=1 δXi and νm ≔ 1 m �m i=1 δYi and given a random cost function1 cn,m such that OT(µn, νm, cn,m) estimates the quantity OT(µ, ν, c), our main focus is on characterizing for n, m → ∞ with m/(n + m) → λ ∈ (0, 1) the limit distribution of � nm n + m � OT(µn, νm, cn,m) − OT(µ, ν, c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) This is of particular interest for asymptotic tests about the relation between µ and ν for unknown c based on the OT value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, this enables the derivation of confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As it is practi- cally more relevant, we mainly focus on the scenario where both measures µ and ν are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, we stress that our theory also provides distributional limits for the one-sample case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', when only µ is estimated from data while ν is assumed to be known (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 and Re- mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, although we mostly focus on empirical measures to estimate the underlying measures, our theory also enables the derivation of distributional limits for alternative measure esti- mators, provided that the corresponding distributional limits can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1Here, cn,m is either a direct estimator for c or chosen via an OT-related optimization problem over a parameter class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2 For a fixed cost function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', for cn,m ≡ c for some c ∈ C(X × Y), already various works derived limit distribution results for the empirical OT quantity in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A specific situation arises for probability measures on R with cp(x, y) = |x − y|p for p ≥ 1 (Munk and Czado, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' del Barrio, Giné, and Matrán, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' del Barrio, Giné, and Utzet, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Mason, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' del Barrio, Gordaliza, and Loubes, 2019) where the OT plan can be represented via a quantile coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For this setting, quantile process theory (Csörgö and Horváth, 1993) in combination with integrability conditions on the underlying densities have been exploited to derive distributional limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, on general Euclidean spaces Rd with d ≥ 1 and p-th power costs cp(x, y) = ∥x − y∥p with p > 1 it has been shown by del Barrio and Loubes (2019) and del Barrio, González-Sanz, and Loubes (2021) for probability measures µ, ν with connected support and finite 2p-th moments for n, m → ∞ with m/(n + m) → λ ∈ (0, 1) that � nm n + m � OT(µn, νm, cp) − E � OT(µn, νm, cp) �� ⇝ N(0, σ2 µ,ν), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) where σ2 µ,ν > 0 if and only if µ � ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Here and throughout, “⇝” denotes weak convergence in the sense of Hoffman-Jørgensen (see van der Vaart and Wellner 1996, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Their proof is based on an L2-linearization technique of the OT value and relies on the Efron-Stein inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In general, the centering quantity E[OT(µn, νm, cp)] in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) cannot be replaced by its population quantity OT(µ, ν, cp) which hinders further statistical inference purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, for identical ab- solutely continuous probability measures µ = ν on Rd with sufficiently many moments it follows for d > 2p by Fournier and Guillin (2015) and Weed and Bach (2019) that E � OT(µn, νm, cp) � ≍ min(n, m)−p/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for different measures µ � ν on Rd which are absolutely continuous and sub-Weibull it has been shown for d ≥ 5 by Manole and Niles-Weed (2021) that E � OT(µn, νm, cp) � − OT(µ, ν, cp) ≍ min(n, m)− min(p,2)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' These rates are also minimax optimal (up to logarithmic factors) over appropriate collections of identical measures µ = ν (Singh and Póczos, 2018) as well as different measures µ � ν (Manole and Niles-Weed, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, this demonstrates that estimation of the OT value suffers from the curse of dimensionality and showcases that it is in general for d ≥ 5, due to the dominance of the bias, not possible to replace E[OT(µn, νm, cp)] with OT(µ, ν, cp) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Nevertheless, according to the recently discovered lower complexity adaptation principle for empirical OT (Hundrieser, Staudt, and Munk, 2022), fast convergence rates are still achieved if one of the population measures, µ or ν, is supported on a sufficiently low dimensional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Based on this observation, Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) proved for compactly supported µ, ν on Rd, with µ supported on a finite set or a smooth submanifold of dimension ˜d < 2 min(p, 2) using the functional delta method (Römisch, 2006), � nm n + m � OT(µn, νm, cp) − OT(µ, ν, cp) � ⇝ sup f∈Scp(µ,ν) √ λGµ( f cpcp) + √ 1 − λGν( f cp), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) where Scp(µ, ν) is the set of optimizers of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) and Gµ, Gν denote µ-, ν-Brownian bridges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', centered Gaussian processes with covariance structure characterized by Cov[Gµ( f cc), Gµ(gcc)] = � f ccgccdµ − � f ccdµ � gccdµ for f, g ∈ C(X) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) and likewise for Gν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The asymptotic theory laid out in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) also provides a unified framework for distributional limits of the empirical OT value under discrete population measures (Sommerfeld 3 and Munk, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Tameling, Sommerfeld, and Munk, 2019) and the semi-discrete setting (del Barrio, González-Sanz, and Loubes, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The central contribution of this work is to extend such distributional limits from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) to settings where the cost function is not fixed and additionally may depend on the underlying measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We focus on the following two special instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (A) The cost estimator cn,m, centered by its population counterpart c and suitably rescaled, weakly converges in C(X × Y) to a tight limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', √nm/(n + m)(cn,m − c) ⇝ Gc in C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B) There exists a collection {cθ}θ∈Θ of costs such that for any µ ∈ P(X), ν ∈ P(Y) the corre- sponding cost function cµ,ν ≔ cθ is selected according to an optimization problem of the OT value over Θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', either θ ∈ arg maxθ∈Θ OT(µ, ν, cθ) or θ ∈ arg minθ∈Θ OT(µ, ν, cθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' These two settings are natural and treat a wide spectrum of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Furthermore, they are strongly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It is noteworthy that setting (B) could be treated in the framework of (A) by estimating the optimal θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, this approach requires the existence of a unique population cost function and weak convergence of the cost process as a random element in C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since we are only interested in the empirical infimal or supremal OT value it is instead more natural to rely on an alternative approach which does not require uniqueness of the population cost function or weak convergence of the cost process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (A) we allow the cost function to be estimated from the given data and thus capture the asymptotic dependency between the cost estimator and the empirical measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, this enables an analysis of the empirical OT cost when the cost estimator is parametrized by a plug-in estimator, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', a maximum likelihood procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notably, setting (A) also allows the cost function to be estimated from independent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Overall, this setting covers many scenarios with “extrinsically estimated costs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We refer to Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (B) the motivation slightly differs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Here, the selected cost function depends on the OT problem itself and often brings invariance of the OT problem with respect to a class of transformation parametrized by Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' One could describe this as OT with “intrinsically estimated costs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Examples of this setting are provided in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under suitable assumptions we show in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 for setting (A) that � nm n + m � OT(µn, νm, cn,m) − OT(µ, ν, c) � ⇝ inf π∈Π⋆c (µ,ν) π(Gc) + sup f∈Sc(µ,ν) √ λGµ( f cc)+ √ 1−λGν( f c), where Π⋆ c (µ, ν) represents the set of optimizers for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) for µ, ν with costs c and π(Gc) ≔ � Gcdπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (B) we only state below the distributional limit for supremal costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' a similar distri- butional limit also occurs for infimal costs (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Upon defining the set S+(Θ, µ, ν) = arg maxθ∈Θ OT(µ, ν, cθ) of maximizers we show in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 that � nm n + m � sup θ∈Θ OT(µn, νm, cθ) − sup θ∈Θ OT(µ, ν, cθ) � ⇝ sup θ∈S+(Θ,µ,ν) sup fθ∈Scθ(µ,ν) √ λGµ( f cθcθ θ )+ √ 1−λGν( f cθ θ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In addition to these distributional limits we show for both settings (A) and (B) consistency of a bootstrap principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This is of practical importance since quantiles of the respective distributional limits are difficult to express explicitly due to their dependency on the collection of primal and dual optimizers for population measures and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Our proof technique for the distributional limit under setting (A) differs from previous ap- proaches and might be of interest in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, due to the estimation of the cost function, we cannot rely on any of the techniques from the references mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Instead, we derive certain lower and upper bounds on the OT value which fulfill appropriate (semi-)continuity 4 properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In conjunction with a Skorokhod representation for the empirical process jointly with the cost process, this enables us to prove that the law of the empirical OT value with estimated costs is asymptotically stochastically dominated from above and below by the asserted limit distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the analysis of setting (B) we show under suitable assumptions on the cost family {cθ}θ∈Θ and the underlying probability measures, that the empirical OT process √n(OT(µn, νm, cθ)−OT(µ, ν, cθ))θ∈Θ weakly converges in C(Θ) to a tight random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We prove this result by invoking the func- tional delta method in conjunction with a general result on Hadamard directional differentiability for extremal-type functionals uniformly over a compact parameter space (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The latter can be viewed as an extension of Römisch (2006, Proposition 1) or Fang and Santos (2019, Lemma S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) to processes over Θ and relies on Dini’s theorem (Toma 1997, Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Central for this differentiability result is a certain continuity condition among the sets of maximizing ele- ments for varying parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the OT process it is fulfilled, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', if for every θ ∈ Θ the set of dual optimizer Scθ(µ, ν) is unique (up to constant shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A similar assumption has been imposed by Xi and Niles-Weed (2022) for weak convergence of the empirical sliced OT process, which can be viewed as a special instance of our results for general OT processes, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The distribu- tional limits for the empirical infimal and supremal OT value over θ ∈ Θ then follow by another application of the functional delta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Outline We begin our exposition by deriving in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 an appropriate dual formulation of the OT value which proves useful for our subsequent considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We then proceed with our main contributions, distributional limits for the empirical OT value under weakly converging costs in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 as well as for the empirical OT value under extremal-type costs in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' These asymptotic results are complemented with consistency results of bootstrap resampling schemes in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We discuss our assumptions for the distributional limits and the bootstrap principles in Section 3 and provide sufficient conditions for their validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Statistical applications of our theory are provided in Section 4, where we also derive a deterministic (first-order) stability result for the OT cost under joint perturbations of measures and cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In Section 5 we explicitly construct functionals which enables us to “elevate” the regularity of cost estimators to that of their population counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We employ them in the proofs of our main results which are stated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' All remaining proofs as well as auxiliary results and lemmata are relegated to the Appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notation and probability spaces Given a set T denote by ℓ∞(T) the Banach space of bounded functionals on T equipped with uniform norm ∥ϕ∥ ≔ supt∈T |ϕ(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, if T is equipped with a topology τ denote by C(T) the Banach space of real valued, bounded, continuous functions on T equipped with uniform norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' If dT denotes a metric on T, then we define by Cu(T, dT), or Cu(T) when the metric dT is clear from context, the space of real-valued, bounded, uniformly continuous functions on (T, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Endowed with the uniform norm, it is a Banach space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A real-valued function class F on X is always be equipped with uniform norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This specifies the Banach space Cu(F ) which is a closed subset of the Banach space ℓ∞(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for ε > 0 the covering number N(ε, T, d) denotes the minimal number of sets with diameter 2ε to cover T, and we write x ≲ y when there exists a constant C > 0 with x ≤ Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For a topological space X the set P(X) denotes the collection of Borel probability measures on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Integration � fdµ of a real-valued Borel measurable function f : X → R with respect to µ ∈ P(X) is abbreviated by µ( f) or µ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, we denote by f#µ the pushforward of µ under f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We define all random variables on the same probability space (Ω, A, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We further assume a product structure of that space to define samples and the random weights of the bootstrap, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', Ω = Ω0 × Ω1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' and P = P0 ⊗ P1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' so that the samples only depend on (Ω0, P0), the weights of the first bootstrap replicate on (Ω1, P1) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The law of a random variable X is denoted by L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We finally assume that there exist infinite sequences of measurable maps X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' from 5 (Ω0, P0) to X, respectively, and that samples of cardinality n are obtained from the infinite sequence by projection of the first n coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Outer probability measures are denoted by P∗ (see van der Vaart and Wellner, 1996, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Denoting by BL1 the set of real-valued functions on a metric space (T, dT) which are bounded by one in uniform norm and such that | f(x) − f(y)| ≤ dT(x, y) for any x, y ∈ T, we define the bounded Lipschitz metric between two probability measures µ, ν as dBL(µ, ν) := sup f∈BL1 |µ( f) − ν( f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For a set A and a function f, we write f(A) ≔ { f(a) | a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For two subsets A, B of a vector space, A + B := {a + b | a ∈ A, b ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2 Main Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Preliminaries For our theory on distributional limits for the empirical OT value under estimated cost functions we consider throughout compact Polish spaces X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Given a continuous cost function c ∈ C(X×Y) and probability measures µ ∈ P(X), ν ∈ P(Y) there always exist optimizers to both primal and dual problem (Villani, 2008, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' According to Villani (2003, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='13), dual optimizers can always be selected from the function class Hc := � h : X → R ���� ∃g: Y → [− ∥c∥∞ , ∥c∥∞], h(·) = inf y∈Y c(·, y) − g(y) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) which yields for any µ ∈ P(X), ν ∈ P(Y) the alternative dual representation of the OT value, OT(µ, ν, c) = sup h∈Hc µ(hcc) + ν(hc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) The function class Hc is uniformly bounded and each element exhibits the same modulus of con- tinuity as c, hence it is compact in C(X) by the Theorem of Arzelà-Ascoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) was exploited by Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) for distributional limits of the empirical OT value under a fixed cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For our purposes, we require a dual formulation over a fixed function class which holds for more than a single cost function and to circumvent potential measurability issues we seek a function class which is compact in C(X) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, let B > 0 and consider a concave modulus of continuity w: R+ → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for a continuous metric dX on X we define the compact function class F (B, w) ⊆ C(X), F (B, w) ≔ � f : X → R ���� ∥ f∥∞ ≤ 2B, | f(x) − f(x′)| ≤ w(dX(x, x′)) for all x, x′ ∈ X � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) which will be utilized for a dual representation of the OT value under suitable costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 (Dual formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c ∈ C(X × Y) with ∥c∥∞ ≤ B and |c(x, y) − c(x′, y)| ≤ w (dX(x, x′)) for all x, x′ ∈ X, y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for F ≔ F (B, w) the following inclusions hold Hc ⊆ F cc ⊆ Hc + [−2B, 2B] and Hc c ⊆ F c ⊆ Hc c + [−2B, 2B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for arbitrary probability measures µ ∈ P(X) and ν ∈ P(Y) it follows that OT(µ, ν, c) = sup f∈F µ( f cc) + ν( f c) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) and the set of dual optimizers Sc(µ, ν) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4), referred to as Kantorovich potentials, is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Overall, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 justifies the use of the function class F = F (B, w) for a dual OT formulation and enables us to state conditions of distributional limits in terms of F instead of potentially varying collections of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Distributional Limits under Weakly Converging Costs For the distributional limits in all the statements below, we consider independent and identically distributed random variables {Xi}n i=1 ∼ µ⊗n and independent {Yi}m i=1 ∼ ν⊗m defined on the probability space put forward in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Based on these samples, we define empirical measures µn ≔ 1 n �n i=1 δXi and νm ≔ 1 m �m i=1 δYi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' All the subsequent asymptotic results are to be understood for n, m → ∞ with m/(n + m) → λ ∈ (0, 1), which we do not recall each time for space considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Our main result on the limit law for the empirical OT value under weakly converging costs is given as follows for the two-sample case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The one-sample case is discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 (OT under weakly converging costs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c ∈ C(X × Y) and consider an estimator cn,m ∈ C(X×Y) for c such that cn,m(x, y) is measurable for each (x, y) ∈ X×Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let w: R+ → R+ be a concave modulus of continuity for c with w(δ) > 0 for δ > 0 such that |c(x, y)−c(x′, y)| ≤ w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume for µ ∈ P(X), ν ∈ P(Y) the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (JW) For the function class F = F (2 ∥c∥∞ + 1, 2w) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) joint weak convergence occurs, � nm n + m \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µn − µ νm − ν cn,m − c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √ λ Gµ √ 1 − λ Gν Gc \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 in ℓ∞(F cc) × ℓ∞(F c) × C(X × Y), where (Gµ, Gν, Gc) is a tight random variable and Gµ, Gν have covariance structure as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, suppose either one of the following two assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (OP) There exists a unique OT plan π ∈ Π⋆ c (µ, ν) between µ and ν for the cost function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (Sup) The empirical processes Gµ n ≔ √n(µn − µ) and Gν m ≔ √m(νm − ν) fulfill the convergence sup f∈F Gµ n( f cn,mcn,m − f cc) P∗ −−→ 0 and sup f∈F Gν m( f cn,m − f c) P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows that � nm n + m � OT(µn, νm, cn,m) − OT(µ, ν, c) � ⇝ inf π∈Π⋆c (µ,ν) π(Gc) + sup f∈Sc(µ,ν) √ λ Gµ( f cc)+ √ 1 − λ Gν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A key insight of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 is that the limit distribution for the estimated OT value can be decomposed into two terms: the fluctuation of the cost estimators evaluated at the collection of OT plans and the Kantorovich potentials evaluated at the limit of the empirical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under uniqueness of primal and dual optimizers for the population OT problem we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 (OT under weakly converging costs and uniqueness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 assume (JW)and (OP), and suppose that the set of Kantorovich potentials Sc(µ, ν) for µ, ν with cost function c is unique (up to a constant shift)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for π ∈ Π⋆ c (µ, ν) and f ∈ Sc(µ, ν), it follows that � nm n + m � OT(µn, νm, cn,m) − OT(µ, ν, c) � ⇝ π(Gc) + √ λGµ( f cc) + √ 1 − λGν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) In particular, if (Gµ, Gν, Gc) is a jointly centered Gaussian process in ℓ∞(F cc)×ℓ∞(F c)×C(X×Y), the weak limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) is centered normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2By this we mean, for any f, g ∈ Sc(µ, ν) the difference f − g is constant on supp(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 7 The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and relies on careful lower and upper bounds for the empirical OT value due to the primal (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) and dual formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4), as well as arguments from empirical process theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the course of this, a key argument is the application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 for cn,m and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notably, we do not demand that the cost estimator cn,m is suitably bounded or exhibits a similar modulus of continuity as c itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Instead, we construct by Corol- lary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 an alternative cost estimator cn,m such that the conditions, ∥cn,m∥∞ ≤ 2 ∥c∥∞ + 1 as well as |cn,m(x, y) − cn,m(x′, y)| ≤ 2w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y, are fulfilled deterministically and √nm/(n + m)∥cn,m − cn,m∥∞ P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The latter implies by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 that � nm n + m � OT(µn, νm, cn,m) − OT(µn, νm, cn,m) � ≤ � nm n + m∥cn,m − cn,m∥∞ P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It thus suffices to show the assertion for cn,m where the dual formulation from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 involving the function class F (2 ∥c∥∞ + 1, 2w) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We call cn,m a regularity elevation of cn,m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' details on different kinds of regularity elevations are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The notion of regularity elevations also proves to be useful for showing the validity of condition (Sup) as outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We like to comment on a few aspects of the derived distributional limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) The assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and sufficient conditions for their validity are discussed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Effectively, (JW) delimits the theory to settings of low dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In such settings (Sup) is often also valid as long as the population cost is sufficiently regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) Our proof technique for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 also asserts distributional limits for the one-sample setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', when µ is estimated by µn and ν is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For this setting, (JW) reduces to the condition √n �µn − µ cn − c � ⇝ �Gµ Gc � in ℓ∞(F cc) × C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, in (Sup) we only require that sup f∈F Gµ n( f cncn − f cc) P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, √n � OT(µn, ν, cn) − OT(µ, ν, c) � ⇝ inf π∈Π⋆c (µ,ν) π(Gc) + sup f∈Sc(µ,ν) Gµ( f cc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) In case of a fixed cost function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', when selecting cn = c, the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 reduce to F cc being µ-Donsker and F c being ν-Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 this is equiv- alent to Hc and Hc c being Donsker for µ and ν (van der Vaart and Wellner, 1996, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7), respectively, matching conditions (C) and (S2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 in Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) which imply that � nm n + m � OT(µn, νm, c) − OT(µ, ν, c) � ⇝ sup f∈Sc(µ,ν) √ λGµ( f cc) + √ 1 − λGν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iv) Our proof technique also yields distributional limits for the estimated OT value when instead of empirical measures µn and νm one considers measurable estimators ˜µn ∈ P(X), ˜νm ∈ P(Y), respectively, that fulfill ˜µn ⇝ µ and ˜νm ⇝ ν in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This would mean to replace the empirical measures µn and νm in Assumptions (JW) and (Sup) by ˜µn and ˜νm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In addition, instead of the scaling rate √nm/(n + m) our proof technique theory also permits a different scaling rate an,m which diverges to infinity for n, m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 8 (v) In Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 we prove that the OT value is Gateaux differentiable in (µ, ν, c) for admis- sible directions in (∆µ, ∆ν, ∆c) ∈ (P(X) − µ) × (P(Y) − ν) × C(X × Y) with derivative, (∆µ, ∆ν, ∆c) �→ inf π∈Π⋆c (µ,ν) π(∆c) + sup f∈Sc(µ,ν) ∆µ( f cc) + ∆ν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, the asymptotic distribution described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 may also be interpreted as a derivative of the OT value with respect to the triple (µ, ν, c) evaluated at the limit process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 via an application of the functional delta method would amount to showing Hadamard directional differentiability of the OT value (Römisch, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, this turns out be a challenging issue without imposing additional assumptions on the measure and cost estimators, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (vi) In case of a centered normal limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) the limit variance is given by Var �π(Gc)� + λ VarX∼µ � f cc(X)� + (1 − λ) VarY∼µ � f c(Y)� + 2 √ λ Cov �π(Gc), Gµ( f cc)� + 2 √ 1 − λ Cov �π(Gc), Gν( f c)�, where we used that the random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn and Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Yn are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In partic- ular, the limit law degenerates if both Kantorovich potentials ( f cc, f c) are (µ, ν)-almost surely constant and cn,m converges to c with a faster rate than (nm/(n + m))−1/2, uniformly on the support of the OT plan π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For a sharp characterization of the occurrence of almost surely constant Kantorovich potentials we refer to Section 4 of Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) where the authors showcase that for most cost functions of practical interest a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' constancy typically does not occur if the underlying measures are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Distributional Limits under Extremal-Type Costs As noted in the introduction, could the empirical infimal or supremal OT value over a fixed collec- tion of cost functions also be analyzed using the previously described framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, as part of this approach, we would require the existence of a single underlying population cost function as well as weak convergence of the cost estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To broaden the scope of our theory, we follow in this subsection a different route to derive limiting distributions where such conditions are not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, we first prove a uniform distributional limit for the empirical OT process indexed over the collection of cost functions before relying on a delta method to characterize the distributional limits for the respective infimal and supremal statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the subsequent assertions we again adhere to the sampling convention provided at the be- ginning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The one-sample case is discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 (OT process uniformly over compact Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let Θ be a compact Polish space and con- sider a continuous map c: Θ → C(X × Y), θ �→ cθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let w: R+ → R+ be a modulus of conti- nuity such that supθ∈Θ |cθ(x, y) − cθ(x′, y)| ≤ w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume for µ ∈ P(X), ν ∈ P(Y) the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (Don) For the function class F = F (supθ∈Θ ∥cθ∥∞ , w) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) the collection � θ∈Θ F cθcθ is µ- Donsker and � θ∈Θ F cθ is ν-Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (KP) For any θ ∈ Θ, the set of Kantorovich potentials Scθ(µ, ν) ⊆ F for the OT problem between µ and ν and cost cθ is unique (up to a constant shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, upon selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Θ, it follows that � nm n + m � OT(µn, νm, cθ) − OT(µ, ν, cθ) � θ∈Θ ⇝ � √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ ) � θ∈Θ in C(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 9 The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 is based on Hadamard directional differentiability of the OT cost pro- cess, which follows from a general sensitivity analysis for extremal-type functions uniformly over a compact parameter space (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The assertion for the empirical OT process then follows by invoking the functional delta method (Römisch, 2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' the proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' From the above result, given any functional Φ: C(Θ) → R that is Hadamard directionally differentiable at the function OT(µ, ν, c(·)) ∈ C(Θ), Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 yields by another application of the functional delta method, the distributional limit � nm n + m � Φ�(OT(µn, νm, cθ))θ∈Θ � − Φ�(OT(µ, ν, cθ))θ∈Θ �� ⇝ DH OT(µ,ν,c(·))Φ �� √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ )� θ∈Θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Here, DH OT(µ,ν,c(·))Φ denotes the directional Hadamard derivative of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This enables the derivation of the limit distribution for the infimal mapping using Fang and Santos (2019, Lemma S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) (see also Cárcamo, Cuevas, and Rodríguez 2020, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 (OT infimum over compact Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, upon selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Θ, it follows that � nm n + m � inf θ∈Θ OT(µn, νm, cθ) − inf θ∈Θ OT(µ, ν, cθ) � ⇝ inf θ∈S−(Θ,µ,ν) √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ ), where S−(Θ, µ, ν) = arg minθ∈Θ OT(µ, ν, cθ) denotes the set of minimizers of OT(µ, ν, cθ) over Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In case only (Don) holds, one can still infer the limit law for the empirical supremal OT value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 (OT supremum over compact Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 and only as- sume (Don).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows that � nm n + m � sup θ∈Θ OT(µn, νm, cθ) − sup θ∈Θ OT(µ, ν, cθ) � ⇝ sup θ∈S+(Θ,µ,ν) fθ∈Scθ(µ,ν) √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ ), where S+(Θ, µ, ν) = arg maxθ∈Θ OT(µ, ν, cθ) denotes the set of maximizers of OT(µ, ν, cθ) over Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proofs of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 are documented in Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, in some contexts the compactness assumption on Θ might be too restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The follow- ing result provides an extension to non-compact spaces Θ and focuses on the infimal statistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' an analogue statement also holds for the supremal statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Its proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 (OT infimum over general Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let Θ be a Polish space and consider a continuous map c: Θ → C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let µ ∈ P(X), ν ∈ P(Y) and suppose there is a compact set K ⊆ Θ such that S−(Θ, µ, ν) ⊆ K, there is a sequence of minimizers θn,m ∈ S−(Θ, µn, νm) with limn,m→∞ P∗(θn,m � K) = 0, and that the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 hold with Θ replaced by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the assertion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 on the empirical infimal OT value over Θ remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A few comments are in order concerning the weak limits for the empirical OT cost process as well as the respective infimal and supremal statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) In the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 the parameter space Θ is compact and c: Θ → C(X × Y) is continuous, therefore the range c(Θ) is also compact in C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, by the Theorem of Arzelà-Ascoli, we conclude that supθ∈Θ ∥cθ∥∞ < ∞ and there exists a suitable modulus of continuity for all cost functions uniformly on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 10 (ii) Both assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 and sufficient conditions are discussed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assumption (Don) appears natural in order to control the empirical OT process uniformly over Θ, whereas (KP) is to ensure that the limit process is supported in C(Θ) and stays tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Our proof technique suggests that (KP) can be slightly lifted, but not much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For instance, one could demand that Kantorovich potentials Scθ(µ, ν) which attain the supremum in the derivative can be approximated by Kantorovich potentials Scθ′(µ, ν) for θ′ in the immediate vicinity of θ, as required in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, if Θ ≔ {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , θK} is a finite set equipped with discrete topology, then (KP) can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) The results also extend to the one-sample setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', when µ is estimated by µn and ν is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the one-sample version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 it suffices to assume in (Don) that the function class ∪θ∈ΘF cθcθ is µ-Donsker in conjunction with (KP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Upon selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Θ, the limit distribution is then given for n → ∞ by √n � OT(µn, ν, cθ) − OT(µ, ν, cθ) � θ∈Θ ⇝ � Gµ( f cθcθ θ ) � θ∈Θ in C(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under identical assumptions, the one-sample analogue of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the validity of the one-sample result in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 it suffices that ∪θ∈ΘF cθcθ is µ-Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iv) The obtained weak limits highlight an intimate dependency of limit distributions to the col- lection of Kantorovich potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 the limit process is centered Gaussian due to Assumption (KP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For fixed θ ∈ Θ the limiting random variables degenerates to a Dirac measure at zero if the respective Kantorovich potentials are (µ, ν)-almost surely con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, the limit distribution in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 is also centered normal if Kantorovich potentials ( f cθcθ, f cθ) for µ, ν and cθ coincide (up to a constant shift) on supp(µ) × supp(ν) for any θ ∈ S−(Θ, µ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under analogous assumptions for θ ∈ S+(Θ, µ, ν) the limit distribution in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 is centered normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, assuming (KP), this condition is fulfilled if S−(Θ, µ, ν) or S+(Θ, µ, ν) consist of a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The resulting limit distributions degenerate if Kantorovich potentials are (µ, ν)-almost surely constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A sharp characterization of almost surely constant potentials is detailed in Section 4 of Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Bootstrap Principle for Optimal Transport Costs Since the limit distributions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 involve the set of Kantorovich potentials (and OT plans), under non-unique optimizers there is little hope for an explicit, closed-form de- scription of the quantiles for these distributions, which is required for further practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To circumvent this issue we suggest the use of a k-out-of-n bootstrap procedure with k = o(n) whose consistency is shown in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For simplicity, we state the subsequent results for equal sample sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', n = m as well as bootstrap samples of equal size k = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under differing sample sizes n � m one would select bootstrap samples of size k = o(n), l = o(m) such that l/(l + k) ≈ m/(n + m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Below, we always consider the same bootstrap approach that we now introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the two sequences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' random variables {Xi}n i=1 ∼ µ⊗n, {Yi}n i=1 ∼ ν⊗n, with respective empirical measures µn, νn, consider another sequence of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' bootstrap random variables {Xb i }k i=1 ∼ µ⊗k n , {Yb i }k i=1 ∼ ν⊗k n and define the bootstrap empirical measures µb n,k ≔ 1 k �k i=1 δXb i and νb n,k ≔ 1 k �k i=1 δYb i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, we write in the subsequent statement cn for the cost estimator and cb n,k for the bootstrap cost estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 (Bootstrap for OT under weakly converging costs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, assume (JW) and either (OP) or (Sup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let cb n,k ∈ C(X × Y) be the bootstrap cost estimator such that cb n,k(x, y) is measurable for all (x, y) ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 11 (JW)∗ The bootstrap empirical processes are conditionally on X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Yn consistent in the space ℓ∞(F cc) × ℓ∞(F c) × C(X × Y) for n, k → ∞ with k = o(n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', dBL \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edL \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √ k \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µb n,k − µn νb n,k − νn cb n,k − cn \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ��������� X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Yn \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , L \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µn − µ νn − ν cn − c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In case of setting (Sup) additionally assume the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (Sup)∗ The unconditional bootstrap empirical processes Gµ n,k ≔ √ k(µb n,k−µ) and Gν n,k ≔ √ k(νb n,k− ν) fulfill sup f∈F Gµ n,k( f cb n,kcb n,k − f cc) P∗ −−→ 0, sup f∈F Gν n,k( f cb n,k − f c) P∗ −−→ 0 for n, k → ∞, k = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows for n, k → ∞ with k = o(n) that dBL � L � √ k � OT(µb n,k, νb n,k, cb n,k) − OT(µn, νn, cn) �����X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Yn � , L � √n (OT(µn, νn, cn) − OT(µ, ν, c)) � � P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Despite not relying on the functional delta method for the derivation of the limit distribution of the empirical OT value under weakly converging costs, we obtain a similar bootstrap principle as Dümbgen (1993, Proposition 2) by employing an equivalent formulation for bootstrap consistency (Bücher and Kojadinovic, 2019) in conjunction with the use of a Skorokhod representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The full proof is provided in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' When employing the functional delta method, the n-out-of-n bootstrap is not consis- tent if the Hadamard directional derivative is not linear (Dümbgen, 1993, Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Although Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 does not build on a differentiability result, we show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 that the OT functional is Gateaux directional differentiability with a derivative that is non-linear if primal or dual optimizers are non-unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since Gateaux directional differentiability is implied by Hadamard directional differentiability, this suggests in the regime of non-unique optimizers the inconsistency of the naive n-out-of-n bootstrap for the empirical OT cost under weakly converging costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Verification of the bootstrap consistency in the settings of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It is a direct consequence of consistency of the k-out-of-n bootstrap empirical processes with k = o(n) (van der Vaart and Wellner, 1996, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='13) and the functional delta method for the bootstrap (Dümbgen, 1993, Proposition 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, we omit the proof of the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='12 (Bootstrap for OT process, supremum, and infimum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces and Θ a compact topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider a continuous map c: Θ → C(X × Y), θ �→ cθ, let µ ∈ P(X), ν ∈ P(Y) and assume (Don).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) (OT process in C(Θ)) Then, under (KP), it follows for n, k → ∞ with k ≤ n that dBL � L � √ k � OT(µb n,k, νb n,k, cθ) − OT(µn, νn, cθ) � θ∈Θ ����X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Yn � , L � √n (OT(µn, νn, cθ) − OT(µ, ν, cθ)) � θ∈Θ � P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) (OT infimum over Θ) Then, under (KP), it follows for n, k → ∞ with k ≤ o(n) that dBL � L � √ k � inf θ∈Θ OT(µb n,k, νb n,k, cθ) − inf θ∈Θ OT(µn, νn, cθ) ������X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Yn � , L � √n � inf θ∈Θ OT(µn, νn, cθ) − inf θ∈Θ OT(µ, ν, cθ) �� � P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 12 (iii) (OT supremum over Θ) Then, it follows for n, k → ∞ with k = o(n) that dBL � L � √ k � sup θ∈Θ OT(µb n,k, νb n,k, cθ) − sup θ∈Θ OT(µn, νn, cθ) �������X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Yn � , L � √n � sup θ∈Θ OT(µn, νn, cθ) − sup θ∈Θ OT(µ, ν, cθ) �� � P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notably, we also obtain consistency of the n-out-of-n bootstrap for setting (i) since (KP) implies linearity of the Hadamard directional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3 Discussion of Assumptions In this section we discuss the assumptions on the distributional limits and the bootstrap consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We also provide sufficient conditions for their validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' All the proofs are deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Assumptions (JW) and (JW)∗: Joint Weak Convergence For the empirical OT value under estimated costs we demand in (JW) and (JW)∗ weak convergence of the empirical processes in ℓ∞(F cc) and ℓ∞(F c), where F = F (2 ∥c∥∞ + 1, 2w) is selected as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This requires F cc and F c to be µ- and ν-Donsker, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, we demand weak convergence of the estimated cost function in C(X × Y) to ensure that any sequence of OT plans for µn, νm and cn,m tends towards an OT plan in Π⋆ c (µ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, we stress the necessity of joint weak convergence in (JW) and (JW)∗ as the limit distribution is determined by the random variable (Gµ, Gν, Gc) and thus characterized by their dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Even though apparently unavoidable, these conditions are somewhat restrictive and delimit the theory to low dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This is to be expected as estimation of the OT value (under population costs) suffers from the curse of dimensionality (Manole and Niles-Weed, 2021), leading to slow convergence rates when both population measures µ, ν exhibit high-dimensional support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, in view of the recently discovered lower complexity adaptation principle (Hundrieser, Staudt, and Munk, 2022), it suffices that one measure, µ or ν, is supported on a low dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The following proposition provides bounds on the covering numbers (see the notation section for a definition) of F c and F cc under uniform norm which leads to a universal Donsker property for both function classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 (Universal Donsker property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c ∈ C(X×Y) be a continuous cost function with ∥c∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume one of the three settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , xN} is a finite space (and no additional assumption on c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) There exists a pseudo metric3 ˜dX on X such that N(ε, X, ˜dX) ≲ ε−β for ε > 0 sufficiently small and some β ∈ (0, 2) and c(·, y) is 1-Lipschitz under ˜dX for all y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) X = �I i=1 ζi(Ui) for I ∈ N compact, convex subsets Ui ⊆ Rdi, di ≤ 3 with non-empty interior and maps ζi : Ui → X such that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , I} the function c(ζi(·), y) is (γi, 1)-Hölder4 on Ui for some γi ∈ (di/2, 2] for all y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3A non-negative function d: M × M → R+ on a set M is a pseudo-metric if the three conditions d(x, x) = 0, d(x, y) = d(y, x) and d(x, y) ≤ d(x, z) + d(z, y) are fulfilled for any x, y, z ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4 A function f : U → R on a convex set U ⊆ Rd with non-empty interior is (γ, Λ)-Hölder with modulus Λ ≥ 0 and γ ∈ (0, 1] if ∥f ∥∞ < Λ and |f (x) − f (y)| ≤ Λ ∥x − y∥γ for any x, y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, f is called (γ, Λ)-Hölder for γ ∈ (1, 2] if every partial derivative of f is (γ − 1, Λ)-Hölder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' If U is not open, we assume the existence of an extension ˜f of f onto an open convex set containing U such that ˜f is (γ, Λ)-Hölder thereon, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hundrieser, Staudt, and Munk (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 13 Let B ≥ 0 and consider a modulus of continuity w: R+ → R+ with respect to a metric dX on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for each setting there exists some α < 2 such that for ε > 0 sufficiently small, log N(ε, F c, ∥·∥∞) = log N(ε, F cc, ∥·∥∞) ≲ ε−α for F = F (B, w), where the hidden constant depends for (i) on N, for (ii) on N(ε, X, ˜dX), and for (iii) on (ζi, Ui)I i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, the function classes F c and F cc are universal Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The bounds for the covering numbers stated in the above proposition are essential for the weak convergence of the empirical processes √n(µn − µ) and √m(νm − ν) and represent an important tool for verifying (JW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In order to clarify the assumptions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, we showcase them in a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We additionally refer to Hundrieser, Staudt, and Munk (2022, Section 3) and Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022, Section 5) for more illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Suppose that X and Y are compact subsets of R3 and let c : R3 × R3 → R be twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By enlarging X to a compact, convex set and since c can be rescaled such that c(·, y) is (2, 1)-Hölder on X, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 is applicable in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To state sufficient conditions for (JW) and (JW)∗ we assume that the population cost as well as the empirical and bootstrap estimators are determined by the underlying measures via a Hadamard directionally differentiable functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For simplicity, we consider in the subsequent proposition random variables {Xi}n i=1 ∼ µ⊗n, {Yi}n i=1 ∼ ν⊗n of identical sample size n with empirical measures µn, νn, and bootstrap samples {Xb i }k i=1 ∼ µ⊗k n , {Yb i }k i=1 ∼ ν⊗k n of size k = k(n) = o(n) with correspond- ing bootstrap empirical measures µb n,k, νb n,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 (Joint weak convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let FX, FY be bounded function classes on X and Y, respectively, and assume there is a functional Φc : P(X) × P(Y) ⊆ ℓ∞(FX) × ℓ∞(FY) → C(X × Y) such that, for all n, k ∈ N, c = Φc(µ, ν), cn = Φc(µn, νn), and cb n,k = Φc(µb n,k, νb n,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' If Φc is Hadamard directionally differentiable at (µ, ν) tangentially to P(X)×P(Y), and if FX ∪F cc is µ-Donsker while FY ∪ F c is ν-Donsker, then both (JW) and (JW)∗ are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We like to point out that if the functional Φc is additionally continuous with respect to the topology induced by weak convergence on P(X)×P(Y), it follows that cn(x, y) and cb n,k(x, y) are measurable for each (x, y) ∈ X × Y and, due to compactness of X and Y, measurable in C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Assumption (OP): Uniqueness of Optimal Transport Plans The subject of uniqueness of OT plans between probability measures and a given cost function is of long-standing interest and has been addressed by various authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' General conditions for continuous settings were stated in Gangbo and McCann (1996) and Levin (1999), building on previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The subject has since been covered in depth in Chapters 9 and 10 of the reference textbook by Villani (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' further advances have been made since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To guarantee the uniqueness of the OT plan, many works resort to the so-called Twist condition which demands for differentiable costs the injectivity of the map y → ∇xc(x, y) for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The following proposition formalizes a uniqueness criterion based on this condition and should fulfill the reader’s needs for many practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The result can be deduced from Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='28 and Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='33 in Villani (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume that X, Y are compact Polish spaces where X ⊆ Rd is a Euclidean sub- set with non-empty interior and µ is absolutely continuous with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume that c is locally Lipschitz on X × Y, that c(·, y) is differentiable on int(X) for each y ∈ Y and that y �→ ∇xc(x, y) is injective for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the OT plan π⋆ ∈ Π(µ, ν) is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 14 Even though in certain cases weaker conditions can yield uniqueness (Ahmad, Kim, and Mc- Cann, 2011), these more general conditions are typically considerably more difficult to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Nev- ertheless, unless the cost function exhibits some kind of symmetry or is constant in some region, uniqueness of OT plans is often to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, for fixed measures there is a residual set of cost functions such that for any such costs the OT plan is unique (McCann and Rifford, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In finite discrete settings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', when both underlying measures are supported on finitely many points, results on the uniqueness of OT plans are mostly based on the theory of finite-dimensional linear programs and we refer to Klatt, Munk, and Zemel (2022, Section 6) for a detailed account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Among others, they provide sufficient conditions for uniqueness of OT plans which solely depend on the cost function and the support points but are independent of the weights of the measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For Euclidean based costs their condition is fulfilled for Lebesgue-almost every arrangement of support points of µ and ν, it is however violated if the support points obey some regular or repetitive pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Assumptions (Sup) and (Sup)∗: Control of Supremum over Empirical Process Our assumptions on the suprema of the empirical processes ensure that the fluctuation on the set of feasible dual potentials caused by estimation of the cost function is asymptotically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let us also point out that the suprema in (Sup) and (Sup)∗ are (Borel) measurable by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This implies that the convergence in outer probability occurs, in fact, in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, following along the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 and due to measurability of cn,m it follows for fixed f ∈ F = F (2 ∥c∥∞ + 1, 2w) that both maps ω �→ Gµ n( f cc) and ω �→ Gµ n( f cn,mcn,m) are measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In conjunction with Gµ n((·)cc), Gµ n((·)cn,mcn,m) ∈ Cu(F , ∥·∥∞) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 and compactness of (F , ∥·∥∞) the measurability of the Gµ n((·)cc − (·)cn,mcn,m) as well as its supremum follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the following, we derive sufficient conditions for the validity of Assumption (Sup) (as well as Assumption (Sup)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Based on empirical process theory, in order to suitably control the suprema sup f∈F Gµ n( f cn,m,cn,m − f cc) and sup f∈F Gν n( f cn,m − f c) a canonical route would be to impose metric entropy bounds for F cn,m,cn,m ∪ F cc and F cn,m ∪ F c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Such bounds, however, would impose certain regularity requirements on the cost estimator cn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, in order not to narrow our scope concerning cost estimators, we employ the same ideas as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and approximate the cost estimator cn,m by a more regular cost estimator ˜cn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The subsequent result formalizes these considerations for our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Its proof relies on techniques developed by van der Vaart and Wellner (2007) for empirical processes indexed over estimated function classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces and consider a continuous cost function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) Assume (JW) for random elements cn,m ∈ C(X×Y) and take random elements ˜cn,m ∈ C(X×Y) with √nm/(n + m)∥cn,m − ˜cn,m∥∞ P→0 for n, m = m(n) → ∞ and m/(n + m) → λ ∈ (0, 1) such that for ε > 0 sufficiently small, log N(ε, F cc, ∥·∥∞) + sup n∈N log N(ε, F ˜cn,m ˜cn,m, ∥·∥∞) ≲ ε−α with α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) Then Assumption (Sup) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) Assume (JW) and (JW)∗ for random elements cb n,k ∈ C(X × Y) and let ˜cb n,k ∈ C(X × Y) be random elements with √ k∥cb n,k − ˜cb n,k∥∞ P→0 for n, k = k(n) → ∞ and k = o(n) such that for ε > 0 sufficiently small, log N(ε, F cc, ∥·∥∞) + sup n∈N log N(ε, F ˜cb n,k ˜cb n,k, ∥·∥∞) ≲ ε−α with α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) Then Assumption (Sup)∗ is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 15 As a straightforward corollary of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 we find that (Sup) and (Sup)∗ are fulfilled if the cost estimators cn,m and cb n,k fulfill certain deterministic regularity conditions once n, m, k are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the large sample regime we then choose ˜cn,m ≔ cn,m and ˜cb n,k ≔ cb n,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces, consider a continuous cost function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume (JW) for cn (and (JW)∗ for cb n,k) and that c, cn (and cb n,k) each fulfill one of the three conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 for n ≥ N, k ≥ K with random variables N, K ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, (Sup) (and (Sup)∗) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, if the population cost c and the estimators cn, cn,k are determined by some parameter θ ∈ Θ and estimators θn, θn,k, such that the regularity properties of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 are met uniformly in an open neighborhood of θ and if the estimators are consistent, then Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 asserts the validity of Assumptions (Sup) and (Sup)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, under mild additional assumptions on the space X and the cost function c, we can state a functional Ψ: C(X × Y) → C(X × Y) such that ˜cn ≔ Ψ(cn) fulfills the entropy bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) while satisfying √n ���˜cn,m − cn,m ���∞ P→ 0 for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We call such a functional Ψ a regularity elevation functional since it lifts the degree of regularity of the cost estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Details on regularity elevations are deferred to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces and consider a continuous cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume (JW) (and (JW)∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Suppose that c fulfills one of the three conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under (ii) or (iii) further assume the subsequent condition (ii)’ or (iii)’, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii)’ The weak limit Gc is almost surely continuous with respect to (X, ˜dX) × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii)’ For each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' I} the set Ui ⊆ Rdi is convex and compact, the map ζi : Ui → ζi(Ui) is a homeomorphism, and ci ≔ c(ζi(·), ·): Ui × Y → R is continuously differentiable in u on Ui × Y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', the derivative ∇uci : int(Ui) × Y → Rd can be continuously extended to Ui × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, there exists a continuous partition of unity5 {ηi}I i=1 on X with supp(ηi) ⊆ ζi(Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, Assumption (Sup) (and (Sup)∗) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Assumption (Don): Donsker Property Uniformly over Θ For the distributional limits by Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) on the empirical OT value under a fixed cost function c, the authors effectively assume that the function classes F cc and F c are µ- and ν-Donsker, respectively (Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4(iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, for the uniform convergence result from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 it is natural that we demand the µ- and ν-Donsker property for the unions ∪θ∈ΘF cθcθ and ∪θ∈ΘF cθ for F = F (supθ∈Θ ∥c∥∞ , w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The validity of this condition can be ensured under assumptions on the complexity of the domain X in conjunction with regularity conditions imposed on the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 (Universal Donsker property over Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces and let (Θ, dΘ) be a metric space such that log N(ε, Θ, dΘ) ≲ ε−α for α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Suppose that c: (Θ, dΘ) → C(X × Y), θ �→ cθ is 1-Lipschitz and assume supθ∈Θ ∥cθ∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider one of the three settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , xN} is a finite space (and no additional assumption on c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) For any θ ∈ Θ there exists a pseudo metric ˜dθ,X on X such that supθ∈Θ N(ε, X, ˜dθ,X) ≲ ε−β for β < 2 and cθ(·, y) is 1-Lipschitz under ˜dθ,X for all y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) X = �I i=1 ζi(Ui) for I ∈ N compact, convex subsets Ui ⊆ Rdi, di ≤ 3 with non-empty interior and maps ζi : Ui → X so that for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , I} the function cθ(ζi(·), y) is (γi, 1)-Hölder on Ui (recall footnote 4) for some γi ∈ (di/2, 2] for all y ∈ Y, θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 5A collection {ηi}I i=1 is a continuous partition of unity if ηi ∈ C(X), ηi ≥ 0 for each i and �I i=1 ηi ≡ 1 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 16 Then, for each setting, there exists some α < 2 such that log N(ε, ∪θ∈ΘF cθcθ, ∥·∥∞) ≲ ε−α and log N(ε, ∪θ∈ΘF cθ, ∥·∥∞) ≲ ε−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, ∪θ∈ΘF cθcθ, ∪θ∈ΘF cθ are universal Donsker, and Assumption (Don) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 is a simple consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 in combination with the subsequent lemma whose proof is deferred to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces and let (Θ, dΘ) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Suppose c: (Θ, dΘ) → C(X × Y) is 1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows for any ε > 0 that max � N�ε, ∪θ∈ΘF cθ, ∥·∥∞ �, N�ε, ∪θ∈ΘF cθcθ, ∥·∥∞ �� ≤ N �ε 4, Θ, dΘ � sup θ∈Θ N �ε 2, F cθcθ, ∥·∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Assumption (KP): Uniqueness of Kantorovich Potentials The uniform weak limit of the empirical OT process from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 demonstrates a close relation to the collection of Kantorovich potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, for the limit to be supported on C(Θ) a certain continuity property on the Kantorovich potentials Scθ(µ, ν) with respect to θ is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assumption (KP) on the uniqueness of Kantorovich potentials represents a sufficient condition to ensure this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The recent work by Staudt, Hundrieser, and Munk (2022) thoroughly analyzes the topic of uniqueness in Kantorovich potentials and highlights that it is often expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, for differentiable costs and assuming that one probability measure is supported on the closure of a connected open set on a smooth manifold, Kantorovich potentials are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As Example 3 in their work showcases, uniqueness also occurs under continuous costs if one measure is discrete while the other has connected support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In case both measures have disconnected support, then uniqueness can still be guaranteed if potentials on restricted OT sub-problems are unique and if there exists, in the language of Staudt, Hundrieser, and Munk, a non-degenerate OT plan, meaning that all connected components of both measures are linked via that OT plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The existence of such OT plans can be guaranteed under mild conditions on the underlying measures (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3)) and intuitively demands that the OT problem cannot be divided into distinct sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The following statement is a direct consequence of Staudt, Hundrieser, and Munk (2022), which we have included for ease of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c: Rd × Rd → R be a differentiable cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider probability mea- sures µ, ν ∈ P(Rd) with compact support and suppose supp(µ) = � i∈I Xi and supp(ν) = � j∈J Yj for finitely many disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume each set Xi is either (i) the closure of a connected open set, (ii) the closure of a connected open set in a smooth compact submanifold of Rd, or (iii) a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, if min(|I|, |J|) ≥ 2, suppose for all non-empty, proper I′ ⊂ I and J′ ⊂ J that � i∈I′ µ(Xi) � � j∈J′ ν(Y j), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) Then, Kantorovich potentials for µ, ν and c are unique (up to a constant shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It follows from Theorem 1 in Staudt, Hundrieser, and Munk (2022), and relies on continuity of Kantorovich potentials due to the compactness assumption in conjunction with uniqueness on subproblems (Corollary 2) and the existence of non-degenerate plans (Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 17 4 Applications In this section we employ our theory from Section 2 to obtain novel insights about various OT related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' All proofs for this section are deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Optimal transport based One-Sample Goodness-of-Fit-Testing Hallin, Mordant, and Segers (2021) proposed to use the Wasserstein distance between a sample measure and a reference measure for goodness-of-fit testing under group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the following, we briefly recall the setting for compactly supported measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let ν0 ∈ P(Rd) be compactly supported, define Y as the convex hull of supp(ν0), and let GΘ = {gϑ : ϑ ∈ Θ} be a group of measurable transformations gϑ : Rd → Rd that is parametrized by ϑ ∈ Θ ⊆ Rk for k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume that the map x �→ gϑ(x) is continuous for every ϑ ∈ Θ and that the mappings ϑ �→ gϑ and gϑ �→ (gϑ)#ν0 are bijective (this implies the identifiability of the model parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hallin, Mordant, and Segers, 2021 consider the subsequent testing problem: Let GΘ be a group and define M = {gϑ#ν0 : gϑ ∈ GΘ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Given an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' sample {Xi}n i=1 from some unknown µ ∈ P(X) with X ⊂ Rd compact, the aim is to test H∗ 0 : µ ∈ M against H∗ 1 : µ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) Note that the parameter ϑ∗ under H0, such that (gϑ∗)#ν0 = µ, is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To construct a test for the above hypothesis, which is for instance of particular interest in the analysis of location-scale families, the authors propose to rely on the (2-)Wasserstein distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', they propose a test based on an empirical version of OT � µ, ν0, ���g−1 ϑ∗ (·) − · ���2� = inf π∈Π(µ,ν0) � ���g−1 ϑ∗ (x) − y ���2 dπ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For this purpose, the unknown measure µ is replaced by µn and the cost function c(x, y) = ∥g−1 ϑ∗ (x) − y∥2 by cn(x, y) = ∥g−1 ϑn (x)−y∥2, where ϑn ∈ Θ denotes a suitable estimator for ϑ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Thus, the proposed test statistic is given as OT � µn, ν0, ���g−1 ϑn (·) − · ���2� = inf π∈Π(µn,ν0) � ���g−1 ϑn (x) − y ���2 dπ(x, y), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) which amounts to solving an OT problem with an estimated cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, we can apply our theory to derive the limiting distribution of √n � OT � µn, ν0, ���g−1 ϑn (·) − · ���2� − OT � µ, ν0, ���g−1 ϑ∗ (·) − · ���2�� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) under the null hypothesis H∗ 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In addition, we are able to extend this to testing whether H∗ 0 holds approximately, which is often preferable in practice (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', Munk and Czado, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Dette and Munk, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Dette and Wu, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For this purpose, we fix an estimation procedure for ϑ∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', we choose a specific estimator ϑn (taking values in Θ) for estimating ϑ∗ and denote its population quantity by ϑo ∈ Θ (under H∗ 0 we assume ϑ∗ = ϑo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, we consider the subsequent testing problem: Let GΘ be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Given an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' sample {Xi}n i=1 from some unknown µ ∈ P(X) with X ⊂ Rd compact, the aim is to test for some prespecified ∆ > 0 the hypothesis H0 : W2(µ, (gϑo)#ν0) ≤ ∆ versus H1 : W2(µ, (gϑo)#ν0) > ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) 18 In order to construct a test for the above problem, we have to derive the distributional limits of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) under the assumption that µ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, we employ the theory from Sections 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The first step for the derivation of distributional limits of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) is to establish Hölder regularity (recall Footnote 4) for costs induced by CΘ : Θ → C(X × Y), ϑ �→ ((x, y) �→ ∥g−1 ϑ (x) − y∥2) near ϑo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y ⊆ Rd be compact and denote by C(X, Rd) the space of continuous functions from X to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume that KΘ : Θ ⊆ Rk → C(X, Rd), ϑ �→ (x �→ g−1 ϑ (x)) is continuous near ϑo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, there is an open (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' relative topology) neighborhood U ⊆ Θ of ϑo and some Λ ≥ 0 such that for any x ∈ X and ϑ ∈ U the cost function CΘ(ϑ)(x, ·) ≔ ∥g−1 ϑ (x) − ·∥2 is (2, Λ)-Hölder on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Next, we verify that Hadamard differentiability of KΘ at ϑo implies Hadamard differentiability of the cost parametrizing map CΘ : Θ → C(X × Y), ϑ �→ ((x, y) �→ ∥g−1 ϑ (x) − y∥2) at ϑo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, we additionally impose the following assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (G) For ϑo there exists mϑo > 0 such that for all ϑ′ ∈ Θ in some neighborhood of ϑo, sup x∈Rd ���g−1 ϑ′ (x) − g−1 ϑo (x) ��� 1 + ���g−1 ϑo (x) ��� ≤ mϑo ���ϑ′ − ϑo��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This condition is fulfilled, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', for location-scale families and affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A global version of the above assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', where the condition is to be fulfilled for any ϑ and not only ϑ0, has been used by Hallin, Mordant, and Segers, 2021 to ensure the consistency of their goodness- of-fit test described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume that the function KΘ : Θ → C(X, Rd), ϑ �→ (x �→ g−1 ϑ (x)) is Hadamard differentiable at ϑo tangentially to Θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', for any sequence (ϑo + tnhn)n∈N ⊆ Θ such that tn ց 0 and hn → h ∈ Rk as n → ∞, lim n→∞ ����� KΘ(ϑo + tnhn) − KΘ(ϑo) tn − DH |ϑoKΘ(h) �����∞ = 0, where DH |ϑoKΘ(h): X → Rd is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, if Assumption (G) is satisfied, CΘ is Hadamard differentiable at ϑo tangentially to Θ with derivative DH |ϑoCΘ(h) ∈ C(X × Y) given by DH |ϑoCΘ(h): X × Y → R, (x, y) �→ 2 � DH ϑoKΘ(h)(x), g−1 ϑo (x) − y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, if √n(ϑn − ϑo) ⇝ Gϑ for n → ∞, we obtain for cn ≔ CΘ(ϑn) and c ≔ CΘ(ϑ) that √n(cn − c) ⇝ Gc ≔ � 2 � DH ϑoKΘ(Gϑ)(x), g−1 ϑo (x) − y �� (x,y)∈X×Y in C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) Under the conditions in the proposition above, our main result from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 yields a (typi- cally) non-degenerate limiting distributions for the statistic OT (µn, ν, cn) under the assumption that µ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, this allows us to construct an asymptotic level α test for the null hypotheses given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) (see Munk and Czado, 1998 for the precise construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let ν0 ∈ P(Rd) for d ≤ 3 be compactly supported and define Y as the convex hull of supp(ν0), and let X ⊆ Rd be compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume that (G) holds, and suppose that GΘ : Θ → C(X, Rd), ϑ �→ (x �→ g−1 ϑ (x)) is continuous near ϑo and Hadamard differentiable at ϑo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Define for Λ ≥ 0 from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 the function class F ≔ � f : Y → R ���� ∥f∥∞ ≤ Λ + 1, | f(y) − f(y′)| ≤ 2Λ ���y − y′��� for all y, y′ ∈ Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 19 Then, the function class F CΘ(ϑo) on X is universal Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' random variables {Xi}n i=1 ∼ µ⊗n consider a measurable estimator ϑn and suppose for n → ∞ joint weak convergence, √n � µn − µ ϑn − ϑo � ⇝ �Gµ Gϑ � in ℓ∞(F CΘ(ϑo)) × Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) Then, for cn ≔ CΘ(ϑn) and c ≔ CΘ(ϑ) and by denoting the limit from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) as Gc, it follows that √n (OT (µn, ν0, cn) − OT (µ, ν0, c)) ⇝ inf π∈Π⋆c (µ,ν0) π(Gc) + sup f∈Sc(µ,ν0) Gµ( f c), where Sc(µ, ν0) represents the set of optimizers for sup f∈F µ( f c) + ν0( f cc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A few comments on the distributional limits are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) Given that the function class F CΘ(ϑo) is universal Donsker and thus µ-Donsker, and assuming that √n(ϑn − ϑo) converges in distribution, the requirement of joint convergence as required in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) is very mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, if √n(ϑn − ϑo) can be expressed asymptotically in terms of a suitable linear functional of an empirical process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', if it admits an asymptotic influence function ψ ∈ L2(µ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' van der Vaart 1998, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 58), joint convergence follows since the union F cc ∪ {ψ} is µ-Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) We like to point out that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 also remains valid if µ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, under this as- sumption it follows that (g−1 ϑo )#µ = ν0 which implies that the corresponding OT plan between µ and ν0 is given by π = (Id, g−1 ϑo (·))#µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) the process Gc vanishes along the sup- port of π and the first term in the limit degenerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, if the support of ν0 is connected since then Kantorovich potentials are unique up to a constant shift (Staudt, Hundrieser, and Munk, 2022, Corollary 2) and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' constant (Hundrieser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2022, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Con- sequently, for this setting the corresponding limit distribution is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In contrast, if ν0 has disconnected support, non-constant Kantorovich potentials exist (Staudt, Hundrieser, and Munk, 2022, Lemma 11) which results in a non-degenerate limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) The elements presented for the one-sample case can also be generalized to the case where both empirical measures undergo a transformation, either separately or jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' One might think of choosing the Mahalanobis distance (x − y)⊤Σ−1(x − y) as a cost function where Σ−1 has to be estimated and could, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', be a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As the OT value is not invariant with respect to affine transformations, rescaling the variables would ensure that no component has an overwhelming impact on the cost function compared to the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Optimal Transport with Embedded Invariances In a similar spirit to the previous section, another strand of the literature (Alvarez-Melis, Jegelka, and Jaakkola, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Grave, Joulin, and Berthet, 2019) aims at making OT invariant to a class of transformation T , with τ: Rd → Rd continuously differentiable for each τ ∈ T , by considering inf τ∈T OT � µ, ν, ∥· − τ(·)∥2� = inf τ∈T inf π∈Π(µ,ν) � X×Y ∥x − τ(y)∥2dπ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) This distance is useful in many contexts, among which the word embedding problem or protein alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' If the class of transformations considered is the set of rotations, analyses relying on that distance is coined Wasserstein–Procrustes Analysis (Grave, Joulin, and Berthet, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Jin, Liu, and Xia, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 provides the required tools for statistical inference for the empirical version of the optimization problem in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 20 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider a set of transformations (T , dT ) that is a compact metric space with log N(ε, T , dT ) ≲ ε−α for α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y ⊆ Rd be compact subsets and assume that the functional c : (T , dT ) → C(X × Y), τ �→ cτ, with cτ(x, y) = ∥x − τ(y)∥2, is L-Lipschitz for some L ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume for X and {ct}t∈T any of the settings from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 and take µ ∈ P(X), ν ∈ P(Y) such that the support of µ or ν is the closure of a connected open set in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for {Xi}n i=1 ∼ µ⊗n and {Yi}m i=1 ∼ ν⊗m, respectively, with n, m → ∞ such that m/(n + m) → λ ∈ (0, 1), it holds that � nm n + m � inf τ∈T OT(µn, νm, cτ) − inf τ∈T OT(µ, ν, cτ) � ⇝ inf τ∈S−(T ,µ,ν) √ λ Gµ( f cτcτ τ ) + √ 1 − λ Gν( f cτ τ ), where fτ ∈ Scτ(µ, ν) denotes a Kantorovich potential between µ and ν and cost cτ for τ ∈ S−(T , µ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As previously noted, one can relax the requirement that T is compact to the assumption that the sequence of estimated optimal transformation τn is contained within a compact set with probability tending to one (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the setting of T consisting of diffeomorphisms we have by Lemma 1 of Nies, Staudt, and Munk, 2021 inf τ∈T OT � µ, ν, ∥· − τ(·)∥2� = inf τ∈T OT � µ, (τ−1)#ν, ∥· − ·∥2� = inf τ∈T W2 2 � µ, (τ−1)#ν � , for which convergence of empirical minimizers τn can be verified for various settings using results by Bernton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 (Wasserstein–Procrustes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The above proposition can be applied under mild regularity assumptions on the measures to the special orthogonal group T ≔ SO(d) for d ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, upon choosing X, Y as compact, convex sets of Rd setting (iii) of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 is fulfilled, asserting (Don).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, if the support of µ or ν is the closure of a connected open set in Rd, then (KP) holds and the distributional limits of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Sketched Wasserstein Distance for Mixture Distributions Recently, Delon and Desolneux (2020) and Bing, Bunea, and Niles-Weed (2022) investigated a distance between (Gaussian) mixtures distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' These distributions are ubiquitous in statistics and machine learning, see McLachlan, Lee, and Rathnayake (2019) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' One way of understanding that distance is to start from the Wasserstein distance between discrete measures but instead of using a cost function between points, one replaces the points by distributions and one must thus choose a cost between distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Before formally defining that concept, recall that, for a set of distributions A := (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , AK) of finite cardinality K, a mixture r is a convex combination of components from A given by a vector α ∈ ∆K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', r = �K i=1 αiAi, where ∆K is the probability simplex in RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Given a distance d: A × A → R+ between mixture components of A, the aforementioned authors define the Sketched Wasserstein distance between two mixture distributions with weights α and β as W(α, β, d) ≔ inf π∈Π(α,β) K � k,ℓ=1 πk,ℓd(Ak, Aℓ), where the infimum is taken over elements of the set of couplings Π(α, β) = � π ∈ ∆K×K ������ �K ℓ=1 πk,ℓ = αk, for all k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , K} �K k=1 πk,ℓ = βℓ, for all ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , K} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Understanding the fluctuations of an estimator for this distance can be achieved using the theory developed in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This is formalized in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 21 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let (αn, βn, dn) ∈ ∆K ×∆K ×RK2 + be measurable estimators for α, β, d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for a positive sequence (an)n∈N with limn→∞ an = ∞, assume for n → ∞ that an \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed αn − α βn − β dn − d \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 = an \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed (αn,k − αk)K k=1 (βn,k − βk)K k=1 (dn(Ak, Aℓ) − d(Ak, Aℓ))K l,k=1 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gα Gβ Gd \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 in R2K+K2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8) where (Gα, Gβ, Gd) represents a tight (possibly non-Gaussian) random variable on R2K+K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, an � W(αn, βn, dn) − W(α, β, d) � ⇝ inf π∈Π⋆ d (α, β)⟨π, Gd⟩ + sup f∈Sd(α,β) ⟨ f dd, Gα⟩ + ⟨ f d, Gβ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proof follows along the same approach as for showing Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and is therefore omit- ted, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In this context, the requirement of weak convergence for the measure estimators (αn, βn) ⇝ (α, β) in probability follows from our assumption in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8) since the popula- tion measures and its estimators are supported on finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In Bing, Bunea, and Niles-Weed (2022), they obtain distributional limits in the case where the asymptotic fluctuation of the cost is negligible in comparison to the estimated measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Their results are recovered by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7, which in addition covers the setting where the cost is esti- mated on the same data and converges at the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, we stress that the case of Gaussian mixtures, a particularly relevant one in applications, is also covered by our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Nonetheless, the lack of distributional results for estimators of the mixture parameters in that case still hinders further developments and would be of interest for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Sliced Optimal Transport Our theory from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 also enables the analysis of sliced OT quantities and complement or extend available results from the literature (Goldfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Manole, Balakrishnan, and Wasserman, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Xi and Niles-Weed, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Xu and Huang, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the following, we formalize this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For two Borel probability measures µ, ν ∈ P(Rd) the average-sliced and max-sliced Wasserstein distances of order 1 ≤ p < ∞ are defined, respectively, as W p(µ, ν) ≔ �� Sd−1OT(pθ #µ, pθ #ν, | · − · |p) dσ(θ) � 1 p and W p(µ, ν) ≔ max θ∈Sd−1 � OT(pθ #µ, pθ #ν, | · − · |p) � 1 p, where pθ : Rd → R is the projection map x �→ θT x and σ represents the uniform distribution on the unit sphere Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Note by Lemma 1 in Nies, Staudt, and Munk (2021) for any θ ∈ Sd−1 that OT(pθ #µ, pθ #ν, | · − · |p) = OT(µ, ν, |pθ(·) − pθ(·)|p), which enables to view the sliced Wasserstein quantities in the framework of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 and asserts by Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let p ≥ 1, d ≥ 2, and define for θ ∈ Sd−1 the cost cθ : Rd × Rd → R, (x, y) �→ |pθ(·) − pθ(·)|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, take compactly supported probability measures µ, ν ∈ P(Rd) with empirical measures µn, νm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For all assertions, we let n, m → ∞ with m/(n + m) → λ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) Assume that the set of Kantorovich potentials Scθ(µ, ν) is unique (up to a constant shift) for any θ ∈ Sd−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows upon selecting fθ ∈ Scθ(µ, ν) for any θ ∈ Sd−1 that � nm n + m � OT(µn, ν, cθ)−OT(µ, ν, cθ) � θ∈Sd−1⇝ � √ λGµ( f cθcθ θ )+ √ 1 − λGν( f cθ θ ) � θ∈Sd−1 in C(Sd−1) 22 (ii) Assume the same as in (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows that � nm n + m � W p p(µn, νm) − W p p(µ, ν) � ⇝ � Sd−1 √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ ) dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) Without imposing the assumption on uniqueness of Kantorovich potentials, it follows that � nm n + m � W p p(µn, νm) − W p p(µ, ν) � ⇝ sup θ∈S+(Sd−1, µ,ν) fθ∈Scθ(µ,ν) √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Comparing Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 to the literature for p > 1, results in Goldfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) and Xi and Niles-Weed (2022) are recovered under slightly weaker assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the analysis of both types of empirical sliced Wasserstein distances Goldfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) require the underlying mea- sures to have compact, convex support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for the uniform central limit theorem by Xi and Niles-Weed (2022) of the sliced OT process, they assume for each u ∈ Sd−1 that one of the pro- jected measures has compact, connected support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' These conditions are sufficient for the uniqueness of Kantorovich potentials, but it can also be guaranteed for measures with disconnected support (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11 and more generally Staudt, Hundrieser, and Munk 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8(ii) also complements results by Manole, Balakrishnan, and Wasserman (2022) on the trimmed sliced Wasserstein distance as we do not require the existence of a density but the underlying measures to be compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the special case p = 1, unlike in our results, distributional limits by Goldfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022) and Xu and Huang (2022) for the average- and max-sliced Wasserstein distance do not require uniqueness of the Kantorovich potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, their theory remains valid for non-compactly supported measures by imposing suitable moment-conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Crucial to their approach is the spe- cial characterization of the 1-Wasserstein distance as an integral probability metric over Lipschitz functions (Villani, 2008, Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5), a property which we do not exploit in our general theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Still, under uniqueness of Kantorovich potentials, which occurs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', if one measure is discrete while the other has connected support and is absolutely continuous (Staudt, Hundrieser, and Munk, 2022, Example 3), Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8(i) asserts weak convergence for the sliced OT process in C(Sd−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Stability analysis of Optimal Transport In addition to statistical applications, our theory for the empirical OT value under weakly converg- ing costs enables a deterministic stability analysis of the OT problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) under joint perturbations of the costs and the measures, which may be of independent interest, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', from the viewpoint of optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, we prove in the following Gateaux differentiability of the OT value in (µ, ν, c) ∈ P(X) × P(Y) × C(X × Y) for all admissible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This extends well-known sta- bility results for finite-dimensional linear programs (Gal and Greenberg, 1997, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) which covers the OT problem for probability measures supported on finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let µ ∈ P(X), ν ∈ P(Y) and c ∈ C(X × Y) be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Define for t > 0 sufficiently small the quantities µt = µ + t∆µ and νt = ν + t∆ν, where ∆µ ∈ (P(X) − µ) and ∆ν ∈ (P(Y) − ν), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, let ct = c + t∆c for some ∆c ∈ C(X × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows that lim tց0 1 t (OT(µt, νt, ct) − OT(µ, ν, c)) = inf π∈Π⋆c (µ,ν) π(∆c) + sup f∈Sc(µ,ν) ∆µ( f cc) + ∆ν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 (On Hadamard directional differentiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since the set of admissible directions (P(X) − µ) × (P(Y) − ν) × C(X × Y) is not a normed vector space, we are in general unable to infer 23 Hadamard directional differentiability by additionally proving Lipschitzianity of the OT problem with respect to the measures µ, ν and the cost function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Invoking the same proof strategy as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 would require us to show for any se- quence (µn, νn, cn) = (µ + tn∆µ n, ν + tn∆ν n, c + tn∆c n) ∈ P(X) × P(Y) × C(X × Y) with tn ց 0 and (∆µ n, ∆ν n, ∆c n) → (∆µ, ∆ν, ∆c) in ℓ∞(F cc) × ℓ∞(F c) × C(X × Y) that sup f∈F ���∆µ n( f cncn − f cc) + ∆ν n( f cn − f c) ��� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) Showing this remains a challenge and would enable us to omit conditions (Sup) and (Sup)∗ in the formulations of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Another challenge in such an attempt is that any such sequence (µn, νn) does not necessarily converge weakly for n → ∞ to (µ, ν), which is relevant for our proof, since the topology induced by ℓ∞(F cc) × ℓ∞(F c) may be too weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Though it is likely possible to show Hadamard directional differentiability of the OT problem jointly in the measures and the cost by selecting a sufficiently strong norm that metrizes weak con- vergence of measures, the functional delta method would inevitably require the empirical process to weakly converge in this norm and impose additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' A similar trade-off for the choice of the norm is natural and known in the literature (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Dudley 1990, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='76;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Jourdain and Tse, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 5 Regularity Elevation Functionals In this section, we construct regularity elevation maps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', continuous maps Ψ: C(X × Y) → C(X × Y) such that for measurable estimators cn with √n(cn − c) ⇝ Gc for n → ∞, it follows that (i) √n�cn − Ψ(cn)� P→ 0 and (ii) Ψ(cn) fulfills certain regularity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) Based on Lipschitzianity of the OT value with respect to the cost function (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2), condi- tion (i) allows us to substitute a cost estimator with one that enjoys certain regularity properties, effectively "elevating" its level of regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Such maps prove useful in our work at two particu- lar instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For one, it enables us to assume in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 that cost estimators are suitably bounded and exhibit the same modulus of continuity as the population cost function (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This represents an important step to rely on Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, the notion of regularity elevations also represents a useful tool to prove Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8, for which we employ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 and set ˜cn ≔ Ψ(cn) for a suitable regularity elevation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Insofar, these maps serve as an effective tool for the theoretical analysis of distributional limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The subsequent result provides a first set of conditions to ensure that condition (i) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Its proof as well as the proof of all subsequent results of this section are detailed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X, Y be compact Polish spaces and let cn ∈ C(X×Y) be a (Borel measurable) random sequence such that an(cn − c) ⇝ L in C(X × Y) for some c ∈ C(X × Y) and (an)n∈N such that an → ∞ for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let U ⊆ C(X × Y) be a linear subspace such that L is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, if Ψ: C(X × Y) → C(X × Y) is continuous near c, Hadamard directionally differentiable at f with a derivative such that DH c Ψ|U = IdU and Ψ(c) = c, it follows for n → ∞ that an �cn − Ψ(cn)� P→0 for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notably, in case Ψ is Hadamard differentiable with DH f Ψ = IdC(X×Y), one may select U = C(X × Y) and the condition on the limit L becomes vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To conclude various types of useful regularity properties, as required in (ii) of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1), we thus define in the following subsections various maps such that the conditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Additionally, we provide suitable metric entropy bounds for F Ψ(˜c)Ψ(˜c) independent of ˜c ∈ C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Regularity Elevation to Deterministic Boundedness Consider compact Polish spaces X, Y and let c ∈ C(X × Y) be a continuous cost function such that ∥c∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We define the regularity elevation functional for boundedness as Ψbdd : C(X × Y) → C(X × Y), ˜c �→ � (x, y) �→ max(min(˜c(x, y), 2), −2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the above setting, Ψ = Ψbdd fulfills Ψ(c) = c, is continuous, and it is Hadamard differentiable at c with DH |cΨ = IdC(X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, if X is a finite space, we ob- tain for any uniformly bounded function class G on Y that sup ˜c∈C(X×Y) log N(ε, GΨ(˜c), ∥·∥∞) ≲ | log(ε)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, for our analysis of the empirical OT value under estimated cost functions we can assume without loss of generality that cost estimators are deterministically bounded by a constant that depends on the population cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the following we prove a similar insight for the modulus of continuity for cost estimators on compact (pseudo-)metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Regularity Elevation to Concave Modulus of Continuity and Lipschitzianity Consider compact Polish spaces X, Y and let ˜dX be a continuous (pseudo-)metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Denote by ˜X the space X equipped with the topology induced by ˜dX which is also compact (Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) but potentially does not satisfy the Hausdorff property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c ∈ C( ˜X × Y) be a cost function such that ∥c∥∞ ≤ 1 and consider a concave modulus w: R+ → R+ with w(δ) > 0 for δ > 0 such that |c(x, y) − c(x′, y)| ≤ w( ˜dX(x, x′)) for any x, x′ ∈ ˜X, y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) If c(·, y) is 1-Lipschitz under ˜dX, then select w(t) ≔ t and if c(·, y) is (γ, 1)-Hölder for γ ∈ (0, 1] (recall footnote 4), choose w(t) ≔ tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The regularity elevation functional for w ◦ dX is then given by Ψw◦ ˜dX mod : C(X × Y) → C( ˜X × Y), ˜c �→ � (x, y) �→ inf x′∈X ˜c(x′, y) + 2w( ˜dX(x, x′)) � Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the above setting, Ψ = Ψw◦ ˜dX mod ◦ Ψbdd fulfills Ψ(c) = c, it is continuous near c, and it is Hadamard directionally differentiable at c with DH |c Ψ|C( ˜X×Y) = IdC( ˜X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for any uniformly bounded function class G on Y it holds that sup ˜c∈C(X×Y) log N(ε, GΨ(˜c), ∥·∥∞) ≲ N(ε/8, X, w ◦ ˜dX)| log(ε)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' An appealing consequence of the above considerations is that they allow us to construct a reg- ularity elevated estimator ˜cn,m from cn,m such that H˜cn,m ⊆ F ˜cn,m ˜cn,m, for F = F (2 ∥c∥∞ + 1, 2w) defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3), holds deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c ∈ C(X × Y), set B ≔ ∥c∥∞ + 1/2 and let w: R+ → R+ be a concave modulus with w(δ) > 0 for δ > 0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) holds for a metric dX on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume for a random sequence cn ∈ C(X × Y) that an(cn − c) ⇝ Gc in C(X × Y) with an → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the random sequence cn ≔ B · Ψw◦dX/B mod Ψbdd(cn/B) ∈ C(X × Y) satisfies an ∥cn − cn∥∞ P→ 0 for n → ∞ and deterministically fulfills ∥cn∥∞ ≤ 2B = 2 ∥c∥∞ + 1, relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2), and the inclusion Hcn ⊆ F cncn(2 ∥c∥∞ + 1, 2w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Regularity Elevation to Hölder Functions of Order γ ∈ (1, 2] Since we are able to leverage for convergence rates of the empirical OT value (recall Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1(i)) the regularity of the underlying cost function up to Hölder degree γ ≤ 2, we provide in this subsection a corresponding regularity elevation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As the setting for γ ≤ 1 can be treated using the theory from previous subsection, we only focus on the regime of γ ∈ (1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider a convex, compact set X ⊆ Rd with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let c ∈ C(X × Y) be a cost function such that ∥c∥∞ ≤ 1 and assume c is continuously differentiable in x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', suppose that ∇xc: int(X)×Y → Rd can be continuously extended to X×Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, suppose that c(·, y) is (γ, 1)- Hölder for each y ∈ Y for γ ∈ (1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We define the regularity elevation map for Hölder functions of order γ ∈ (1, 2] by Ψc,γ Hol : C(X × Y) → C(X × Y), ˜c �→ � (x, y) �→ inf x′∈X ˜c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 √ d ���x − x′���γ� Notably, it is crucial that the scalar product term involves the partial derivative of the respective (population) cost function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, we like to point out that the image under Ψc,γ Hol does not necessarily lead to (γ, 1)-Hölder functions but nonetheless ensures suitable metric entropy bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the above setting with X ⊆ Rd convex and compact, Ψ = Ψc,γ Hol ◦ Ψbdd fulfills Ψ(c) = c, it is continuous near c, and it is Hadamard differentiable at c with DH |c Ψ = IdC(X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for any uniformly bounded function class G on Y we obtain that sup ˜c∈C(X×Y) log N(ε, GΨ(˜c), ∥·∥∞) ≲ ε−d/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Combination of Regularity Elevations Finally, we also outline a constructive way to combine regularity elevation maps defined on different spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This is important since it enables to leverage regularity properties of the population cost function for different regions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, let X, Y be compact Polish spaces and assume existence of a collection of homeomor- phisms ζi : Ui → ζi(Ui) for 1 ≤ i ≤ I such that X = �I i=1 ζi(Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume there exists a parti- tion on unity {ηi}I i=1 on X with supp(ηi) ⊆ ζi(Ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider a continuous cost function c: X×Y → R and let ci : Ui ×Y → R, (u, y) �→ c(ζi(u), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume there exist maps Ψi : C(Ui ×Y) → C(Ui ×Y) such that Ψi(ci) = ci and where Ψi is continuous near ci and Hadamard differentiable at ci with derivative DH |ciΨi = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Using these maps we define the combination of regularity elevations as Ψcom : C(X × Y) → C(X × Y), ˜c �→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed(x, y) �→ I� i=1 ηi(x)Ψi � ˜c(ζi(·), ·) � (ζ−1 i (x), y) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, by continuity of the partition of one ηi as well as the functionals Ψi and ζi, ζ−1 i for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , I} it follows that the range of this functional is indeed contained in C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the above setting, Ψ = Ψcom fulfills Ψ(c) = c, it is continuous near c, and it is Hadamard differentiable at c with DH |cΨ = IdC(X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for any uniformly bounded function class G on Y we obtain sup ˜c∈C(X×Y) log N(ε, GΨ(˜c), ∥·∥∞) ≤ I� i=1 sup ˜c∈C(X×Y) log N(ε, GΨi(˜c(ζi(·),·)), ∥·∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 26 6 Proofs of Main Results In this section, we provide the full proofs of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 for the dual representation of the OT value, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 for the distributional limit of the empirical OT value under weakly converging costs, as well as Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 for empirical OT with extremal- type costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proofs for all auxiliary results of this subsection are deferred to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1: Dual Representation of Optimal Transport Value The subsequent auxiliary lemma establishes an important property of cost-transformations which is essential throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 (Lipschitz property of cost-transformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For arbitrary functions f, ˜f : X → R and cost functions c, ˜c: X × Y → R, it follows that ���f c − ˜f ˜c���∞ ≤ ��� f − ˜f ���∞ + ∥c − ˜c∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, upon selecting the constant functions ˜f, ˜c ≡ 0, it follows that ∥f c∥∞ ≤ ∥ f∥∞ + ∥c∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For any h ∈ Hc there exists g: Y → [− ∥c∥∞ , ∥c∥∞] with h = gc, and hence − ∥c∥∞ − sup y∈Y g(y) ≤ h(x) = inf y∈Y c(x, y) − g(y) ≤ ∥c∥∞ − sup y∈Y g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In consequence, we find that ∥h∥∞ ≤ 2 ∥c∥∞ ≤ 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for arbitrary x, x′ ∈ X and ε > 0, consider y′ ∈ Y such that h(x′) ≥ c(x′, y′) − g(y′) − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows that h(x) − h(x′) = � inf y∈Y c(x, y) − g(y) � − � inf y∈Y c(x′, y) − g(y) � ≤ c(x, y′) − g(y′) − c(x′, y′) + g(y′) + ε ≤ w(dX(x, x′)) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since ε > 0 can be chosen arbitrarily small, we obtain that |h(x) − h(x′)| ≤ w(dX(x, x′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This yields Hc ⊆ F and thus Hc c ⊆ F c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, by Santambrogio (2015, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='34) we infer Hc = Hcc c ⊆ F cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To show the remaining inclusions note for f ∈ F that − ∥c∥∞ − sup x∈X f(x) ≤ f c ≤ ∥c∥∞ − sup x∈X f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, the function g ≔ f c + supx∈X f(x) fulfills ∥g∥∞ ≤ ∥c∥∞, and since ∥ f∥∞ ≤ 2B, we find that f cc(x) = ( f c)c = (g)c + sup x∈X f(x) ∈ Hc + [−2B, 2B], which yields F cc ⊆ Hc+[−2B, 2B] as well as F c = F ccc ⊆ Hc c +[−2B, 2B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To show representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4), we combine the inclusions Hc ⊆ F ⊆ C(X) with the alternative dual representations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the final claim, take a maximizing sequence { fn}n∈N for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) which admits by com- pactness of F a converging subsequence { fnk}k∈N with uniform limit f ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 it follows that { f c nk}k∈N and { f cc nk }k∈N also uniformly converge to f c and f cc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We thus obtain that µ( f cc) + ν( f c) = limk→∞ µ( f cc nk ) + ν( f c nk) = OT(µ, ν, c) which shows that f ∈ F is a maximizing element hence the set of optimizers Sc(µ, ν) for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proofs for Distributional Limits under Weakly Converging Costs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 For the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 the following auxiliary results are crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We start with lower and upper bound on the difference between OT values for varying costs and probability measures which are a consequence of the OT problem having a representation in terms of an infimum over feasible couplings as well as a supremum over feasible potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 27 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 (Lower and upper bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Define for B > 0 and a concave modulus of continuity w: R+ → R+ the collection C(B, w) ≔ �¯c ∈ C(X × Y) ��� ∥¯c∥∞ ≤ B, |¯c(x, y) − ¯c(x′, y)| ≤ w(dX(x, x′)) for all x, x′ ∈ X, y ∈ Y� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for costs c, ˜c ∈ C(B, w) and probability measures µ, ˜µ ∈ P(X), ν, ˜ν ∈ P(Y) it holds that inf π∈Π⋆ ˜c (˜µ,˜ν) π(˜c − c) + sup f∈Sc(µ,ν) (˜µ − µ) f cc + (˜ν − ν) f c ≤ OT(˜µ, ˜ν, ˜c) − OT(µ, ν, c) ≤ min � inf π∈Π⋆c (˜µ,˜ν) π(˜c − c) + sup f∈Sc(˜µ,˜ν) (˜µ − µ) f cc + (˜ν − ν) f c, inf π∈Π⋆c (µ,ν) π(˜c − c) + sup f∈S˜c(˜µ,˜ν) (˜µ − µ) f cc + (˜ν − ν) f c + sup f∈F (˜µ − µ)( f ˜c˜c − f cc) + (˜ν − ν)( f ˜c − f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' � In particular, for fixed measures or fixed costs it follows that |OT(µ, ν, ˜c) − OT(µ, ν, c)| ≤ ∥˜c − c∥∞, |OT(˜µ, ˜ν, c) − OT(µ, ν, c)| ≤ sup f∈F cc ���(˜µ − µ) f ��� +sup f∈F c ���(˜ν − ν) f ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To employ the lower and upper bounds of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 we addition- ally require a number of continuity and measurability properties which are captured in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notably, we equip P(X) × P(Y) with the bounded Lipschitz norm, which turns it into a Polish metric space and metrizes weak convergence of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 (Continuity and Measurability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let µ ∈ P(X), ν ∈ P(Y), and c ∈ C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Take a concave modulus of continuity w: R+ → R+ for c and set C ≔ C(2 ∥c∥∞ + 1, 2w) (for the definition of C(2 ∥c∥∞ + 1, 2w) see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, recall the function class F = F (2 ∥c∥∞ + 1, 2w) introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) and define the functions T1 : P(X) × P(Y) × C → R, (µ′, ν′, c′) �→ OT(µ′, ν′, c′), T2 : P(X) × P(Y) × C × C(X × Y) → R, (µ′, ν′, c′, hc) �→ inf π∈Π⋆ c′(µ′,ν′) π(hc), T3 : P(X) × P(Y) × C × Cu(F )2 → R, (µ′, ν′, c′, hµ, hν) �→ sup f∈Sc′(µ′,ν′) hµ( f) + hν( f), T4 : Cu(F )4 → R, (hµ, ˜hµ, hν, ˜hν) �→ sup f∈F hµ( f) − ˜hµ( f) + hν( f) − ˜hν( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, T1 and T4 are continuous, T2 is lower semi-continuous, and T3 is upper semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' If Π⋆ c′(µ′, ν′) is unique, T2 is continuous at (µ′, ν′, c′, hc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for fixed (µ′, ν′, c′) the map T2 is continuous in hc while T3 is continuous in (hµ, hν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, each function Ti for 1 ≤ i ≤ 4 is Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The previous two assertions fully deal with deterministic statements on the OT functional and related terms that arise from corresponding bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The following two results provide the relevant tools to control the stochastic aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, for our proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 we consider a Skorokhod representation of the random sequence detailed in (JW) which additionally fulfills the property that µn and νn weakly converge to µ and ν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For this purpose, we state the following measurability assertions and joint weak convergence statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 (Measurability of empirical process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For a Polish space X consider a totally bounded function class G ⊆ C(X) under uniform norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 28 (i) Any probability measure µ ∈ P(X) defines via evaluation a uniformly continuous functional on G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', µ ∈ Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) A map ω �→ µ(ω) ∈ P(X) ⊆ Cu(G) is Borel measurable if and only if for any g ∈ G the evaluation map ω �→ µ(ω)(g) is Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) The empirical process √n(µn −µ) and the bootstrap empirical process √ k(µb n,k −µn) are both Borel measurable random variables in Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 (Joint weak convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, assume (JW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for n, m → ∞, weak convergence in the Polish space Cu(F )2 ×C(X × Y) × P(X) × P(Y) to a tight limit occurs �� Gµ n( f cc), Gν m( f c) � f∈F , Gc n,m, µn, νm � ⇝ �� Gµ( f cc), Gν( f c) � f∈F , Gc, µ, ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) If (Sup) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 is also valid, then, for n, m → ∞, it follows in the Polish space Cu(F )4 × C(X × Y) × P(X) × P(Y) that �� Gµ n( f cc), Gµ n( f cn,mcn,m), Gν m( f c), Gν m( f cn,m) � f∈F , Gc n,m, µn, νn, � ⇝ �� Gµ( f cc), Gµ( f cc), Gν( f c), Gν( f c) � f∈F , Gc, µ, ν, � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) Each sequence element for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) as well as the weak limit are Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 (Skorokhod representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' When dealing with weak convergence of empirical pro- cesses in non-separable spaces, special care is required due to potential measurability issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' How- ever, since the different maps of interest are defined between Polish spaces and measurable, we circumvent such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, since the random variables from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 converge weakly to a tight limit with separable support, the conditions of Billingsley (1999, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) are met and a (measurable) Skorokhod representation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' With these tools at our disposal, we now proceed with the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Invoking Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4, as √nm/(n + m)(cn,m − c) =: Gc n,m ⇝ Gc in the space C(X×Y), there exists cn,m such that the inclusion Hcn,m ⊆ F cn,mcn,m (recall the function classes from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) holds deterministically for F = F (2 ∥c∥∞ + 1, 2w) and √n(cn,m − cn,m) P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The latter implies by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 that � nm n + m �OT(µn, νm, cn,m) − OT(µn, νm, cn,m)� P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, to prove the assertion it suffices by Slutzky’s lemma to show that � nm n + m � OT(µn, νm, cn,m) − OT(µ, ν, c) � ⇝ inf π∈Π⋆c (µ,ν)π(Gc) + sup f∈Sc(µ,ν) √ λGµ( f cc) + √ 1 − λGν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) Without loss of generality, we may therefore assume cn,m = cn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, set λn ≔ m/(n+m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, 29 by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, the subsequent lower and upper bounds follow, inf π∈Π⋆cn,m(µn,νm) π(Gc n,m) + sup f∈Sc(µ,ν) � λn Gµ n( f cc) + � 1 − λn Gν m( f c) ≤ � nm n + m(OT(µn, νm, cn,m) − OT(µ, ν, c)) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) ≤ min � inf π∈Π⋆c (µn,νm) π(Gc n,m) + sup f∈Sc(µn,νm) � λn Gµ n( f cc) + � 1 − λn Gν m( f c), inf π∈Π⋆c (µ,ν) π(Gc n,m) + sup f∈Scn,m(µn,νm) � λn Gµ n( f cc) + � 1 − λn Gν m( f c) + sup f∈F � λn � Gµ n( f cn,mcn,m) − Gµ n( f cc) � + � 1 − λn �Gν m( f cn,m) − Gν m( f c)� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For each setting (OP) and (Sup) we show that the upper and lower bounds asymptotically converge in distribution to the limit in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3), which then asserts that the empirical OT value also tends to this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, we take for the random variables of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 a Skorokhod representation on a probability space (Ω, A, P) (Billingsley, 1999, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 70) which is well-defined by Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, under (OP) we take the Skorokhod representation such that �� ˜Gµ n( f cc), ˜Gν m( f c) � f∈F , ˜Gc n,m, ˜µn, ˜νm � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' −−→ �� ˜Gµ( f cc), ˜Gν( f c) � f∈F , ˜Gc, µ, ν � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) in Cu(F )2 × C(X × Y) × P(X) × P(Y), whereas under (Sup) we choose it such that �� ˜Gµ n( f cc), ˜Gµ n( f ˜cn,m ˜cn,m), ˜Gν m( f c), ˜Gν m( f ˜cn) � f∈F , ˜Gc n,m, ˜µn, ˜νm � a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' −−→ �� ˜Gµ( f cc), ˜Gµ( f cc), ˜Gν( f c), ˜Gν( f c) � f∈F , ˜Gc, µ, ν � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) in Cu(F )4 × C(X × Y) × P(X) × P(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We also set ˜cn,m ≔ c + ˜Gc n,m/ √nm/(n + m) which a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' converges to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the subsequent argument recall the functions T1, T2, T3, T4 from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 and their (semi- ) continuity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Furthermore, note that an application of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 in combination with the arguments of the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 (i) yields that the maps F → R, f �→ Gµ n( f cc), f �→ Gµ n( f cn,mcn,m), f �→ Gν m( f c), f �→ Gν m( f cn,m) are uniformly continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', elements in Cu(F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For both settings (OP) and (Sup) it follows by measurability of the maps T1, T2, T3 for each n, m ∈ N that � nm n + m(OT(µn, νm, cn,m) − OT(µ, ν, c)) d= � nm n + m(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) inf π∈Π⋆cn,m(µn,νm) π(Gc n,m) + sup f∈Sc(µ,ν) � λn Gµ n( f cc) + � 1 − λn Gν m( f c) d= inf π∈Π⋆ ˜cn,m(˜µn,˜νm) π( ˜Gc n,m) + sup f∈Sc(µ,ν) � λn ˜Gµ n( f cc) + � 1 − λn ˜Gν m( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under (OP) we also notice that inf π∈Π⋆c (µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='νm) π(Gc n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) + sup f∈Sc(µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='νm) � λn Gµ n( f cc) + � 1 − λn Gν m( f c) d= inf π∈Π⋆c (˜µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='˜νm) π( ˜Gc n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) + sup f∈Sc(˜µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='˜νm) � λn ˜Gµ n( f cc) + � 1 − λn ˜Gν m( f c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 30 whereas under (Sup) we additionally employ measurability of T4 to infer for each n ∈ N that inf π∈Π⋆c (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν) π(Gc n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) + sup f∈Scn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m(µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='νm) � λn Gµ n( f cc) + � 1 − λn Gν m( f c) + sup f∈F � λn � Gµ n( f cc) − Gµ n( f cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='mcn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) � + � 1 − λn �Gν m( f c) − Gν m( f cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m)� d= inf π∈Π⋆c (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν) π( ˜Gc n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) + sup f∈S˜cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m(˜µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='˜νm) � λn ˜Gµ n( f cc) + � 1 − λn ˜Gν m( f c) + sup f∈F � λn � ˜Gµ n( f cc) − ˜Gµ n( f ˜cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m ˜cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) � + � 1 − λn � ˜Gν m( f c) − ˜Gν m( f ˜cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, it suffices to work with the Skorokhod representation to obtain the weak limit for the empir- ical OT value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Invoking Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, identical lower and upper bounds on the quantity of interest, √nm/(n + m)(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)), as for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) can be concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To obtain a suitable bound on the limit inferior of √nm/(n + m)(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) take for both (OP) and (Sup) a measurable set A ∈ A of full measure such that the convergence from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) is fulfilled thereon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for each ω ∈ A it follows by lower semi-continuity of T2 jointly with continuity of T3 under fixed (µ, ν, c) that lim inf n,m→∞ inf π∈Π⋆ ˜cn,m(˜µn,˜νm) π( ˜Gc n,m) + sup f∈Sc(µ,ν) � λn ˜Gµ n( f cc) + � 1 − λn ˜Gν m( f c) ≥ inf π∈Π⋆c (µ,ν) π( ˜Gc) + sup f∈Sc(µ,ν) √ λ ˜Gµ( f cc) + √ 1 − λ ˜Gν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under (OP), we find for each ω ∈ A by continuity of T2 at (µ, ν, c, Gc n,m) as a consequence of (OP) and upper semi-continuity of T3 that lim sup n,m→∞ inf π∈Π⋆c (˜µn,˜νm) π( ˜Gc n,m) + sup f∈Sc(˜µn,˜νm) � λn ˜Gµ n( f cc) + � 1 − λn ˜Gν m( f c) ≤ π⋆( ˜Gc) + sup f∈Sc(µ,ν) √ λ ˜Gµ( f cc) + √ 1 − λ ˜Gν( f c) = inf π∈Π⋆c (µ,ν) π( ˜Gc) + sup f∈Sc(µ,ν) √ λ ˜Gµ( f cc) + √ 1 − λ ˜Gν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under (Sup),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' we note for each ω ∈ A by continuity of T2 in hc for fixed (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' upper semi- continuity of T3 and continuity of T4 that lim sup n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m→∞ inf π∈Π⋆c (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν)π( ˜Gc n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) + sup f∈S˜cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m(˜µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='˜νm) � λn ˜Gµ n( f cc) + � 1 − λn ˜Gν m( f c) + sup f∈F � λn � ˜Gµ n( f cc) − ˜Gµ n( f cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='mcn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) � + � 1 − λn � ˜Gν m( f c) − ˜Gν m( f cn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='m) � ≤ inf π∈Π⋆c (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν) π( ˜Gc) + sup f∈Sc(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν) √ λ ˜Gµ( f cc) + √ 1 − λ ˜Gν( f c) + sup f∈F √ λ � ˜Gµ( f cc) − ˜Gµ( f cc) � + √ 1 − λ � ˜Gν( f c) − ˜Gν( f c) � = inf π∈Π⋆c (µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν) π( ˜Gc) + sup f∈Sc(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ν) √ λ ˜Gµ( f cc) + √ 1 − λ ˜Gν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As the lower bound and the upper bounds for √nm/(n + m)(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) asymptotically match for all ω ∈ A, it follows under both (OP) and (Sup) that lim n,m→∞ � nm n + m(OT(˜µn, ˜νm, ˜cn,m)−OT(µ, ν, c)) = inf π∈Π⋆c (µ,ν) π( ˜Gc)+ sup f∈Sc(µ,ν) √ λ ˜Gµ( f cc)+ √ 1 − λ ˜Gν( f c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 31 As the set A has full measure we obtain that � nm n + m(OT(˜µn, ˜νm, ˜cn,m) − OT(µ, ν, c)) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' −−→ inf π∈Π⋆c (µ,ν) π( ˜Gc) + sup f∈Sc(µ,ν) √ λ ˜Gµ( f cc) + √ 1 − λ ˜Gν( f c), where the limit has by measurability of T2 and T3 the same Borel law as the limit in the assertion, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 Before turning to the proof of the bootstrap consistency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10, we introduce an important result on the convergence of the bootstrap empirical measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For a Polish space X let µ ∈ P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' random variables {Xi}n i=1 ∼ µ⊗n to define the empirical measure µn = n−1 �n i=1 δXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, consider k(n) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' random variables {Xb k}n i=1 ∼ µ⊗k(n) n to define the bootstrap empirical measure µb n,k := 1 k(n) �k(n) j=1 δXb i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, provided that k(n) → ∞ as n → ∞, it follows under n → ∞ that µb n,k weakly converges to µ, in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The above lemma is a corollary of Theorem 2 in (Beran, Le Cam, and Millar, 1987) and was added to ease further referencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We can now prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 on bootstrap consistency under weakly converging costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By Assumptions (JW) and (JW)∗, and by measurability of the empirical and bootstrap empirical processes (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) we infer using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2(c) ⇒ (a) in Bücher and Kojadinovic (2019) for two bootstrap versions (µ(1) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ν(1) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' c(1) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (µ(2) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ν(2) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' c(2) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k) based on independent bootstrap samples {X(1) i }k i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' {X(2) i }k i=1 ∼ µ⊗k n and {Y(1) i }k i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' {Y(2) i }k i=1 ∼ ν⊗k n for n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' k → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' k = o(n) that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µn − µ νn − ν cn − c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 √ k \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µ(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − µn ν(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − νn c(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − cn \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gµ Gν Gc \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) Gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 in � Cu(F cc) × Cu(F c) × C(X × Y) �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since k = o(n) we also obtain by Slutzky’s lemma that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µn − µ νn − ν cn − c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 √ k \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µ(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − µ ν(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − ν c(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≕ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gµ n Gν n Gc n \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k Gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gµ Gν Gc \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) Gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Herein, the triples (Gµ, Gν, Gc), (Gµ,(1), Gν,(1), Gc,(1)), and (Gµ,(2), Gν,(2), Gc,(2)) are independent and have identical law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Notably, invoking Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 we may assume without loss of generality that the empirical and bootstrap cost function cn and c(i) n,k for i ∈ {1, 2} deterministically satisfy the relation F¯c ⊆ F ¯c¯c, ¯c ∈ {cn, c(i) n,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by Varadarajan (1958) we know that µn ⇝ µ, νn ⇝ ν a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' for n → ∞, and by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 we infer for i ∈ {1, 2} that µ(i) n,k ⇝ µ, ν(i) n,k ⇝ ν in probability for 32 n, k → ∞, k = o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, Slutzky’s lemma asserts that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � Gµ n, Gν n, Gc n, µn, νn �T � Gµ,(i) n,k , Gν,(i) n,k , Gc,(i) n,k , µ(i) n,k, ν(i) n,k �T i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � Gµ, Gν, Gc, µ, ν �T � Gµ,(i), Gν,(i), Gc,(i), µ, ν �T i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) in �Cu(F cc) × Cu(F c) × C(X × Y) × P(X) × P(Y)�3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, using an analogous argument as for the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 we conclude that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed �� Gµ n( f cc), Gν n( f c) � f∈F , Gc n, µn, νn �T �� Gµ,(i) n,k ( f cc), Gν,(i) n,k ( f c) � f∈F , Gc,(i) n,k , µ(i) n,k, ν(i) n,k �T i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed �� Gµ( f cc), Gν( f c) � f∈F , Gc, µ, ν �T �� Gµ,(i)( f cc), Gν,(i)( f c) � f∈F , Gc,(i), µ, ν �T i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8) in the Polish space �Cu(F )2 × C(X × Y) × P(X) × P(Y)�3, and we use under Assumption (OP) a Skorokhod representation for the process in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Under Assumptions (Sup) and (Sup)∗, by measurability of cn and cn,k as maps to C(X × Y), Lipschitzianity under c-transformation (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) and Slutzky’s lemma we conclude weak con- vergence of the random variables \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed �� Gµ n( f cc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gµ n( f cncn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν n( f c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν n( f cn) � f∈F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gc n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' µn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' νn �T �� Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ( f cc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ( f c(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ( f c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ( f c(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k) � f∈F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' µ(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ν(i) n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k �T i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed �� Gµ( f cc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gµ( f cc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν( f c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν( f c) � f∈F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ν �T �� Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i)( f cc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i)( f cc),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i)( f c),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i)( f c) � f∈F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ν �T i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) in the Polish space �Cu(F )4 × C(X × Y) × P(X) × P(Y)�3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the random variables from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) we now take a Skorokhod representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To denote the random elements from the Skorokhod representation, we equip to the respective random variable with a tilde, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', we write ˜µn for the representation of µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Following the same proof technique as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 we thus conclude with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n � OT(˜µn, ˜νn, ˜cn) − OT(µ, ν, c) � √ k � OT(˜µ(i) n,k, ˜ν(i) n,k, ˜c(i) n,k) − OT(µ, ν, c) � i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' −−→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed infπ∈Π⋆c (µ,ν) π( ˜Gc) + sup f∈Sc(µ,ν) ˜Gµ( f cc) + ˜Gν( f c) � infπ∈Π⋆c (µ,ν) π( ˜Gc,(i)) + sup f∈Sc(µ,ν) ˜Gµ,(i)( f cc) + ˜Gν,(i)( f c) � i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consequently, we infer for the original random variables and using that k = o(n) that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n � OT(µn, νn, cn) − OT(µ, ν, c) � √ k � OT(µ(i) n,k, ν(i) n,k, c(i) n,k) − OT(µn, νn, cn) � i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed infπ∈Π⋆c (µ,ν) π(Gc) + sup f∈Sc(µ,ν) Gµ( f cc) + Gν( f c) � infπ∈Π⋆c (µ,ν) π(Gc,(i)) + sup f∈Sc(µ,ν) Gµ,(i)( f cc)+ Gν,(i)( f c) � i=1,2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since the three components in the limit have identical distributions and are independent, the asser- tion follows at once from Bücher and Kojadinovic (2019, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 (a) ⇒ (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Proofs for Distributional Limits under Extremal-Type Costs Before we proceed with the proofs for the results from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 which rely on an application of the functional delta method, we provide a simple result on the support of the limiting processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Its proof is deferred to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For a Polish space X let µ ∈ P(X) and consider a bounded, measurable function class ˜F on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) The contingent cone of P(X) at µ is given by TµP(X) = Cl{ µ′−µ t |t > 0, µ′ ∈ P(X)} ⊆ ℓ∞( ˜F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) For any ∆ ∈ TµP(X) and f, f ′ ∈ ˜F with f − f ′ ≡ κ for some κ ∈ R it holds that ∆( f) = ∆( f ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) If ˜F is µ-Donsker, then the tight limit Gµ of the empirical process √n(µn − µ) in ℓ∞( ˜F ) is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' contained in TµP(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 The result follows by an application of the functional delta method (Römisch, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Without loss of generality, we assume that X = supp(µ) and Y = supp(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This ensures that Kantorovich potentials are by (KP) unique on the full domains X and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assumption (Don) in conjunction with independence of the underlying random variables from µ and ν ensure by van der Vaart and Wellner (1996, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) that the joint process √nm/n + m(µn − µ, νm − ν) weakly converge in ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 the limit is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' contained in TµP(X) × TνP(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It remains to show that the map (OT(·, ·, cθ))θ∈Θ : P(X) × P(Y) ⊆ ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ) → C(Θ), (µ, ν) �→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edθ �→ sup f∈F µ( f cθ,cθ) + ν( f cθ) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 is Hadamard directionally differentiable at (µ, ν) tangentially to P(X) × P(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the language of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, take F and Θ as they are and set V ≔ ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ), U ≔ P(X) × P(Y), E((µ, ν), f, θ) ≔ µ( f cθcθ) + ν( f cθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, Assumption (EC) follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, while (Lip) and (Lin) are simple to verify by definition of V and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by Assumption (KP) the condition of point (ii) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 holds, since the evaluations of E in f with (∆µ, ∆ν) ∈ TµP(X)×TνP(Y) are invariant under constant shifts (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8), and since Kantorovich potentials are unique on X and Y up to a constant shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This establishes (DC), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 Since Θ is a compact Polish space, it follows by Fang and Santos (2019, Lemma S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) (see also Cárcamo, Cuevas, and Rodríguez 2020, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) that the infimal mapping, I : C(Θ) → R, h �→ inf θ∈Θ h(θ), is Hadamard directionally differentiable at OT(µ, ν, c(·)) ∈ C(Θ) with derivative given by DH OT(µ,ν,c(·))I : C(Θ) → R, ∆h �→ inf θ∈S−(Θ,µ,ν) ∆h(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, applying the functional delta method (Römisch, 2006) for the infimal mapping I onto the uniform weak limit for the empirical OT process from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 asserts the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 From the dual formulation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) the supremal OT value over Θ is given by sup θ∈Θ OT(·, ·, cθ): P(X) × P(Y) ⊆ ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ) → R, (µ, ν) �→ sup (f,θ)∈F ×Θ µ( f cθcθ) + ν( f cθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The results of Appendix A readily apply, with the choices for V, U, and E as in the proof of Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' the only difference being that the supremum is taken over F ×Θ instead of F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, (EC), (Lip), and (Lin) are valid, whereas (DC) is now trivially fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Overall, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 as- serts that supθ∈Θ OT(·, ·, cθ) is Hadamard directionally differentiable tangentially to P(X) × P(Y) with derivative DH |(µ,ν) sup θ∈Θ OT(·, ·, cθ): TµP(X) × TνP(Y) → R, (∆µ, ∆ν) �→ sup θ∈S+(Θ,µ,ν) fθ∈Scθ(µ,ν) ∆µ( f cθcθ θ ) + ∆ν( f cθ θ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Combined with weak convergence of √nm/n + m(µn − µ, νm − ν) in ℓ∞(∪θ∈ΘF cθcθ) × ℓ∞(∪θ∈ΘF cθ) by (Don) in conjunction with the independence of the underlying samples (van der Vaart and Well- ner, 1996, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6), and the inclusion of the limit in TµP(X) × TνP(Y) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8, the functional delta method (Römisch, 2006) implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In addition to the proof presented above, it is also possible to show Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 with similar arguments to those found in the proof of Fang and Santos (2019, Lemma S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) or Cárcamo, Cuevas, and Rodríguez (2020, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, their statements only provide sufficient conditions for Hadamard directional differentiability for tangentially to the space C(∪θ∈ΘF cθcθ) × C(∪θ∈ΘF cθ), whereas the supremal OT value is defined only on the strict subset P(X) × P(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 Define ∆(µn, νm, K) ≔ infθ∈Θ OT(µn, νm, cθ) − infθ∈K OT(µn, νm, cθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, P∗�∆(µn, νm, K) � 0� ≤ P∗��∆(µn, νm, K) � 0� ∩ �θn,m ∈ K�� + P∗(θn,m � K), and as the first summand in the above display is null while limn→∞ P∗(θn,m � K) = 0, the right- hand side converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, invoking Slutzky’s Lemma (van der Vaart and Wellner, 1996, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 that � nm n + m � inf θ∈Θ OT(µn, νm, cθ) − inf θ∈Θ OT(µ, ν, cθ) � = � nm n + m∆(µn, νm, K) + � nm n + m � inf θ∈K OT(µn, νm, cθ) − inf θ∈K OT(µ, ν, cθ) � ⇝ 0 + inf θ∈S−(K,µ,ν) √ λGµ( f cθcθ θ ) + √ 1 − λGν( f cθ θ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The claim now follows at once after observing that S−(K, µ, ν) = S−(Θ, µ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ Acknowledgements: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hundrieser and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Weitkamp gratefully acknowledge support from the DFG Research Training Group 2088 Discovering structure in complex data: Statistics meets opti- mization and inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Mordant gratefully acknowledges support from the DFG CRC 1456 Mathematics of the Experiment A04 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Munk gratefully acknowledges support from the DFG CRC 1456 A04, C06 and the Cluster of Excellence 2067 MBExC Multiscale bioimaging–from molecular machines to networks of excitable cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 35 References Ahmad, Najma, Hwa Kil Kim, and Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' McCann (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Optimal transportation, topology and uniqueness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bulletin of Mathematical Sciences 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 13–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Albano, Paolo (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Some properties of semiconcave functions with general modulus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Journal of mathematical analysis and applications 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 217–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Alvarez-Melis, David, Stefanie Jegelka, and Tommi S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Jaakkola (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Towards optimal trans- port with global invariances”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The 22nd International Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Frankowska (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Set-valued analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Modern Birkhäuser Classics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Springer.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 159–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bernton, Espen, Pierre E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Jacob, Mathieu Gerber, and Christian P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Robert (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “On parameter estimation with the Wasserstein distance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Information and Inference: A Journal of the IMA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 657–676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Billingsley, Patrick (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Convergence of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' New York: Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bing, Xin, Florentina Bunea, and Jonathan Niles-Weed (2022).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Springer, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 371–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bonneel, Nicolas, Julien Rabin, Gabriel Peyré, and Hanspeter Pfister (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Sliced and Radon Wasserstein Barycenters of measures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Journal of Mathematical Imaging and Vision 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bücher, Axel and Ivan Kojadinovic (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “A note on conditional versus joint unconditional weak convergence in bootstrap consistency results”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Journal of Theoretical Probability 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1145–1165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Cárcamo, Javier, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' del Barrio, Eustasio, Evarist Giné, and Carlos Matrán (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Central limit theorems for the Wasserstein distance between the empirical and the true distributions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The Annals of Proba- bility 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' del Barrio, Eustasio, Alberto González-Sanz, and Jean-Michel Loubes (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Central limit theo- rems for general transportation costs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='06379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' — (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Central limit theorems for semidiscrete 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 817–849.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 36 del Barrio, Eustasio and Jean-Michel Loubes (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Central limit theorems for empirical trans- portation cost in general dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The Annals of Probability 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 926–951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Delon, Julie and Agnes Desolneux (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “A Wasserstein-type distance in the space of Gaussian mixture models”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' SIAM Journal on Imaging Sciences 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 936–970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Deshpande, Ishan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Max-sliced Wasserstein distance and its use for GANs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proceed- ings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 10648– 10656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Dette, Holger and Axel 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+page_content=' 569–592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Fang, Zheng and Andres Santos (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Inference on directionally differentiable functions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The Review of Economic Studies 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 377–412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Fournier, Nicolas and Arnaud Guillin (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “On the rate of convergence in Wasserstein distance of the empirical measure”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Probability Theory and Related Fields 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 707–738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Gal, Tomas and Harvey J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Greenberg (1997).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Substrate specificity of thioredoxins and glutaredoxins–towards a functional classification”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Heliyon 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='12, e02943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Giné, Evarist and Richard Nickl (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Mathematical foundations of infinite-dimensional statisti- cal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Cambridge university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Goldfeld, Ziv, Kengo Kato, Gabriel Rioux, and Ritwik Sadhu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Statistical inference with regularized optimal transport”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='04283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Grave, Edouard, Armand Joulin, and Quentin Berthet (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Unsupervised alignment of embed- dings with Wasserstein Procrustes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The 22nd International Conference on Artificial Intelli- gence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' PMLR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1880–1890.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Guntuboyina, Adityanand and Bodhisattva Sen (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1328–1371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hundrieser, Shayan, Marcel Klatt, Thomas Staudt, and Axel Munk (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “A unifying approach to distributional limits for empirical optimal transport”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='12790.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Klatt, Marcel, Axel Munk, and Yoav Zemel (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Limit Laws for Empirical Optimal Solutions in Stochastic Linear Programs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Annals of Operations Research 315, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 251–278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Kolmogorov, Andrei 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 199–211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Toma, Vladimír (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Strong convergence and Dini theorems for non-uniform spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Annales mathématiques Blaise Pascal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' American Mathematical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' — (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Optimal transport: old and new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Cambridge Series in Statistical and Probabilistic Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Weed, Jonathan and Francis Bach (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bernoulli 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4A, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2620–2648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Weitkamp, Christoph Alexander, Katharina Proksch, Carla Tameling, and Axel Munk (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Dis- tribution of distances based object matching: asymptotic inference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Journal of the American Statistical Association, in press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Wiesel, Johannes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Measuring association with Wasserstein distances”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Bernoulli 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2816–2832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Xi, Jiaqi and Jonathan Niles-Weed (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Distributional convergence of the sliced Wasserstein process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='00156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Xu, Xianliang and Zhongyi Huang (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' “Central limit theorem for the sliced 1-Wasserstein distance and the max-sliced 1-Wasserstein distance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='14624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 39 A Uniform Hadamard Directional Differentiability of Extremal- Type Functionals A number of results in this work rely on the notion of Hadamard directional differentiability and the functional delta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' More precisely, both the result on the weak convergence of the empirical OT process from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 and the formulation of regularity elevation functionals from Section 5 rely on this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Although, these two findings are conceptually rather unrelated, their proof techniques are based on a more general insight which we lay out in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let (V, ∥·∥V) be a normed vector space and consider sets F and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Additionally, consider a real-valued function E : V ×F ×Θ → R which assigns each triple (v, f, θ) to a some objective value E(v, f, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We are interested in sensitivity results for extremal-type functionals Ψ(v) ≔ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F E(v, f, θ) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 θ∈Θ and ˜Ψ(v) ≔ � inf f∈F E(v, f, θ) � θ∈Θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Herein, Θ provides the collection of feasible parameters which affect the optimization problem, while F represents the collection of feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The space V denotes another set of param- eters that determine the optimization problem and exhibit a vector space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Overall, these optimization problems characterize the general structure of processes indexed over Θ which are pointwise defined as the supremum or infimum over a collection F and depend on some parameter in V with an additive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For our sensitivity analysis under perturbations of v it suffices to focus only on Ψ since inf f∈F E(v, f, θ) = − sup f∈F −E(v, f, θ) for any (v, θ) ∈ V × Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In the following, we first establish sufficient conditions in terms of E for the continuity properties of Ψ and the underlying sets of optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 (Continuity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let (V, ∥·∥V) be a normed vector space, consider compact topological spaces F and Θ whose topologies are generated by (pseudo-)metrics dF and dΘ, respectively, and assume that E : V × F × Θ → R satisfies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (EC) For any v ∈ V the functional E(v, ·, ·): F × Θ → R is continuous (Lip) There exists some L ≥ 0 such that for any ( f, θ) ∈ F × Θ the functional E(·, f, θ): V → R is L-Lipschitz with respect to ∥·∥V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, Range(Ψ) ⊆ C(Θ) and the functional Ψ: V → C(Θ) is L-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for any (v, θ) ∈ V × Θ the set of optimizers S (v, θ) ≔ { f ∈ F | sup f ′∈F E(v, f ′, θ) = E(v, f, θ)} is non-empty, and for fixed v ∈ V the set-valued map (θ, t) ∈ Θ × R+ �→ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' t) ≔ \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 f ∈ F ���� sup f ′∈F E(v, f ′, θ) ≤ E(v, f, θ) + t \uf8fc\uf8f4\uf8f4\uf8fd\uf8f4\uf8f4\uf8fe is upper semi-continuous in terms of inclusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', for θn → θ and tn → t any sequence fn ∈ S (v, θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' tn) admits a converging subsequence ( fnk)k∈N in F with limit f ∈ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By Assumption (EC) and compactness of Θ × F it follows for any v ∈ V that E(v, ·, ·) is uniformly continuous, hence the function wE,v : R+ → R+, t �→ sup dΘ(θ,θ′)≤t dF (f, f ′)≤t |E(v, f, θ) − E(v, f ′, θ′)| 40 is finite for all t ≥ 0 and fulfills limtց0 wE,v(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For θ, θ′ ∈ Θ we thus find that �������sup f∈F E(v, f, θ) − sup f∈F E(v, f, θ′) ������� ≤ sup f∈F |E(v, f, θ) − E(v, f, θ′)| ≤ wE,v(dΘ(θ, θ′)), which implies for v ∈ V that Ψ(v) ∈ C(Θ) and therefore Range(Ψ) ⊆ C(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the Lipschitzianity of Ψ, note by Assumption (Lip) for any v, v′ ∈ V that ���Ψ(v) − Ψ(v′) ���C(Θ) = sup θ∈Θ �������sup f∈F E(v, f, θ) − sup f∈F E(v′, f, θ) ������� ≤ sup θ∈Θ f∈F |E(v, f, θ) − E(v′, f, θ)| ≤ L ���v − v′���V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To see that S (v, θ) � ∅, note that the function E(v, ·, θ): F → R is continuous for any (v, θ) ∈ V × Θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' hence, by compactness of F the supremum over F is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It remains to prove the assertion on upper semi-continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider converging sequences tn → t ≥ 0 and θn → θ ∈ Θ and take a sequence fn ∈ S (v, θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By compactness of F a converging subsequence ( fnk)k∈N exists with limit f ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by Assumption (EC) and since sup f∈F E(v, f, ·) = Ψ(v)(·) ∈ C(Θ) we obtain that f ∈ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' t) since E(v, f, θ) + t = lim k→∞ E(v, fnk, θnk) + tnk ≥ lim k→∞ sup f∈F E(v, f, θnk) = sup f∈F E(v, f, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ With these tools at our disposal, we can state our general sensitivity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 (Differentiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume in the setting of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Conditions (EC) and (Lip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let v ∈ V and consider a convex set U ⊂ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Denote by TvU ≔ Cl{ v−v t | t > 0, v ∈ U} ⊆ V its contingent cone at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume the following: (Lin) For any ( f, θ) ∈ F × Θ the function ∆|vE(·, f, θ): V → R, v �→ E(v + v, f, θ) − E(v, f, θ) is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (DC) For any h ∈ TvU the function θ ∈ Θ �→ sup f∈S (v,θ) ∆|vE(h, f, θ) is lower semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the functional Ψ: V → C(Θ), v �→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F E(v, f, θ) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 θ∈Θ is Hadamard directionally differentiable at v tangentially to U with derivative given by DH |vΨ: TvU → C(Θ), h �→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed sup f∈S (v,θ) ∆|vE(h, f, θ) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 θ∈Θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 can be viewed as an extension of Römisch (2006, Proposition 1) and Fang and Santos (2019, Lemma S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) to a uniform perturbation result over the parameter space Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Addition- ally, our result does not require regularity properties on the full domain V but only a convex set U, an appealing property which we exploit in the context of our analysis for the OT process (where we choose U = P(X) × P(Y)) as well as regularity elevations (see proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assumptions (EC), (Lip), and (Lin) are fairly straightforward and often simple to verify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The first two conditions also appear to be necessary to infer that Range(Ψ) ⊆ C(Θ) and Lipschitzianity of Ψ: V → C(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assumption (DC) is more technical and requires some knowledge on the set of opti- mizers S (v, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 reveals, is the functional θ ∈ Θ �→ sup f∈S (v,θ) E(h, f, θ) under the assumptions of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 always upper semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, the sole purpose of (DC) is to ensure Range(DH |v Ψ) ⊆ C(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Sufficient conditions for its validity are stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 41 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume the setting of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then under either of the following conditions Assumption (DC) of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) For any θ ∈ Θ and h ∈ TvU there exists f ∈ S (v, θ) with sup f ′∈S (v,θ) ∆|vE(h, f ′, θ) = ∆|vE(h, f, θ) such that any converging sequence θn → θ admits a sub-sequence (θnk) and a converging sequence fnk ∈ S (v, θnk) with fnk → f in F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) For any θ ∈ Θ and h ∈ TvU it holds that ∆|vE(h, f, θ) = ∆|vE(h, f ′, θ) for any f, f ′ ∈ S (v, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let θn → θ and consider an element f ∈ S (v, θ) such that ∆|vE(h, f, θ) = sup f ′∈S (v,θ) ∆|vE(h, f, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (i) take an arbitrary subsequence θnk and take another subse- quence θnkl such that fnkl ∈ S (v, θnkl) converges to f for l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, by (EC), lim l→∞ ∆|vE(h, fnkl , θnkl) = ∆|vE(h, f, θ) = sup f ′∈S (v,θ) ∆|vE(h, f ′, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This implies by monotonicity of the limit inferior and Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that lim inf n→∞ sup f ′∈S (v,θ) ∆|vE(h, f ′, θn) ≥ lim inf n→∞ ∆|vE(h, fn, θn) ≥ sup f ′∈S (v,θ) ∆|vE(h, f ′, θ), which asserts the validity of Assumption (DC) of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (ii) take fn ∈ S (v, θn) and consider by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 a converging subsequence fnk with limit f ∈ S (v, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, it holds that ∆|vE(h, f, θ) = sup f ′∈S (v,θ) ∆|vE(h, f ′, θ) and the assertion follows from (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proof strategy is inspired by Römisch (2006) who performs a sensitivity analysis for when Θ is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To extend the claim for a compact topological space Θ we employ the subsequent version of Dini’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 (Dini’s Theorem, Toma 1997, Corollary 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let Θ be a compact topological space and consider a decreasing fn : Θ → R sequence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', fn ≥ fn+1 for all n ∈ N) of upper semi- continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, assume that fn pointwise converges to a (lower semi-)continuous function f : Θ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, fn converges to f uniformly on Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Take a positive null sequence tn ց 0 with tn > 0 for all n ∈ N and let h ∈ TvU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, take a sequence hn ∈ V such that vn ≔ v + tnhn ∈ U for all n ∈ N and hn → h in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For any θ ∈ Θ, we then observe by (Lin) and (Lip) for any n ∈ N the lower bound Ψ(vn)(θ) − Ψ(v)(θ) = sup f∈F E(vn, f, θ) − sup f∈F E(v, f, θ) ≥ sup f∈S (v,θ) E(vn, f, θ) − E(v, f, θ) ≥ sup f∈S (v,θ) ∆|vE(tnhn, f, θ) ≥ tn sup f∈S (v,θ) ∆|vE(h, f, θ) − 2tnL ∥h − hn∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) Analogously, we obtain the upper bound Ψ(vn)(θ) − Ψ(v)(θ) = sup f∈F E(vn, f, θ) − sup f∈F E(v, f, θ) ≤ sup f∈S (vn,θ) E(vn, f, θ) − E(v, f, θ) ≤ tn sup f∈S (vn,θ) ∆|vE(h, f, θ) + 2tnL ∥h − hn∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) 42 Note that S (vn, θ) ⊆ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 2L ∥vn − v∥V) since any f ∗ ∈ S (vn, θ) fulfills by (Lip) the bound E(v, f ∗, θ) ≥ E(vn, f ∗, θ) − L ∥vn − v∥V = sup f∈F E(vn, f, θ) − L ∥vn − v∥V ≥ sup f∈F E(v, f, θ) − 2L ∥vn − v∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, it follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) upon defining εn ≔ supk≥n 2L ∥vk − v∥V that Ψ(vn)(θ) − Ψ(v)(θ) ≤ tn sup f∈S (v,θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2L∥vn−v∥V) ∆|vE(h, f, θ) + 2tnL ∥h − hn∥V ≤ tn sup f∈S (v,θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='εn) ∆|vE(h, f, θ) + 2tnL ∥h − hn∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) we thus obtain for any θ ∈ Θ that sup f∈S (v,θ) ∆|vE(h, f, θ) − 2L ∥h − hn∥V ≤ Ψ(vn)(θ) − Ψ(v)(θ) tn ≤ sup f∈S (v,θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='εn) ∆|vE(h, f, θ) + 2L ∥h − hn∥V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To conclude the claim we show that the lower and upper bound uniformly converge on Θ for n → ∞ to the DH |v Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since ∥hn − h∥V → ∞, it suffices to prove for the functions Φ ≔ DH |v Ψ: Θ → R, θ �→ sup f∈S (v,θ) ∆|vE(h, f, θ), Φn : Θ → R, θ �→ sup f∈S (v,θ,εn) ∆|vE(h, f, θ), that limn→∞ ∥Φ − Φn∥C(Θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For this purpose, we employ Dini’s theorem (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In this context note, since (εn)n∈N is a decreasing null-sequence, for all n ∈ N and any θ ∈ Θ that S (v, θ) ⊆ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' εn+1) ⊆ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' εn) and consequently Φ(θ) ≤ Φn+1(θ) ≤ Φn(θ) ≤ 2 sup θ∈Θ sup f∈F E(h, f, θ) < ∞, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) where the upper bound is finite due to Assumption (EC) and compactness of F × Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, let us show for any θ ∈ Θ that limn→∞ Φn(θ) = Φ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Take a sequence fn ∈ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' εn) such that Φn(θ) ≤ ∆|vE(h, fn, θ) + 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider a converging subsequence ( fnk)k∈N with limit f∞ ∈ S (v, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, by (EC) it follows that lim sup k→∞ Φnk(θ) ≤ lim k→∞ ∆|vE(h, fnk, θ) + 1/nk = ∆|vE(h, f∞, θ) ≤ sup f∈S (v,θ) ∆|vE(v, f, θ) = Φ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Recalling (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4), it thus follows that limn→∞ Φn(θ) = Φ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To conclude the assertion with Dini’s theorem it remains to show upper-continuity of Φn and of Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' recall by Assumption (DC) that Φ is already lower semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, let ε ≥ 0 and consider a converging sequence θn → θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Select fn ∈ S (v, θn, ε) such that sup f∈S (v,θn,ε) ∆|vE(h, f, θn) ≤ ∆|vE(h, fn, θn) + 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Take a subsequence fnk and select by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 another converging subse- quence fnkl with limit f∞ ∈ S (v, θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Using Assumption (EC) it thus follows that lim l→∞ ∆|vE(h, fnkl, θnkl) + 1/nkl = ∆|vE(h, f∞, θ) ≤ sup f∈S (v,θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ε) ∆|vE(h, f, θ) Invoking monotonicity of the limit superior and Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 we thus obtain that lim sup n→∞ sup f∈S (v,θn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ε) ∆|vE(h, f, θ) ≤ lim sup l→∞ ∆|vE(h, fn, θn) + 1/n ≤ sup f∈S (v,θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='ε) ∆|vE(h, f, θ), Hence, by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 we conclude that Φn is upper semi-continuous and that Φ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Dini’s theorem (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) thus implies limn→∞ ∥Φ − Φn∥∞ = 0, asserting the Hadamard direc- tional differentiability of Ψ at v tangentially to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, note that the range of DH v Ψ is indeed contained in C(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 43 B Proofs for Section 3: Sufficient Criteria for Assumptions B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 it follows that F c ⊆ Hc c + [−2B, 2B] and F cc ⊆ Hc + [−2B, 2B] with Hc defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Invoking Hundrieser, Staudt, and Munk (2022, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) and Santambrogio (2015, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='34) we obtain for any ε > 0 that N(ε, F c, ∥·∥∞) = N(ε, F cc, ∥·∥∞) ≤ �2B ε � N(ε/2, Hc c, ∥·∥∞) = �2B ε � N(ε/2, Hc, ∥·∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the function class Hc, the asserted uniform metric entropy bounds are available in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Appendix A of Hundrieser, Staudt, and Munk (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Note by uniform boundedness of the cost function that Hc and Hc c are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The assertion on the universal Donsker property then follows from van der Vaart and Wellner (1996, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 By assumption the functional Φc : P(X) × P(Y) → ℓ∞(F cc) × ℓ∞(F c) × C(X × Y) (µ, ν) �→ (µ, ν, Φc(µ, ν)), where the domain is viewed as a subset of ℓ∞(FX ∪F cc)×ℓ∞(FY ∪F c), is Hadamard differentiable at (µ, ν) tangentially to P(X) × P(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, since FX ∪ F cc is µ-Donsker it follows that √n/2(µn − µ) ⇝ Gµ in ℓ∞(FX ∪ F cc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Likewise, since FY ∪ F c is ν-Donsker it follows that √n/2(νn − ν) ⇝ Gν in ℓ∞(FY ∪ F c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, by independence of the random variables {Xi}n i=1 and {Yi}n i=1 it follows from van der Vaart and Wellner, 1996, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 that the joint empirical processes √n/2(µn − µ, νn − ν) weakly converge in ℓ∞(FX ∪ F cc) × ℓ∞(FY ∪ F c) to (Gµ, Gν), contained in TµP(X) × TνP(Y) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We thus conclude by the functional delta method (Römisch, 2006) for Φc that (JW) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by the Donsker property and independence of the random variables, we also infer by van der Vaart and Wellner (1996, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='13) in the space ℓ∞(FX ∪ F cc) × ℓ∞(FY ∪ F c) that dBL � L � √ k �µb n,k − µn νb n,k − νn �������X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Yn � , L �Gµ Gν �� P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by the functional delta method for conditionally weakly converging random variables Düm- bgen (1993) for Ψc we infer that dBL \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edL \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √ k \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µb n,k − µn νb n,k − νn cb n,k − cn \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ��������� X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , Xn, Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Yn \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 , L \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed µn − µ νn − ν cn − c \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 P∗ −−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 Before, we start to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6, we establish an auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let X and Y be compact Polish spaces and consider c ∈ C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) For any function g : X → R and any constant κ, it holds that (g + κ)c = gc − κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) Let B > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for any g : X → R and ∆c ∈ C(X × Y) with ∥g∥∞ + 2 ∥c + ∆c∥∞ ≤ B it holds that g(c+∆c)(c+∆c)cc ∈ Hc + [−B, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 44 The proof of the above lemma can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The proof is strongly inspired by van der Vaart and Wellner, 2007, Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 and employs standard empirical process arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In order to simplify the notation, we only consider the case n = m and write cn instead of cn,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Note that the claim for n � m follows by the analogous arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To show (i) first note by triangle inequality and using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that sup f∈F |Gµ n( f cncn − f cc)| ≤ sup f∈F |Gµ n( f cncn − f ˜cn ˜cn)| + sup f∈F |Gµ n( f ˜cn ˜cn − f cc)| ≤ 4 √n ∥cn − ˜cn∥∞ + sup f∈F |Gµ n( f ˜cn˜cn − f cc)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) The first term converges by assumption for n → ∞ in probability to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the latter term note by (JW) and the assumption on ˜cn that √n/2(˜cn − c) ⇝ Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By tightness of the law of Gc there exists for any ε > 0 a compact set K ⊆ C(X × Y) such that P(Gc ∈ K) > 1 − ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' thus for any δ > 0 the set Kδ of elements in C(X × Y) with distance less than δ > 0 to K fulfills lim inf n→∞ P � � n/2(˜cn − c) ∈ Kδ� ≥ P(Gc ∈ Kδ) > 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) By compactness of K there exists a finite δ/2-covering {h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , hp} which implies that Kδ/2 ⊆ �p i=1 B(hi, δ), where B(h, δ) denotes the open ball of radius δ around h in the space C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We thus obtain � � n/2(˜cn − c) ∈ Kδ/2� ⊂ p � i=1 � ˜cn ∈ B(c + 2n−1/2hi, δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by Santambrogio (2015, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='34) it follows for any f ∈ F and ¯c ∈ C(X × Y) that f ¯c¯c = f ¯c¯c¯c¯c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Therefore, by triangle inequality, sup f∈F |Gµ n( f ˜cn ˜cn − f cc)| = sup f∈F |Gµ n( f ˜cn ˜cn˜cn ˜cn − f cccc)| ≤ sup f∈F |Gµ n( f ˜cn ˜cn˜cn ˜cn − f ˜cn ˜cncc)| + sup f∈F |Gµ n( f ˜cn ˜cncc − f cccc)| ≤ sup f∈F ˜cn ˜cn |Gµ n( f ˜cn˜cn − f cc)| + sup f∈F |Gµ n( f ˜cn˜cncc − f cccc)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) Assuming √n/2(˜cn − c) ∈ Kδ/2, it follows for the first term in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) that sup f∈F ˜cn˜cn |Gµ n( f ˜cn ˜cn − f cc)| ≤ sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p sup ∥h−hi∥∞<δ |Gµ n( f (c+2h/ √n)(c+2h/ √n) − f cc)| ≤ sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p sup ∥h−hi∥∞<δ |Gµ n( f (c+2h/ √n)(c+2h/ √n) − f (c+2hi/ √n)(c+2hi/ √n))| + sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n( f (c+2hi/ √n)(c+2hi/ √n) − f cc)| ≤ 8δ + sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n( f (c+2hi/ √n)(c+2hi/ √n) − f cc)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) Here, we used in the last inequality Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 to infer ���� f (c+2h/ √n)(c+2h/ √n) − f (c+2hi/ √n)(c+2hi/ √n)����∞ ≤ 4 ∥hi − h∥∞ / √n ≤ 4δ/ √n 45 in conjunction with Gµ n(g) = √n(µn − µ)(g) ≤ 2 √n ∥g∥∞ for any measurable function g on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Now, define for 1≤ i ≤ p the function class ˜Gi n ≔ ˜Gi n(hi) ≔ � f (c+2hi/ √n)(c+2hi/ √n) − f cc��� f ∈ F ˜cn ˜cn� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For each 1 ≤ i ≤ p and any ε > 0 we then observe that log N(ε, ˜Gi n, ∥·∥∞) ≤ log N(ε, F ˜cn ˜cn(c+2hi/ √n)(c+2hi/ √n), ∥·∥∞) + log N(ε, ˜F ˜cn ˜cncc, ∥·∥∞) ≤2 log N(ε, F ˜cn˜cn, ∥·∥∞), where the last step follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 in Hundrieser, Staudt, and Munk, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In consequence, it follows by Dudley’s entropy integral (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',Wainwright, 2019, Chapter 5) that E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p ����Gµ n( f (c+2hi/ √n)(c+2hi/ √n) − f cc) ���� \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≤ p � i=1 E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup g∈ ˜Gin ���Gµ n(g) ��� \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≲ p � i=1 � 4∥hi∥∞/ √n 0 � log �N(ε, F ˜cn˜cn, ∥·∥∞)�dε ≲ p � i=1 � 4∥hi∥∞/ √n 0 ε−α/2dε ≲ p � i=1 (∥hi∥∞ / √n)1−α/2, where by assumption the hidden constants do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We thus infer conditionally on the event √n/2(˜cn − c) ∈ Kδ/2 for n → ∞ that sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p Gµ n( f (c+2hi/ √n)(c+2hi/ √n) − f cc) P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) For the second term in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) we assume √n/2(˜cn − c) ∈ Kδ/2 and obtain by similar arguments, sup f∈F |Gµ n( f ˜cn ˜cncc − f cccc)| ≤ 8δ + sup f∈F max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) Upon defining the function class ˜Gn ≔ ˜Gn(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , hp) ≔ � f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc��� f ∈ F � (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) we note again by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that any g ∈ Gn fulfills ∥g∥∞ ≤ maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p 4 ∥hi∥∞ / √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for n sufficiently large there exists a constant B > 0 such that Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 is applicable for any f ∈ F , and we obtain f (c+2hi/ √n)(c+2hi/ √n)cc ∈ Hc + [−B, B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, for sufficiently large n, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 for any ε > 0 that N(ε, ˜Gn(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , hp), ∥·∥∞) ≤ �N(ε, F cc + [−B, B], ∥·∥∞)�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 46 Again invoking, Dudley’s entropy integral asserts such for n that E � sup f∈F max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p �����Gµ n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc) ����� � = E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup f∈ ˜Gn ���Gµ n( ˜f) ��� \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≲ � maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p 4∥hi∥∞/ √n 0 � log � N(ε, ˜Gn, ∥·∥∞) � dε ≤ � maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p 4∥hi∥∞/ √n 0 � log (N(ε, F cc + [−B, B], ∥·∥∞))dε ≲ � maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p 4∥hi∥∞/ √n 0 ε−α/2dε ≲ � max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p ∥hi∥∞ / √n �1−α/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This implies conditionally on the event √n/2(˜cn − c) ∈ Kδ/2 for n → ∞ that sup f∈F max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc)| P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8) Concluding, for any ε > 0 it follows for δ ≔ ε/32 > 0 from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2)–(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8) that lim sup n→∞ P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F |Gµ n( f ˜cn˜cn − f cc)| > ε \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≤ lim sup n→∞ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edP \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F |Gµ n( f ˜cn ˜cn − f cc)| > ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' � n/2(˜cn − c) ∈ Kδ/2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + P � � n/2(˜cn − c) � Kδ/2�\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≤ lim sup n→∞ P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F |Gµ n( f ˜cn˜cn − f cc)| > ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' � n/2(˜cn − c) ∈ Kδ/2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + ε ≤ lim sup n→∞ P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed sup f∈F ˜cn ˜cn |Gµ n( f ˜cn ˜cn − f cc)| > ε/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' � n/2(˜cn − c) ∈ Kδ/2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + lim sup n→∞ P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F |Gµ n( f ˜cn˜cncc − f cccc)| > ε/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' � n/2(˜cn − c) ∈ Kδ/2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + ε ≤ lim sup n→∞ P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed sup f∈F ˜cn ˜cn max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n( f (c+2hi/ √n)(c+2hi/ √n) − f cc)| > ε/4, � n/2(˜cn − c) ∈ Kδ/2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + lim sup n→∞ P \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edsup f∈F max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n( f (c+2hi/ √n)(c+2hi/ √n)cc − f cccc)| > ε/4, � n/2(˜cn − c) ∈ Kδ/2 \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 + ε = ε, which shows the convergence in probability of sup f∈F |Gµ n( f ˜cn˜cn− f cc)| to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We thus conclude the convergence in probability for both terms of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' An analogous argument yields the convergence sup f∈F |Gν n( f cn − f c)| P→ 0 for n → ∞, where we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 of Hundrieser, Staudt, and Munk, 2022 to obtain sup n∈N log N(ε, F ˜cn ∪ F c, ∥·∥∞) ≤ sup n∈N � log N(ε, F ˜cn, ∥·∥∞) + log N(ε, F c, ∥·∥∞) � = sup n∈N � log N(ε, F ˜cn ˜cn, ∥·∥∞) + log N(ε, F cc, ∥·∥∞) � ≤ sup n∈N 2 log N(ε, F ˜cn ˜cn ∪ F cc, ∥·∥∞) ≲ ε−α for α < 2, 47 which overall verifies (Sup) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For (ii) note by Bücher and Kojadinovic (2019) and since k = k(n) = o(n) for n → ∞ that √ k(˜cb n,k − c) = √ k(˜cb n,k − cb n,k) + √ k(cb n,k − cn) + � k n √n(cn − c) ⇝ Gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Likewise, it follows for n → ∞ that √ k(µb n,k − µ) ⇝ Gµ in ℓ∞(F cc), √ k(νb n,k − ν) ⇝ Gν in ℓ∞(F c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This means that we can pursue a similar proof strategy as for (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Define Gµ n,k ≔ √ k(µb n,k − µ) and Gν n,k ≔ √ k(νb n,k − ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, we infer from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that sup f∈F |Gµ n,k( f cb n,kcb n,k − f cc)| ≤ sup f∈F |Gµ n,k( f cb n,kcb n,k − f ˜cb n,k ˜cb n,k)| + sup f∈F |Gµ n,k( f ˜cb n,k˜cb n,k − f cc)| ≤ 4 √ k ���cb n,k − ˜cb n,k ���∞ + sup f∈F |Gµ n,k( f ˜cb n,k ˜cb n,k − f cc)|, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9) where the first term converges for n → ∞ in probability to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By Santambrogio (2015, Proposi- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='34) we obtain that sup f∈F |Gµ n,k( f ˜cb n,k ˜cb n,k − f cc)| ≤ sup f∈F ˜cb n,k ˜cb n,k |Gµ n,k( f ˜cb n,k ˜cb n,k − f cc)| + sup f∈F |Gµ n,k( f ˜cb n,k ˜cb n,kcc − f cccc)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by analogous arguments to those for (i) we obtain with probability at least 1 − ε for n sufficiently large that sup f∈F ˜cb n,k ˜cb n,k |Gµ n,k( f ˜cb n,k ˜cb n,k − f cc)| ≤ 8δ + sup f∈F ˜cb n,k ˜cb n,k max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n,k( f (c+2hi/ √ k)(c+2hi/ √ k) − f cc)| (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10) as well as sup f∈F |Gµ n,k( f ˜cb n,k ˜cb n,kcc − f cccc)| ≤ 8δ + sup f∈F max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n,k( f (c+2hi/ √ k)(c+2hi/ √ k)cc − f cccc)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11) Next, we verify that the suprema on the right-hand sides of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11) converge (uncondi- tionally with respect to the µn but conditionally on the set with probability at least 1 − ε) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We note by Dudley’s entropy integral for the bootstrap empirical process √ k(µb n,k − µn) and the empirical process √n(µn − µ) as well as our previous considerations that ≲E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 sup f∈F ˜cb n,k ˜cb n,k max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='p |Gµ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k( f (c+2hi/ √ k)(c+2hi/ √ k) − f cc)| \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = p � i=1 Eµn \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0Eµb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 sup f∈F ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k | √ k(µb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k − µn)( f (c+2hi/ √ k)(c+2hi/ √ k) − f cc)| ������ µn \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + � k nE \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 sup f∈F ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k |Gµ n( f (c+2hi/ √ k)(c+2hi/ √ k) − f cc)| \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≲ p � i=1 Eµn � 4∥hi∥∞/ √ k 0 � log � N(ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' F ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ∥·∥∞) � dε + � k n � 4∥hi∥∞/ √ k 0 � log � N(ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' F ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k ˜cb n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' ∥·∥∞) � dε ≲ p � i=1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed1 + � k n \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 � 4∥hi∥∞/ √ k 0 ε−α/2dε ≲ p � i=1 \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed1 + � k n \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 � ∥hi∥∞ / √ k �1−α/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 48 which tends to zero for n → ∞ with k = k(n) = o(n) since the hidden constants do not depend on n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Recalling the definition of the function class ˜Gk in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) with n replaced by k, we obtain ≲E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup f∈F max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p |Gµ n,k( f (c+2hi/ √ k)(c+2hi/ √ k)cc − f cccc)| \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup f∈ ˜Gk ����Gµ n,k( f) ���� \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, Dudley’s entropy integral in combination with our previous considerations yields E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup f∈ ˜Gk ����Gµ n,k( f) ���� \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≤ Eµn \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0Eµb n,k \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup f∈ ˜Gk | √ k(µb n,k − µn)( f)| ������ µn \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + � k nE \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0sup f∈ ˜Gk |Gµ n( f)| \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≲ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed1 + � k n \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 � maxi=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p 4∥hi∥∞/ √ k 0 � log (N(ε, F cc + [−B, B], ∥·∥∞))dε ≲ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed1 + � k n \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 � max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',p ∥hi∥∞ / √ k �1−α/2 , which goes to zero for n, k(n) → ∞ with k(n) = o(n) (the hidden constants are independent of n, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Using the same arguments as in (i), we conclude that sup f∈F |Gµ n,k( f ˜cb n,k ˜cb n,k − f cc)| P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, analogous arguments yield that sup f∈F |Gν n,k( f cb n,k − f c)| P→ 0, thus showing (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 Define the random variables ˜cn ≔ \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 c if n < N, cn if n ≥ N, and ˜cb n,k ≔ \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3 c if n < N or k < K, cb n,k if n ≥ N and k ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 the cost estimators ˜cn and ˜cb n,k satisfy the entropy bounds in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Tightness of N and K implies that √n ∥˜cn − cn∥∞ P→ 0 and √ k∥˜cb n,k − cb n,k∥∞ P→ 0 for n, k → ∞, which asserts the claim by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 By Assumption (JW) it follows that √nm/(n + m)(cn,m − c) ⇝ Gc, whereas under (JW)∗ we infer from Bücher and Kojadinovic (2019) and k = o(n) that √ k(cb n,k − c) ⇝ Gc unconditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In what follows we state the arguments for (Sup);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' for (Sup)∗ a similar proof strategy applies by replacing the empirical costs process by the bootstrap cost process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' First, assume without loss of generality that the population cost function fulfills ∥c∥∞ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, for all three settings of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 it follows that log N(ε, F cc, ∥·∥∞) ≲ ε−α with α < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (i) we set ˜cn,m ≔ Ψbdd(cn,m), for Ψbdd defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since ∥˜cn,m∥∞ ≤ 2 and F = F (2 ∥c∥∞ + 1, 2w) is uniformly bounded by 6, we obtain that F ˜cn,m is uniformly bounded by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 both conditions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6(i) are met, asserting (Sup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (ii) we take ˜cn,m ≔ Ψ ˜dX mod ◦ Ψbdd(cn,m) for Ψ ˜dX mod from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, ∥˜cn,m∥∞ ≤ 2 and F ˜cn,m is uniformly bounded by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by Assumption (ii)’ it follows with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 that √nm/(n + m)∥˜cn,m − cn,m∥∞ P→ 0 and that sup n∈N log N(ε, F ˜cn,m ˜cn,m, ∥·∥∞) ≲ N(ε/8, X, ˜dX)| log(ε)| ≲ ε−β| log(ε)| ≲ ε−2+(2−β)/2, 49 where we used the covering number assumption on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (Sup) then follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For setting (iii) define ci ∈ C(Ui × Y) as ci(u, y) ≔ c(ζi(u), y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We consider ˜cn,m ≔ Ψcom(cn,m) where Ψcom denotes the combination (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) of regularity elevation functionals Ψi : C(Ui × Y) → C(Ui × Y) defined by Ψi = Ψ∥·∥γi mod ◦ Ψbdd from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 if γi ∈ (0, 1], and Ψi = Ψci,γi Hol ◦ Ψbdd from Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 if γi ∈ (1, 2], where we replace X by Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, by Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 the functional Ψ fulfills the assumptions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and therefore √nm/(n + m)∥˜cn,m−cn,m∥∞ P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, since for any ˜c ∈ C(X × Y) it holds that ∥Ψ(˜c)∥∞ < C for a deterministic constant C ≥ 0 that only depends on the functions ci and the spaces Ui, it follows that F Ψ(˜c) is uniformly bounded by C + 6 and therefore sup n∈N log N(ε, F ˜cn,m ˜cn,m, ∥·∥∞) ≲ I� i=1 sup ˜ci∈C(Ui×Y) log N(ε, F ˜cn,mΨi(˜ci), ∥·∥∞) ≲ max i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=',I ε−di/γi, where we use for the first inequality Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6, and for the second we employ the bounds from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 with N(ε, Ui, ∥·∥γi) ≲ ε−di/γi for 0 < γi ≤ 1 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 for 1 < γi ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The assertion then follows by an application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='10 For ε > 0 suppose that the right-hand side is finite since otherwise the claim is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Set k = N(ε/4, Θ, dΘ) and let {θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , θk} be a minimal ε/4-covering of Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , k let { f i 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , f i ki} be a minimal ε/2-covering of F cθicθi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', ki = N (ε/2, F cθicθi, ∥·∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Once we show that FX(ε) ≔ �k i=1{ f i 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , f i ki} is an ε-covering for � θ∈Θ F cθcθ and that FY(ε) ≔ �k i=1{( f i 1)cθi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , ( f i ki)cθi} is an ε-covering for � θ∈Θ F cθ the claim follows, since |FY(ε)| ≤ |FX(ε)| = k � i=1 N �ε 2, F cθicθi, ∥·∥∞ � ≤ N �ε 4, Θ, dΘ � sup θ∈Θ N �ε 2, F cθcθ, ∥·∥∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, let θ ∈ Θ and f ∈ F cθcθ, and choose ˜f ∈ F with f = ˜f cθcθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Select θi with dΘ(θ, θi) ≤ ε/4 and choose f i li ∈ FX(ε) such that ���� f i li − ˜f cθicθi ����∞ ≤ ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Now, by Lipschitzianity of the cost in θ and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 we infer ��� ˜f cθicθi − ˜f cθcθ���∞ ≤ 2dΘ(θ, θi) ≤ ε/2, and it follows that ��� f i li − f ���∞ = ���f i li − ˜f cθcθ���∞ ≤ ���f i li − ˜f cθicθi���∞ + ��� ˜f cθicθi − ˜f cθcθ���∞ ≤ ε, which verifies that FX(ε) is an ε-covering of ∪θ∈ΘF cθcθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for any f ∈ F cθ there exists ˜f ∈ F with f = ˜f cθ and by Santambrogio (2015, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='34) it follows that ˜f cθ = ˜f cθcθcθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, upon selecting f i li ∈ FX(ε) as above, we find by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that ���( f i li)cθi − ˜f cθi���∞ = ���( f i li)cθi − ˜f cθicθicθi���∞ ≤ ��� f i li − ˜f cθicθi���∞ ≤ ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Again invoking Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 yields ��� ˜f cθi − ˜f cθ���∞ ≤ d(θ, θi) ≤ ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consequently, we find that ���( f i li)cθi − f ���∞ = ���( f i li)cθi − ˜f cθ���∞ ≤ ���( f i li)cθi − ˜f cθi���∞ + ��� ˜f cθi − ˜f cθ���∞ ≤ 3ε 4 ≤ ε, which proves that FY(ε) is an ε-covering of ∪θ∈ΘF cθ and finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 50 C Proofs for Section 4: Applications C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Select U as the pre-image of {˜g ∈ C(X, Rd): ∥˜g − g−1 ϑo ∥∞ < 1} under KΘ, which is open (rela- tive) in Θ due to continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by compactness of X, the collection {g−1 ϑ }ϑ∈U is uniformly bounded on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Invoking the Cauchy-Schwarz inequality and, due to compactness of Y, we infer that {CΘ(ϑ)(x, ·)}ϑ∈U,x∈X is also uniformly bounded on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, since ∇yCΘ(ϑ)(x, y) = 2�g−1 ϑ (x) − y� for y ∈ int(Y) the collection {∇yCΘ(ϑ)(x, ·)} is bounded on Y uniformly over ϑ ∈ U, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Fi- nally, note that HessyCΘ(ϑ)(x, y) = −2Id, independent of ϑ ∈ U, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Thus, by combining these observations, we conclude the existence of Λ ≥ 0 such that the (2, Λ)-Hölder regularity is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 To establish the Hadamard differentiability of CΘ at ϑo note that ����� CΘ(ϑo + tnhn) − CΘ(ϑo) tn − DH |ϑoCΘ(h) �����∞ = sup (x,y)∈X×Y ����� 1 tn � g−1 ϑo+tnhn(x) − g−1 ϑo (x), g−1 ϑo+tnhn(x) + g−1 ϑo (x) − 2y � − 2 � DH ϑoKΘ(h)(x), g−1 ϑo (x) − y ������ ≤ sup (x,y)∈X×Y �������2 � 1 tn � g−1 ϑo+tnhn(x) − g−1 ϑo (x) � − DH ϑoKΘ(h)(x), g−1 ϑo (x) − y ������� + 1 tn ���g−1 ϑo+tnhn(x) − g−1 ϑo (x) ���2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For n → ∞, the first term tends to zero by Hadamard differentiability of KΘ whereas the second term tends to zero by (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, CΘ is Hadamard differentiable at ϑo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The second assertion follows from the functional delta method for Hadamard differentiable functionals (Römisch, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 First note that, in comparison to Sections 2 and 3, the roles of X and Y are interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The universal Donsker property of F CΘ(ϑo) follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1(iii) since d ≤ 3 and CΘ(ϑo)(x, ·) is (2, Λ)-Hölder for some Λ ≥ 0 uniformly in x ∈ X (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, note by measurability of ϑn and continuity of CΘ near ϑo that cn is also measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By joint weak convergence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6) we infer from Hadamard differentiability of CΘ at ϑo (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) using the functional delta method that the one-sample version of (JW) (recall Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4(ii)) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, since ϑn P→ ϑo, as n tends to infinity, we infer from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that the one-sample version of (Sup) is also met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The assertion now follows at once from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Note that by assumption, (T , dT ) and c fulfill the requirements of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Furthermore, note that Assumption (Don) can be established via Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 and Assumption (KP) is implied by the assumptions on the support of µ and ν (Staudt, Hundrieser, and Munk, 2022, Corollary 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, the statement follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 Select X ⊆ Rd as a compact set which contains the supports of µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Note that Sd−1 is a compact Polish space and consider the Lipschitz map cSd−1 : (Sd−1, ∥·∥) → C(X × X), θ �→ cθ|X×X whose modulus depends on X and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By compactness of X and Sd−1 it thus follows from the Theorem of Arzelà-Ascoli that {cθ|X×X}θ∈Sd−1 is uniformly bounded and equicontinuous with a uniform modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 51 Therefore, upon choosing the function class F as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5, Assertion (i) follows by Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 once we verify that Assumption (Don) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, note that log N(ε, Sd−1, ∥·∥) ≲ | log(ε)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, define for θ ∈ Sd−1 the pseudo metric ˜dθ,X(x, y) = |θT x−θTy| on X which fulfills supθ∈Sd−1 N(ε, X, ˜dθ,X) ≲ ε−1 and for any x, x′, y ∈ X, ���cθ(x, y) − cθ(x′, y) ��� ≤ p diam(pθ(X))p−1|θT x − θT x′| ≤ p diam(X)p−1 ˜dθ,X(x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since the upper bound for the Lipschitz modulus does not depend on θ, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9(ii) is appli- cable and we conclude that � θ∈Sd−1 F cθcθ and � θ∈Sd−1 F cθ are universal Donsker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By applying the continuous mapping theorem (van der Vaart and Wellner, 1996, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) for the integration operator over Sd−1 we obtain Assertion (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, Assertion (iii) follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='9 Since X × Y is compact and by continuity of c and ∆c there exists a common modulus of continuity w for {ct(·, y)}y∈Y,t∈[0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, for any t ∈ [0, 1] we have c, ct ∈ C(∥c∥∞ + ∥∆c∥∞ + 1, w) (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 for the definition of C(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consequently, we infer by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 the inequalities, 1 t \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed inf π∈Π⋆ct(µt,νt) π(t∆c) + sup f∈Sc(µ,ν) t∆µ( f cc) + t∆ν( f c) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≤1 t (OT(µt, νt, ct) − OT(µ, ν, c)) ≤1 t \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed inf π∈Π⋆c (µ,ν) π(t∆c) + sup f∈Sct(µt,νt) t∆µ( f cc) + t∆ν( f c) + sup f∈F t∆µ( f ctct − f cc) + t∆ν( f ct − f c) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Next, we observe that ∆µ = ˜µ − µ for some ˜µ ∈ P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This yields using Lipschitzianity under cost transformations with respect to the cost function (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) that sup f∈F ���∆µ( f ctct − f cc) ��� = sup f∈F ���(˜µ − µ)( f ctct − f cc) ��� ≤ 4 ∥ct − c∥∞ = 4t ���∆c���∞ t→0 −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Likewise, it follows that | sup f∈F ∆ν( f ct − f c)| → 0 for t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, since the pair (µt, νt) weakly converges for t ց 0 to (µ, ν) it follows by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 that lim inf tց0 inf π∈Π⋆ct(µt,νt) π(∆c) + sup f∈Sc(µ,ν) ∆µ( f cc) + ∆ν( f c) ≥ inf π∈Π⋆c (µ,ν) π(∆c) + sup f∈Sc(µ,ν) ∆µ( f cc) + ∆ν( f c) as well as lim sup tց0 inf π∈Π⋆c (µ,ν) π(∆c) + sup f∈Sct(µt,νt) ∆µ( f cc) + ∆ν( f c) ≤ inf π∈Π⋆c (µ,ν) π(∆c) + sup f∈Sc(µ,ν) ∆µ( f cc) + ∆ν( f c), which yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 52 D Proofs for Section 5: Regularity Elevation Functionals D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 By the functional delta method (Römisch, 2006) and the assumptions on Ψ and L it follows that an � ( fn − f) (Ψ( fn) − f) � ⇝ � L DH f Ψ(L) � d= �L L � for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The continuous mapping theorem (van der Vaart and Wellner, 1996, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) in combi- nation with measurability of the random elements fn and Ψ( fn) (due to continuity Ψ near f) thus asserts an �Ψ( fn) − fn � P→ 0 for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 First note that Ψ(˜c) ∈ C(X×Y) for any ˜c ∈ C(X×Y) as a concatenation of continuous functions and under ∥˜c∥∞ < 2 that Ψ(˜c) = ˜c, which yields Ψ(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, this shows that Ψ: C(X × Y) → C(X × Y) is continuous near c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For Hadamard differentiability at c consider a positive sequence tn ց 0 and take a converging sequence (hn)n∈N ⊆ C(X × Y) with limit h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since h is bounded and ∥c∥∞ ≤ 1, for n sufficiently large we have ∥c + tnhn∥∞ < 2 and therefore Ψ(c + tnhn) = c + tnhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We then obtain ����� Ψ(c + tnhn) − Ψ(c) tn − h �����∞ = ∥hn − h∥∞ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, since for any ˜c ∈ C(X × Y) it holds that ∥gΨ(˜c)∥∞ ≤ B + 2 where B ≔ supg∈G ∥g∥∞ we find for a finite space X that sup ˜c∈C(X×Y) log N(ε, GΨ(˜c), ∥·∥∞) ≤ |X|(log(B + 2) + | log(ε)|) ≲ | log(ε)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 By condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) it follows for x, x′ ∈ X with ˜dX(x, x′) = 0 that c(x, y) = c(x′, y), whereas under ˜dX(x, x′) > 0 we have by w(δ) > 0 for δ > 0 that c(x, y) ≤ c(x′, y) + w( ˜dX(x, x′)) < c(x′, y) + 2w( ˜dX(x, x′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This asserts for any (x, y) ∈ X × Y that S (c, (x, y)) ≔ arg min x′∈X c(x′, y) + 2w � ˜dX(x, x′) � = {x′′ ∈ X | ˜d(x, x′′) = 0}, and overall yields by ∥c∥∞ ≤ 1 that Ψ(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the second and third claim, recall from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 that Ψbdd : C(X × Y) → C(X × Y) is continuous near c and Hadamard differentiable at c with derivative IdC(X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, it suffices to verify that Ψw◦ ˜dX mod is continuous near c and Hadamard directionally differentiable with DH |c Ψ|C( ˜X×Y) = IdC( ˜X×Y) for which we rely on Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Define the spaces V ≔ C(X × Y), F = X, Θ = ˜X × Y and the functional Ew◦ ˜dX : V × F × Θ = C(X × Y) × X × ( ˜X × Y) �→ R, (˜c, x′, (x, y)) �→ −˜c(x′, y) − 2w( ˜dX(x, x′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 53 For any ˜c ∈ C(X×Y) the function Ew◦ ˜dX(˜c, ·, ·): X×( ˜X×Y) → R is continuous as a sum of continu- ous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for any (x′, (x, y)) ∈ X×( ˜X×Y) note that the function Ew◦ ˜dX(·, x′, (x, y)): C(X× Y) → R is 1-Lipschitz under uniform norm and that ∆cEw◦ ˜dX(˜c, x′, (x, y)) ≔ Ew◦ ˜dX(˜c + c, x′, (x, y)) − Ew◦ ˜dX(c, x′, (x, y)) = −˜c(x′, y) is linear in ˜c ∈ C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 we obtain continuity of the functional Ψw◦ ˜dX mod : C(X × Y) → C( ˜X × Y), ˜c �→ � (x, y) �→ inf x′∈X ˜c(x′, y) + 2w( ˜dX(x, x′)) = − sup x′∈X Ew◦ ˜dX(˜c, x′, (x, y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider the closed sub-vector space U ≔ C( ˜X × Y) ⊆ C(X × Y), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' It remains to show Assumption (DC) of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, note for h ∈ C( ˜X × Y) that h(x, y) + 2w( ˜dX(x, x′)) = h(x′, y) + 2w( ˜dX(x′, x′)) for any x, x′ ∈ S (c, (x, y)) since ˜dX(x, x′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This implies by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 that (DC) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 thus asserts that Ψw◦ ˜dX mod is Hadamard directionally differentiable at c with derivative given by DH |c Ψw◦ ˜dX mod : C(X × Y) → C( ˜X × Y), h �→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed(x, y) �→ inf x′ : ˜dX(x′,x)=0 h(x′, y) = − sup x′ : ˜dX(x′,x)=0 −∆cEw◦ ˜dX(h, x′, (x, y)) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, if h ∈ C( ˜X × Y), then DH |cΨw◦ ˜dX mod (h) = h, which yields DH |c Ψw◦ ˜dX mod |C( ˜X×Y) = IdC( ˜X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the last claim note that any ˜c ∈ C(X × Y) fulfills for (x, y) ∈ X × Y that − ∥˜c∥∞ ≤ Ψw◦ ˜dX mod (˜c)(x, y) ≤ ˜c(x, y) ≤ ∥˜c∥∞ , and hence ∥Ψ(˜c)∥∞ = ∥Ψw◦ ˜dX mod ◦ Ψbdd(˜c)∥∞ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for any x, x′ ∈ X, y ∈ Y we have Ψw◦ ˜dX mod (˜c)(x, y) − Ψw◦ ˜dX mod (˜c)(x′, y) ≤ inf x′′∈X c(x′′, y) + 2w( ˜dX(x′′, x)) − c(x′′, y) − 2w( ˜dX(x′′, x′)) ≤ 2w( ˜dX(x, x′)), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) where we used the reverse triangle inequality since w ◦ ˜dX defines a (pseudo-)metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We thus conclude for any ˜c ∈ C(X × Y) and a bounded function class G with B ≔ supg∈G ∥g∥∞ < ∞ from ∥Ψ(˜c)∥∞ ≤ 2 and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) that the elements of GΨ(˜c) are bounded by B + 2 and 2-Lipschitz under w ◦ ˜dX as an infimum over such 2-Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, GΨ(˜c) ⊆ BL(B+2),2(X, w ◦ ˜dX) where for the latter class uniform metric entropy bounds are available by Kolmogorov and Tikhomirov (1961, Section 9), asserting for any ε > 0 N(ε, BL(B+2),2(X, w ◦ ˜dX), ∥·∥∞) = N(ε/2, BL(B+2)/2,1(X, w ◦ ˜dX), ∥·∥∞) ≲ N(ε/8, X, w ◦ ˜dX)| log(ε)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 We infer from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 that Ψ ≔ B · Ψw◦dX/B mod Ψbdd(·/B) is continuous near c and Hadamard differentiable at c with derivative DH |cΨ = IdC(X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, invoking Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 it follows that an(cn − cn) P→ 0 for n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by definition of Ψbdd and Ψw/B,dX mod it follows that ∥cn∥∞ ≤ 2B and that cn fulfills (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) with w replaced by 2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The inclusion now follows at once from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 54 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Since c is (γ, 1)-Hölder it follows for any x, x′ ∈ X and y ∈ Y as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 of Hundrieser, Staudt, and Munk (2022) by convexity of X that c(x, y) = c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + Rx′(x) with |Rx′(x)| ≤ √ d ���x − x′���γ , and consequently, for x � x′ we obtain c(x, y) < c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 √ d ���x − x′���γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This asserts for any (x, y) ∈ X × Y that S (c, (x, y)) ≔ arg min x′∈X c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 √ d ���x − x′���γ = {x} and yields by ∥c∥∞ ≤ 1 that Ψ(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To show the claim on continuity and Hadamard differentiability it suffices to verify that ΨHol is continuous near c and that is Hadamard differentiable at c with derivative DH |c ΨHol = IdC(X×Y) for which we rely on Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Set V ≔ C(X × Y), F ≔ X and Θ ≔ X × Y and define the functional EHol : V × F × Θ = C(X × Y) × X × (X × Y) → R, (˜c, x′, (x, y)) �→ − � ˜c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 √ d ���x − x′���γ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For any ˜c ∈ C(X × Y) the functional EHol(˜c, ·, ·): X × (X × Y) → R is continuous by continuity of ∇xc(·, ·) and for any (x′, (x, y)) ∈ X × (X × Y) the functional EHol(·, x′, (x, y)): C(X × Y) → R is 1-Lipschitz under uniform norm while ∆cEHol(˜c, x′, (x, y)) = EHol(˜c + c, x′, (x, y)) − EHol(c, x′, (x, y)) = −˜c(x′, y) is linear in ˜c ∈ C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, condition (DC) follows by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 since S (c, (x, y)) = {x} is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 the functional ΨHol : C(X × Y) → C(X × Y), ˜c �→ � (x, y) �→ − sup x′∈X EHol(˜c, x′, (x, y)) � is continuous near c and Hadamard differentiable at c with derivative DH |cΨHol : C(X × Y) → C( ˜X × Y), h �→ � (x, y) �→ h(x, y) = −∆cEHol(h, x′, (x, y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the claim on the uniform metric entropy bound let ˜c ∈ C(X × Y), and assume (after applica- tion of Ψbdd) that ∥˜c∥∞ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Define the collection of functions ( ˜Ex′,y)x′∈X,y∈Y with ˜Ex′,y : X → R, x �→ ˜c(x′, y) + ⟨∇xc(x′, y), x − x′⟩ + 2 √ d ���x − x′���γ , which is (γ, 2)-Hölder on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by Hundrieser, Staudt, and Munk (2022, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5) there exists another collection ( ˜Eσ x′,y)x′∈X,y∈Y,σ∈(0,1] of smooth functions on X such that sup x′∈X y∈Y ���� ˜Ex′,y − ˜Eσ x′,y ����∞ ≤ Kσγ and sup x′∈X y∈Y ���� ˜Eσ x′,y ����C2(X) ≤ Kσγ−2, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) for all σ > 0 and some independent K > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Here, the C2(X)-norm of a twice continuously differen- tiable function g : X ⊂ Rd → R is defined as ∥g∥C2(X) ≔ max |β|≤2 ���Dβg ���∞ , where Dβg = ∂|β|g/∂xβ1 1 · · · xβd d for β ∈ Nd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 55 Note that a function with ∥g∥C2(X) ≤ Γ for Γ > 0 is absolutely bounded by Γ, it is Γ-Lipschitz, and dΓ-semi-concave (for a formal definition see Albano, 2002 or Hundrieser, Staudt, and Munk, 2022), since the Eigenvalues of its Hessian are bounded by d · Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Upon defining c(x, y) ≔ Ψ(˜c)(x, y) = infx′∈X ˜Ex′,y(x) and cσ(x, y) ≔ infx′∈X ˜Eσ x′,y(x) we thus obtain from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) that ���c − cσ���∞ ≤ Kσγ and that cσ is semi-concave of order Γ(σ) ≔ dKσγ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, following along the lines the proofs of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 in Hundrieser, Staudt, and Munk, 2022 we obtain for any ε > 0 and σ(ε) ≔ (ε/4K)1/γ that N(ε, Gc, ∥·∥∞) ≤ N(ε/2, Gcσ(ε), ∥·∥∞) = N \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed ε 2Γ(σ(ε)), Gcσ(ε) 2Γ(σ(ε)), ∥·∥∞ \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ≲ � ε 2Γ(σ(ε)) �−d/2 ≲ ε−d/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Here, we used in the second inequality that Gcσ(ε)/2Γ(σ(ε)) is contained in the collection of functions on X which are absolutely bounded by B ≥ 0, Lipschitz with modulus L ≥ 0 and 1-semi-concave, where B depends on G and L depends on X, in conjunction with uniform metric entropy bounds by Bronshtein (1976) and Guntuboyina and Sen (2013) for convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, since the hidden constants do not depend on ˜c, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 For the first claim note that Ψi(ci) = ci for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , I} and consequently, it follows for x ∈ ζi(Ui) that Ψi(ci)(ζ−1 i (x), y) = c(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, since �I i=1 ηi(x) ≡ 1 it follows that Ψ(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The claim on continuity of Ψ near c follows by continuity of the functionals Ψi : C(Ui × Y) → C(Ui × Y) near ci for each 1 ≤ i ≤ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the claim on Hadamard differentiability of Ψ define for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' , I} the functionals Ψ1 com,i: C(X × Y) → C(Ui × Y), ˜c �→ ((u, y) �→ ˜c(ζi(u), y)) , Ψ2 com,i: C(Ui × Y) → C(ζi(Ui) × Y), ˜c �→ � (x, y) �→ ˜c(ζ−1 i (x), y) � , where both maps assign to the respective spaces of continuous functions since ζ−1 i and ζi are both continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, note for any ˜c ∈ C(X × Y) that Ψ(˜c) = �I i=1 ηi · Ψ2 com,i ◦ Ψi ◦ Ψ1 com,i(˜c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Both functionals Ψ1 com,i and Ψ2 com,i are Hadamard differentiable at ci with derivative DH |ciΨ1 com,i: C(X × Y) → C(Ui × Y), h �→ ((u, y) �→ h(ζi(u), y)) , DH |ciΨ2 com,i: C(Ui × Y) → C(ζi(Ui) × Y), h �→ � (x, y) �→ h(ζ−1 i (x), y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By assumption on Ψi and chain rule we infer that Ψ is Hadamard differentiable at c with derivative DH c Ψ: C(X × Y) → C(X × Y), h �→ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed(x, y) �→ I� i=1 ηi(x)h(ζ−1 i (ζi(x)), y) = I� i=1 ηi(x)h(x, y) = h(x, y) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 and conclude that DH |c Ψ = IdC(X×Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, the bound on the covering numbers is a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 in Hundrieser, Staudt, and Munk, 2022 as they assert for arbitrary ˜c ∈ C(X × Y) that log N(ε, GΨ(˜c), ∥·∥∞) ≤ I� i=1 log N(ε, GΨ(˜c)|ζi(Ui), ∥·∥∞) ≤ I� i=1 log N(ε, GΨi(˜c) ◦ ζi, ∥·∥∞) = I� i=1 log N(ε, GΨi(˜c(ζi(·),·)), ∥·∥∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 56 E Proofs for Section 6: Lemmata of Distributional Limits E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 Assume ∥f − ˜f∥∞ + ∥c − ˜c∥∞ < ∞ since otherwise the claim is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For ˜f and ˜c there exists for y ∈ Y and ε > 0 some x′ ∈ X such that ˜f ˜c(y) ≥ ˜c(x′, y) − ˜f(x′) − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, f c(y) − ˜f ˜c(y) = � inf x∈X c(x, y) − f(x) � − � inf x∈X ˜c(x, y) − ˜f (x) � ≤ c(x′, y) − f(x′) − ˜c(x′, y) + ˜f(x′) + ε ≤ ��� f − ˜f ���∞ + ∥c − ˜c∥∞ + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As ε > 0 can be chosen arbitrarily small, we obtain for any y ∈ Y the inequality f c(y) − ˜f ˜c(y) ≤ ���f − ˜f ���∞ + ∥c − ˜c∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Repeating the argument for f and c asserts the converse inequality and proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2 Let us start by splitting the problem in two different ways, OT(˜µ, ˜ν, ˜c) − OT(µ, ν, c) = (OT(˜µ, ˜ν, ˜c) − OT(˜µ, ˜ν, c)) + (OT(˜µ, ˜ν, c) − OT(µ, ν, c)) = (OT(˜µ, ˜ν, ˜c) − OT(µ, ν, ˜c)) + (OT(µ, ν, ˜c) − OT(µ, ν, c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since c, ˜c ∈ C(2 ∥c∥∞ + 1, 2w), we can employ the dual representation of the OT value from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 with F = F (2 ∥c∥∞ + 1, 2w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, for each bracket in the display above, one can choose to plug-in a feasible plan in the primal formulation or a potential from F in the dual formu- lation to obtain upper and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Doing so, we obtain inf π∈Π⋆ ˜c (˜µ,˜ν) π(˜c − c) ≤ OT(˜µ, ˜ν, ˜c) − OT(˜µ, ˜ν, c) ≤ inf π∈Π⋆c (˜µ,˜ν) π(˜c − c), sup f∈Sc(µ,ν) (˜µ − µ) f cc + (˜ν − ν) f c ≤ OT(˜µ, ˜ν, c) − OT(µ, ν, c) ≤ sup f∈Sc(˜µ,˜ν) (˜µ − µ) f cc + (˜ν − ν) f c, OT(µ, ν, ˜c) − OT(µ, ν, c) ≤ inf π∈Π⋆c (µ,ν) π(˜c − c), OT(˜µ, ˜ν, ˜c) − OT(µ, ν, ˜c) ≤ sup f∈S˜c(˜µ,˜ν) (˜µ − µ) f ˜c˜c + (˜ν − ν) f ˜c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, for the last upper bound we further note that sup f∈S˜c(˜µ,˜ν) (˜µ−µ) f ˜c˜c +(˜ν−ν) f ˜c ≤ sup f∈S˜c(˜µ,˜ν) (˜µ−µ) f cc +(˜ν−ν) f c + sup f∈F (˜µ−µ)( f ˜c˜c − f cc)+(˜ν−ν)( f ˜c − f c), which overall yields the lower and upper bounds for the OT cost under varying measures and costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, the bound under fixed measures µ, ν it follows by Hölder’s inequality for any π ∈ Π(µ, ν) that |π(˜c − c)| ≤ ∥˜c − c∥∞, whereas under a fixed cost function c we have sup f∈Sc(˜µ,˜ν)∪Sc(µ,ν) ���(˜µ − µ) f cc + (˜ν − ν) f c��� ≤ sup f∈F ���(˜µ − µ) f cc��� + sup f∈F ���(˜ν − ν) f c��� = sup f∈F cc ���(˜µ − µ) f ��� + sup f∈F c ���(˜ν − ν) f ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 57 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3 The continuity of T1 is a consequence of Villani (2008, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, any converging sequence (µn, νn, cn) with limit (µ∞, ν∞, c∞) admits a sequence of OT plans πn ∈ Π⋆ cn(µn, νn) which converges weakly along a subsequence, say (πnk)k∈N, to an OT plan π∞ ∈ Π⋆ c∞(µ∞, ν∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, lim sup k→∞ |T1(µnk, νnk, cnk) − T1(µ∞, ν∞, c∞)| = lim sup k→∞ |OT(µnk, νnk, cnk) − OT(µ∞, ν∞, c∞)| = lim sup k→∞ |πnk(cnk) − π∞(c∞)| ≤ lim sup k→∞ |(πnk − π∞)(c∞)| + ���cnk − c∞ ���∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since this holds for any sequence of converging OT plans, continuity of T1 follows from Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the lower semi-continuity of T2 take a sequence (hc,n)n∈N with limit hc,∞ and consider OT plans πn ∈ Π⋆ cn(µn, νn) such that inf π∈Π⋆cn(µn,νn) π(hc,n) ≥ πn(hc,n) − 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, by Villani (2008, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='20) a converging subsequence (πnk)k∈N with limit π∞ ∈ Π⋆ c∞(µ∞, ν∞) exists and it follows that lim inf k→∞ T2(µnk, νnk, cnk, hc,nk) = lim inf k→∞ inf π∈Π⋆cnk (µnk,νnk) π(hc,nk) ≥ lim inf k→∞ πnk(hc,nk) − 1/nk ≥ lim inf k→∞ πnk(hc,∞) − ���hc,∞ − hc,nk ���∞ − 1/nk = π∞(hc,∞) ≥ T2(µ∞, ν∞, c∞, hc,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consequently, by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, lower semi-continuity of T2 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To infer upper semi-continuity of T2, and thus continuity, at (µ∞, ν∞, c∞, hc,∞) under the assumption of a unique OT plan π⋆ ∈ Π⋆ c∞(µ∞, ν∞) note by Villani (2008, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='20) that for any sequence of OT plans πn ∈ Π⋆ cn(µn, νn) there exists a weakly converging subsequence πnk which tends to π⋆ for k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, we con- clude that lim sup k→∞ T2(µnk, νnk, cnk, hc,nk) = lim sup k→∞ inf π∈Π⋆cnk (µnk,νnk) π(hc,nk) ≤ lim sup k→∞ πnk(hc,nk) ≤ lim sup k→∞ πnk(hc,∞) − ���hc,∞ − hc,nk ���∞ = π∞(hc,∞) = T2(µ∞, ν∞, c∞, hc,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This implies by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 the upper semi-continuity of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for fixed (µ′, ν′, c′) the map T2 is continuous in hc since for any ˜hc it holds that |T2(µ, ν, c, hc) − T2(µ, ν, c, ˜hc)| ≤ ���hc − ˜hc ���∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To show upper semi-continuity of T3 take a sequence (hµ,n, hν,n)n∈N with limit (hµ,∞, hν,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Fur- ther, by definition of C, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that any cn ∈ C fulfills Hcn ⊆ F cncn ⊆ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Take a sequence fn ∈ Scn(µn, νn) ⊆ F such that T3(µn, νn, cn, hµ,n, hν,n) ≤ hµ( fn) + hν( fn) + 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 58 By compactness of F there exists a uniformly converging subsequence, say ( fnk)k∈N, with limit f∞ ∈ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Next, we demonstrate that f∞ ∈ Sc∞(µ∞, ν∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, we note OT(µ∞, ν∞, c∞) ≥ µ∞( f c∞c∞ ∞ ) + ν∞( f c∞ ∞ ) = lim k→∞ µnk( f c∞c∞ ∞ ) + νnk( f c∞ ∞ ) ≥ lim k→∞ µnk( f cnkcnk nk ) + νnk( f cnk nk ) − ���� f c∞c∞ ∞ − f cnkcnk nk ����∞ − ���� f c∞ ∞ − f cnk nk ����∞ = lim k→∞ OT(µnk, νnk, cnk) − ����f c∞c∞ ∞ − f cnkcnk nk ����∞ − ���� f c∞ ∞ − f cnk nk ����∞ = OT(µ∞, ν∞, c∞), where the last equality follows by continuity of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, we get f∞ ∈ Sc∞(µ∞, ν∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By continuity of hµ and hν on F and upon denoting the norm on Cu(F ) by ∥·∥F , we infer that lim sup k→∞ T3(µnk, νnk,cnk, hµ,nk, hν,nk) = lim sup k→∞ sup f∈Scnk (µnk,νnk) hµ,nk( f) + hν,nk( f) ≤ lim sup k→∞ hµ,nk( fnk) + hν,nk( fnk) + 1/nk ≤ lim sup k→∞ hµ,∞( fnk) + hν,∞( fnk) + ���hµ,∞ − hµ,nk ���F + ���hν,∞ − hν,nk ���F + 1/nk = hµ,∞( f∞) + hν,∞( f∞) ≤ T3(µ∞, ν∞, c∞, hµ,∞, hν,∞) and consequently, by Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, upper semi-continuity of T3 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Further, for fixed (µ′, ν′, c′) the map T3 is continuous in (hµ, hν) since for another (˜hµ, ˜hν) it holds that |T3(µ, ν, c, hµ, hν) − T3(µ, ν, c, ˜hµ, ˜hν)| ≤ ���˜hµ − ˜hµ ���F cc + ���˜hν − ˜hν ���F c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, for T4 take (h1,µ, ˜h1,µ, h1,ν, ˜h1,ν), (h2,µ, ˜h2,µ, h2,ν, ˜h2,ν) ∈ Cu(F )4 and note that |T4(h1,µ, ˜h1,µ, h1,ν, ˜h1,ν) − T4(h2,µ, ˜h2,µ, h2,ν, ˜h2,ν)| ≤ ���h1,µ − h2,µ ���F + ���˜h1,µ − ˜h2,µ ���F + ���h1,ν − h2,ν ���F + ���˜h1,ν − ˜h2,ν ���F , which asserts continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4 For (i) take f, g ∈ G, then |µ( f) − µ(g)| ≤ ∥f − g∥∞ and hence µ: G → R defines a Lipschitz map under uniform norm which asserts µ ∈ Cu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assertion (ii) follows from Giné and Nickl (2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Finally, (iii) follows from (ii) since for any g ∈ G the evaluations µn(g) = n−1 �n i=1 g(Xi) and µb n,k(g) = k−1 �k i=1 g(Xb i ) are Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='5 We first prove that Assumption (JW) implies for n, m → ∞ with m/(n + m) → λ ∈ (0, 1) that \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed √n � (µn − µ)( f cc) � f∈F √m � (νm − ν)( f c) � f∈F � nm n+m(cn,m − c) \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 = \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � Gµ n( f cc) � f∈F � Gν m( f c) � f∈F Gc n,m \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ⇝ \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed � Gµ( f cc) � f∈F � Gν( f c) � f∈F Gc \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) 59 in the Polish space Cu(F ) × Cu(F ) × C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To this end, consider the map Ψ: Cu(F cc) × Cu(F c) × C(X × Y) → Cu(F ) × Cu(F ) × C(X × Y), (α, β, γ) �→ ��α( f cc)� f∈F , �β( f c)� f∈F , γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' This map is well-defined (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', its range is correct) since for any (α, β) ∈ Cu(F cc) × Cu(F c) there exist moduli of continuity wα, wβ: R+ → R+ such that for f, ˜f ∈ F it follows by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that |α( f cc) − α( ˜f cc)| ≤ wα( ��� f cc − ˜f cc���∞) ≤ wα( ��� f − ˜f ���∞), |β( f c) − β( ˜f c)| ≤ wβ( ��� f c − ˜f c���∞) ≤ wβ( ��� f − ˜f ���∞), which assert that ((α( f cc))f∈F , (β( f c))f∈F , γ) ∈ Cu(F ) × Cu(F ) × C(X × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for any (α, β), (˜α, ˜β) ∈ Cu(F cc) × Cu(F c) we have sup f∈F |α( f cc) − ˜α( f cc)| = sup ˜f ∈F cc |α( ˜f ) − ˜α( ˜f )| and sup f∈F |β( f c) − ˜β( f c)| = sup ˜f∈F c |β( ˜f ) − ˜β( ˜f )|, hence the map Ψ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consequently, Assumption (JW) and the continuous mapping theorem (van der Vaart and Wellner, 1996, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) assert weak convergence (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, by Varadarajan (1958) the empirical measures (µn, νn) weakly converge a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' in P(X)× P(Y) to (µ, ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Note that P(X) × P(Y) is by compactness of X and Y a separable, complete metric space (Bolley, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Invoking Slutzky’s lemma (van der Vaart and Wellner, 1996, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7) in conjunction with (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) we thus obtain the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' In particular, by measurability of cn,m and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4, all involved quantities are Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the second claim note by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that any realization of µn, νm and cn,m leads the pro- cesses Gµ n( f cn,mcn,m) and Gν m( f cn,m) to be 2√n-Lipschitz and 2√m-Lipschitz in f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Thus, they are uniformly continuous in f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, for fixed f ∈ F we can show that the function ˜Gµ n : P(X) × C(X × Y) → R, (˜µ, ˜c) �→ √n(˜µ − µ)( f ˜c˜c) is upper semi-continuous (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=', in particular measurable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Indeed, for ˜µk ⇝ ˜µ in P(X) and ˜ck → ˜c in C(X × Y) it follows by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1, upper semi-continuity of f ˜c˜c and the Portmanteau Theorem (van der Vaart and Wellner, 1996, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) that lim sup k→∞ √n(˜µk − µ)( f ˜ck˜ck) ≤ lim sup k→∞ √n(˜µk − µ)( f ˜c˜c) + 2 √n ��� f ˜ck˜ck − f ˜c˜c���∞ ≤ √n(˜µ − µ)( f ˜c˜c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4(ii) we conclude that (Gµ n( f cn,mcn,m))f∈F is Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Likewise, we conclude (Gν m( f cn,m))f∈F is Borel measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consequently, by (Sup) we infer, for n, m → ∞, that � Gµ n( f cc) − Gµ n( f cn,mcn,m), Gν m( f c) − Gν m( f cn) � f∈F P→ (0, 0) in Cu(F )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The claim now follows by a combination of Slutzky’s lemma and the continuous mapping theorem (van der Vaart and Wellner, 1996, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='6 Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='8 The first claim follows by an observation in Römisch (2006) since the set of probability measures P(X) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For additional insights see Aubin and Frankowska (1990, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1) 60 For the second claim consider a sequence ∆n = (˜µn − µ)/tn with tn > 0 and ˜µn ∈ P(X) such that ∥∆n − ∆∥ ˜F = sup f∈ ˜F |∆n( f) − ∆( f)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, it follows from triangle inequality that ���∆( f) − ∆( f ′) ��� = ���∆n( f) − ∆n( f ′) + (∆ − ∆n)( f) + (∆ − ∆n)( f ′) ��� ≤ ���∆n( f − f ′) ��� + 2 ∥∆ − ∆n∥ ˜F Herein, the first term vanishes since ∆n( f − f ′) = (˜µn − µ)(κ)/tn = 0, whereas the second term converges for n → ∞ to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, ∆( f) = ∆( f ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The third claim relies on Portemanteau’s theorem (van der Vaart and Wellner, 1996, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='4) which asserts using the notion of outer probabilities P∗ that P � Gµ ∈ TµP(X) � ≥ lim sup n→∞ P∗ � √n(µn − µ) ∈ TµP(X) � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='7 Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 We start by proving (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Note for κ ∈ R that ( f + κ)c(y) = inf x∈X c(x, y) − f(x) − κ = f c(y) − κ, which yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' To show assertion (ii), observe by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1 that ���g(c+∆c)(c+∆c)���∞ ≤ ∥g∥∞ + 2 ���c + ∆c���∞ ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) Further, we find that − ∥c∥∞ − sup x∈X g(c+∆c)(c+∆c)(x) ≤ g(c+∆c)(c+∆c)c(y) ≤ ∥c∥∞ − sup x∈X g(c+∆c)(c+∆c)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) Using part (i) of this lemma, we obtain g(c+∆c)(c+∆c)cc = �� g(c+∆c)(c+∆c)�c + sup x∈X g(c+∆c)(c+∆c)(x) �c + sup x∈X g(c+∆c)(c+∆c)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Combining (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='3) with the above equation demonstrates that g(c+∆c)(c+∆c)cc ∈ Hc + [−B, B] and hence yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ F Elementary Analytical Results Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Consider a real-valued sequence (an)n∈N and let K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (i) If for any subsequence (ank)k∈N there exists a subsequence (ankl)l∈N with lim supl→∞ ankl ≤ K, then it follows that lim supn→∞ an ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (ii) If for any subsequence (ank)k∈N there exists a subsequence (ankl )l∈N with lim infl→∞ ankl ≥ K, then it follows that lim infn→∞ an ≥ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' (iii) If for any subsequence (ank)k∈N there exists a subsequence (ankl )l∈N with liml→∞ ankl = K, then it follows that limn→∞ an = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' We only prove (i) and note that (ii) and (iii) can be shown analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Assume that lim supn→∞ an = infn∈N(supm≥n am) ≥ K + ε for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Since (supm≥n am)n∈N is decreas- ing in n, this would imply that supm≥n am ≥ K + ε for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Hence, there would exist a subsequence of (an)n∈N, say (anl)l∈N, with anl ≥ K + ε/2 for all l ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' However, this would assert lim infl→∞ anl ≥ K + ε/2 > K, contradicting the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Thus, lim supn→∞ an ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 61 Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Let (X, dX) be a compact metric space and consider a continuous (pseudo-)metric ˜dX on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, (X, ˜dX) is a compact (pseudo-)metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Moreover, given a Polish space Y it follows that C((X, ˜dX) × Y) ⊆ C((X, dX) × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' The (pseudo-)metric properties are clearly fulfilled for (X, ˜dX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' By continuity of ˜dX under dX the canonical inclusion ι: (X, dX) → (X, ˜dX), x �→ x is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' As the image of a compactum under a continuous map is again compact the first claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' For the second claim, take h ∈ C((X, ˜dX) × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' Then, the composition map X × Y → R, (x, y) �→ h(ι(x), y) is continuous and therefore the canonical embedding h ◦ (ι, IdY) of h is included in C((X, dX) × Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtAzT4oBgHgl3EQfVfzp/content/2301.01287v1.pdf'} +page_content=' □ 62' metadata={'source': 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tomography via +convex optimization +Anatoli Juditsky ∗ +Arkadi Nemirovski † +Michael Zibulevsky‡ +January 10, 2023 +Abstract +Proper X-ray radiation design (via dynamic fluence field modulation, +FFM) allows to reduce effective radiation dose in computed tomogra- +phy without compromising image quality. It takes into account patient +anatomy, radiation sensitivity of different organs and tissues, and location +of regions of interest. We account all these factors within a general convex +optimization framework. +1 +Introduction +Recently, there has been significant research interest in dynamic fluence field +modulation (FFM), which consists in varying the beam shape throughout the +scan allowing for the adaptation of the spatial x-ray distribution to conform to +the patient anatomy (cf., e.g., [1] and references therein). A range of hardware +solutions have been proposed, including dynamic wedges [2, 3, 4], fluid-filled +chambers [5, 6, 7, 8], and slitbased multiple aperture devices [9, 10]. +Until now, to the best of our knowledge, the problem of FFM radiation +planning has been treated within non-convex optimization framework, and only +rough distribution of radiation with relatively small number of parameters (co- +efficients of basis functions) was modeled. In this work we utilize convex opti- +mization techniques to optimize the fine structure of the fluence, namely, the +amount of radiation sent into each projection bin (towards the corresponding +detector pixel). +∗LJK, Universit´e Grenoble Alpes, 700 Avenue Centrale, 38401 Domaine Universitaire de +Saint-Martin-d’H`eres, France anatoli.juditsky@univ-grenoble-alpes.fr +†Georgia +Institute +of +Technology, +Atlanta, +Georgia +30332, +USA, +nemirovs@isye.gatech.edu +‡Department of Computer Science, Technion—Israel Institute of Technology, Haifa 32000, +Israel, mzib@cs.technion.ac.il +This work was supported by Multidisciplinary Institute in Artificial intelligence MIAI @ +Grenoble Alpes (ANR-19-P3IA-0003) and by the Israel Council For Higher Education— +Planning & Budgeting Committee +1 +arXiv:2301.03379v1 [physics.med-ph] 6 Jan 2023 + +The main body of this paper is organized as follows. In Section 2 we present +the CT observation model underlying our derivations, introduce and motivate +the loss index of radiation design responsible for the asymptotical, as the total +amount of radiation grows, quality of the Maximum Likelihood recovery of the +image of interest. We then pose the problem of optimal radiation design as +the problem of optimizing the loss index under general convex constraints on +the design. The resulting optimization problem is convex and thus efficiently +solvable (at least in theory, and to some extent, also in practice). In Section +3, we report on the (in our appreciation, encouraging) results of a “proof of +concept” numerical experiment implementing the radiation design proposed in +Section 2. Section 4 contains some concluding remarks. +2 +CT Radiation Design +2.1 +CT observation scheme +The CT observation scheme we intend to consider is as follows: +1. x ∈ Rn ++ is the “signal” – the discretized body attenuation density, so that +n is the number of voxels in the field of view; +2. w ∈ Rm is the vector of observations; observations are indexed by bins, +m being the total number of bins; +3. A ∈ Rm×n is the “projection matrix”; +4. q ∈ Rn ++ is the vector with entries qi which are the expected numbers of +photons sent to bin i during the stude. +We assume that the i-th observation stemming from signal x is +ωi ∼ Poisson(qi exp{−[Ax]i}, +(1) +where Poisson(µ) is the Poisson distribution with parameter µ ≥ 0: +Probι∼Poisson(µ){ι = k} = µk +k! eµ, k = 1, 2, ... +We assume also that entries ωi in the vector of observations ω are inde- +pendent across i = 1, ..., m. +5. Given observation ω stemming from the unknown signal x, we want to +recover the vector Bx, where B ∈ Rν×n is a given matrix.1 We measure +the recovery error in the standard Euclidean norm ∥ · ∥2 on Rν. +1In our experiments Bx is a truncation of image x towards the region of interest (ROI). +2 + +2.2 +Radiation sensitivity - from voxels to bins +Informally, our goal is to find radiation design q, which provides best image +quality given the whole-body effective dose. Various organs and tissues have +different sensitivity to radiation—effective dose caused by unitary radiation ex- +posure. +This can be reflected in 3D voxel-wise sensitivity map s, which is +standardly used in radiation therapy planning. +We assume that, in addition, an (approximate) attenuation map of the body +is available, and we can compute an auxiliary bin-sensitivity vector c with entries +ci reflecting effective dose obtained from a unit of radiation sent into bin i. +Constant whole body effective dose is provided by the condition +cT q = const, +(2) +Thus our goal is to find radiation design q, which provides the best image quality +under constraint (2). +2.3 +Maximum Likelihood estimate +With A and q given, we intend to recover Bx as B�x(ω), where �x(ω) is the +Maximum Likelihood (ML) estimate. Denoting by aT +i , i = 1, 2, ..., m, the rows +of A, the log-likelihood of observing ω, the signal being x, is +Lq(ω, x) += +�m +i=1[ωi ln(qi exp{−aT +i x}) − qi exp{−aT +i x} − ln(ωi!)] += +�m +i=1[−ωiaT +i x − qi exp{−aT +i x}] + R(q, ω). +Consequently, maximizing Lq(ω, x) in x is the same as maximizing in x the +concave in x function +Lq(x) − [ +� +i +ωiai]T x, Lq(x) = − +� +i +qi exp{−aT +i x} +(3) +and is therefore an efficiently solvable problem; the resulting estimate +�x(ω) ∈ Argmax +x +� +Lq(x) − [ +� +i +ωiai]T x +� +of x is a function of ω depending on q as a parameter. +2.4 +Radiation Design via Loss index minimization +Our goal is “presumably good,” in terms of the performance of the resulting +estimate, selection of q in a given compact convex set Q ⊂ Rm ++. This informal +goal can be stated formally in different ways; the formalization we propose is +based on the following observation. +A. In the “noiseless case” where the observations are ωi = µi, µi = qi exp{−aT +i x∗}, +x∗ being the true signal (instead of being Poisson random variables with pa- +rameters µi), the ML estimate exactly recovers the true signal. In other words, +3 + +setting γ∗ = � +i µiai, we get x+ ∈ Argmaxx[Lq(x) − γT +∗ x]. Indeed, since Lq +is concave and smooth in x, it suffices to verify that ∇xLq(x∗) = γ∗, which is +evident. +B. Now assume that the positive semidefinite Hessian matrix +H := −∇2Lq(x∗) = +� +i +µiaiaT +i +is positive definite, or, which is the same when q > 0, that A is of rank m. Let +Lq(·) be the second order Taylor approximation of Lq(·) around x = x∗. Were +we specifying our estimate of x as +�x = �x(ω) := argmax +x +[Lq(x) − [ +� +i +ωiai]T x], +we would have +∇xLq(�x) = +� +i +ωiai, ∇xL(x∗) = +� +i +µiai, +that is, �x−x∗ = −H−1 � +i +[ωi − µi]ai +� +�� +� +δ +. Hence, taking into account the immediate +relation E{δδT } = H (recall that ωi ∼ Poisson(µi) are independent across i), +E{∥B�x − Bx∗∥2 +2} += +E +� +δT H−1BT BH−1δ +� += E +� +Tr +� +H−1BT BH−1δδT �� += +Tr +� +H−1BT BH−1E +� +δδT �� += Tr +� +H−1BT B +� += +Tr(BH−1BT ). +This computation suggests, as a meaningful “loss index” of q (the smaller it is, +the better for our purposes) the function +I∗(q) = Tr +� +�B +�� +i +qi exp{−aT +i x∗}aiaT +i +�−1 +BT +� +� +(4) +Needless to recall, the above reasoning is informal—our estimate of x is not �x, +it is �x. Nevertheless, it can be proved that when A is of rank m and q becomes +large, specifically, q = t¯q with ¯q > 0, when t → ∞, our informal reasoning yields +correct asymptotics of the expected squared ∥ · ∥2-error of the recovery of Bx2. +We, however, neither present these asymptotic results, nor need them—our +goal at this point is to find a convenient criterion to be optimized over q ∈ Q in +order to get a presumably good measurement design, and to this end no rigorous +justification of our choice is needed. What indeed is important for us, is that +I∗ is a convex function of q > 0, and thus is well suited for minimization. +A bad news is that strictly speaking, we cannot use I∗ as a criterion to +be minimized—this quantity depends, as on a parameter, on the unknown to +2Namely, with q = t¯q, ¯q > 0, one has E{∥B�x − Bx∗∥2 +2} = (1 + o(t))t−1I∗(¯q) as t → ∞. +4 + +us true signal x∗. On the other hand, when replacing in the right hand side +of (4) the unknown quantities ρ∗ +i := exp{−aT +i x∗} with their approximations +ρi which are within factor c from ρ∗ +i , that is c−1ρ∗ +i ≤ ρi ≤ cρ∗ +i , the resulting +right hand side in (4) is within the same factor from the true one. Therefore, +assuming that an approximate attenuation map ρ is available a priori, or that it +can be estimated using observation ω from a pilot run with, say, equal to each +other “moderate” qi = q, we may select q by minimizing over Q the observable +criterion +I(q) = Tr +� +�B +�� +i +qiρiaiaT +i +�−1 +BT +� +� . +(5) +This is, essentially, the course of actions we propose. +2.5 +Regularization +The problem of minimizing I(q) over q ∈ Q is convex and as such can, the- +oretically, be solved to whatever high accuracy in a computationally efficient +manner. At the same time, numerical experimentation shows that this prob- +lem may happen to be ill-conditioned, resulting in difficulties when solving it +numerically. +To overcome, to some extent, these difficulties, we can replace +minimizing over q ∈ Q the objective I(q) given by (5) with minimizing over the +same domain the penalized objective +Iλ(q) = Tr +� +�B +�� +i +qiρiaiaT +i + λIn +�−1 +BT +� +� +(6) +where regularizing coefficient λ is nonnegative; a good value of this coefficient +could be found experimentally. Thus, in the sequel we intend to optimize mea- +surement design by solving convex optimization problem +min +q∈Q +� +� +�Iλ(q) := Tr +� +�B +�� +i +qiρiaiaT +i + λIn +�−1 +BT +� +� +� +� +� +(7) +which can be solved by standard Convex Optimization algorithms. +It turns +out, however, that one can utilize problem’s structure to design a dedicated +algorithm which, as our experiments show, is much faster than the “general +purpose” algorithms. The following underlying observation is essentially well +known: +Proposition 1. Let P be an m×n matrix of rank n, B be a k×n matrix, and q, ρ +be positive m-dimensional vectors. Then, setting W = W[q] := Diag{qiρi, i ≤ +m}, +Iλ(q) := Tr +� +�B +�� +i +qiρiaiaT +i + λIn +�−1 +BT +� +� +5 + += +� +� +� +min +G∈Rk×m +� +λ−1∥GA − B∥2 +F + Tr(GW −1GT ) +� +, +λ > 0 +(8a) +min +G∈Rk×m +� +Tr(GW −1GT ) : GA = B +� +, +λ = 0 +(8b) +where ∥ · ∥F is the Frobenius norm +We provide the proof of Proposition 1 in Appendix A for the sake of com- +pleteness. +Alternating minimization. +Proposition 1 suggests the alternating mini- +mization algorithm for minimizing Iλ(q) over q ∈ Q. +Assume for the sake +of definiteness that λ > 0 (modification for the case of λ = 0 is immediate). By +(8a), the problem of interest (7) can be rewritten equivalently as +min +G∈Rk×n,q∈Q Φ(G, q) := +� +λ−1∥GA − B∥2 +F + ∥GW −1(q)GT ∥2 +� +To solve the latter problem, we start with some positive q0 ∈ Q. At a step t of +the algorithm, given positive qt−1 ∈ Q, +• we minimize Φ(G, qt−1) over G ∈ Rk×m; this is an unconstrained mini- +mization problem with strongly convex quadratic objective, and its solu- +tion Gt is given by explicit formula: +Gt = B[AT W[qt−1]A + λIn]−1AT W[qt−1]; +(9) +• after Gt is computed, we specify qt by minimizing Φ(Gt, q) over q ∈ Q, +which boils down to finding +qt ∈ Argmin +q∈Q +�m +j=1 +∥Colj(Gt)∥2 +2 +qjρj +� +�� +� +=Tr(GtW −1[q]GT +t ) +. +When Q is simple, the latter problem is simple as well. For instance, in +the case of Q = {q ∈ Rm ++ : cT q ≤ 1} with c > 0, assuming that Gt has no +zero columns (which can be ensured by an arbitrarily small perturbation +of Gt), the solution is given by +[qt]j = rj/cj +cT r , rj = ∥Colj[Gt]∥2 +� +cj/ρj, 1 ≤ j ≤ m. +(10) +After Gt, qt are computed, we pass to step t + 1. +3 +Computational experiment +Here we report on a toy “proof of concept” numerical experiment on recovering +n = 32 × 32-pixel Shepp-Logan type phantom with 68 projection angles spread +uniformly over 360 degrees. +6 + +Figure 1: Top row: phantom, sensitivity image and region of interest (ROI). +Bottom row: phantom sinogram, sensitivity of the bins and sinogram of the +ROI. +3.1 +Experimental setup +Unlike conventional tomography, where 180 degree angle span is sufficient, we +consider 360 degree span to account for the fact that effective radiation dose +is generally different when photons are sent along the same line from opposite +directions due to different attenuation between the source and a particular body +point. +In our experiments, the number of bins is m = 2828; we utilize the +baseline (uniform) radiation design qi ≡ 2e5 photons/bin. Figure 1, top row, +shows the attenuation map, radiation sensitivity, and region of interest (ROI) +which we aim to reconstruct. Bottom row shows the corresponding maps in the +sinogram (projection) space. In particular, the middle picture shows radiation +sensitivity in projection space, i.e. effective dose caused by a unit of radiation +sent into each projection bin. One can see a bright curve strip of high sensitivity. +As we will see later, our algorithm will reduce amount of radiation sent into this +strip. +Building the field design vector q in the model presented in the previous +section requires, along with the matrices A and B readily given by the geometry +of bins and the ROI, the knowledge of vectors c and ρ with entries indexed by +bins. The entries in c are proportional to our guess on effective dose from a +single photon sent into the corresponding bin, and entries in ρ are estimations +of the quantities exp{−aT +i x∗}. Both these vectors depend on the actual image +x∗. “In real life” they may be either considered available a priori or should be +obtained from a pilot recovery of x∗ in which the radiation is a small fraction +of R split, say, equally between m bins. In our “proof of concept” experiment +where the goal is to understand potential of radiation design we use “ideal” +values of ρ and c; in particular, we set ρi = ρ∗ +i = exp{−aT +i x∗} . To specify c +7 + +Phantom +Sensitivityimage +ROlimage +250 +0.3 +0.3 +10 +10 +200 +10 +0.2 +150 +0.2 +20 +20 +100 +20 +0.1 +0.1 +50 +30 +30 +30 +0 +10 +20 +30 +10 +20 +30 +10 +20 +30 +Projections (sinogram) +Log10sensitivityofbins +ProjectionsROl +4 +-2.5 +20 +3 +20 +-3 +20 +10 +-3.5 +40 +2 +40 +-4 +40 +5 +1 +-4.5 +60 +60 +60 +0 +A +0 +10 +30 +40 +10 +203040 +10203040we act as follows: first, we utilize x∗ and pixel sensitivities image s depicted +in Figure 1 to compute the effective dose ci caused by a single photon sent +through the bin i. Then, to get c, we scale c to ensure that cT q = 1, where q is +the uniform (“baseline”) design such that the radiation in every bin is R/m. +3.2 +Computations +Design optimization was carried out using alternating minimization (9)–(10) +described in Section 2.5; 5-6 iterations of the process turned out to be sufficient +to get high accuracy solution. An alternative, somehow more time consuming +solution is provided by the first order algorithm—Mirror Descent with simplex +setup, see [11, Lecture 5]. Maximum Likelihood estimate was computed utilizing +Nesterov’s Fast Gradients [12] for minimizing smooth convex functions over the +nonnegative orthant. +3.3 +Numerical results +We present results of two experiments for the phantom image presented in Fig- +ure 1 and two values of the regularization parameter, λ = 0 and λ = 1e3. Fig- +ure 2 shows the obtained optimized radiation design in the penalized problem. +The optimization results in significant reduction of radiation in the sensitive +area and its increase in the ROI-related part of the sinogram. In each experi- +Figure 2: Radiation design for regularization λ=1e3 (right image.) For com- +parison, sensitivity and ROI sinograms are presented in the left and the middle +plots. +ment we compute N = 100 recoveries utilizing baseline and optimized designs; +in Figure 3 we present the histograms of the mean square error (MSE—squared +recovery error per ROI pixel). Figure 4 shows typical recoveries of the ROI +image in the case of λ = 0. One can observe a clear image quality improvement +under the optimized design. +4 +Concluding remarks +We present the first attempt to treat CT radiation design within a convex +optimization framework. Unlike conventional design, it takes into account the +8 + +Log10sensitivityofbins +ROlsinogram +Optimal radiation design +-2.5 +10 +10 +12 +10 +15 +20 +-3 +20 +10 +20 +30 +3.5 +30 +8 +30 +10 +40 +-4 +40 +6 +40 +50 +50 +4 +50 +5 +-4.5 +60 +60 +N +60 +-5 +0 +0 +1020 +3040 +10203040 +10 +20304010-4 +10-3 +10-2 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +4 +5 +6 +7 +8 +9 +10 +10-5 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +Figure 3: Distribution of the MSE in experiments with λ = 0 (left plot) and λ = +1e3 (right plot): blue histogram—baseline design, red histogram—optimized +design. +Figure 4: Original and reconstructed ROI (baseline and optimized designs), +λ = 0. +radiation sensitivity map of the body, therefore minimizing the total effective +dose. +Further challenges and opportunities +3D reconstruction +In the 3D case the number of voxels scales as n3 with +single-dimension grid size, while the number of bins grows as n4. This gives +much more degrees of freedom for the optimal design than in the 2D case. On +the other hand, a faster algorithm is needed to implement radiation design in a +reasonable time. +Our approach is especially attractive for use in radiation therapy rooms, +where a preliminary radiation sensitivity map of a patient is readily available. +It may be used e.g. with kilo-voltage cone-beam CT (kV-CBCT) systems inte- +grated into the gantry of linear accelerators. [13, 14]. +One more point to note: usually cone beam CT gantry moves with max- +imal axial speed so that each body point is exposed to x-ray only once for a +given transaxial angle. It may be better to slow this movement down, allowing +multiple axial x-ray angles for each tranaxial angle. This would improve the re- +construction noise-to-dose ratio in CT systems in both cases, with and without +9 + +Original +Reconstructedbasic +Reconstructedoptimal +2 +0.4 +2 +0.4 +2 +0.4 +4 +0.3 +4 +0.3 +4 +0.3 +6 +0.2 +9 +0.2 +6 +0.2 +8 +0.1 +8 +0.1 +8 +0.1 +10 +10 +0 +10 +2 +4 +6 +8 +10 +2 +6 +8 +10 +6 +10control of radiation pattern. +Radiation design under constraints +In this work the design is optimized +using an asymptotic maximum likelihood reconstruction model under unique +effective dose constraint (2). One can easily impose extra convex constraints on +the radiation vector q (e.g., total radiation dose, maximal dose per pixel, etc). +Furthermore, the proposed design optimization framework can be adapted to +account for the available a priori information about the body image when it is +formulated in the form of convex constraints on the image space (cf., e.g., [15, +Section 6.4.2]). +A +Proof of Proposition 1 +Given m × n matrix P of rank n and λ > 0, consider the optimization problem +min +H∈Rk×m +� +λ−1∥HP − B∥2 +F + ∥H∥2 +F +� +The objective in this unconstrained minimization problem is a strongly convex +quadratic function, so that optimal H is given the unique solution to the Fermat +equation +λ−1P[P T HT − BT ] + HT = 0, +resulting in +H = B[λ−1P][λ−1PP T + Im]−1 = B[P T P + λIn]−1P T , +where the second equality is due to Sherman-Morrison formula. Direct compu- +tation shows that for this H one has +λ−1∥HP − B∥2 +F + ∥H∥2 +F = Tr(B[P T P + λIn]−1BT ). +Setting P = W 1/2A and substituting the optimization variable H with GW −1/2, +we arrive at (8a). Passing in the latter relation to limit as λ → +0 and taking +into account that A is of rank n, we arrive at (8b). +References +[1] G. J. Gang, J. H. Siewerdsen, and J. W. Stayman, “Task-driven optimiza- +tion of fluence field and regularization for model-based iterative reconstruc- +tion in computed tomography,” IEEE transactions on medical imaging, +vol. 36, no. 12, pp. 2424–2435, 2017. +[2] T. P. Szczykutowicz and C. A. Mistretta, “Design of a digital beam at- +tenuation system for computed tomography: Part i. system design and +simulation framework,” Medical physics, vol. 40, no. 2, p. 021905, 2013. +10 + +[3] S. S. Hsieh and N. J. Pelc, “The feasibility of a piecewise-linear dynamic +bowtie filter,” Medical physics, vol. 40, no. 3, p. 031910, 2013. +[4] F. Liu, G. Wang, W. Cong, S. S. Hsieh, and N. J. Pelc, “Dynamic bowtie +for fan-beam ct,” Journal of X-ray Science and Technology, vol. 21, no. 4, +pp. 579–590, 2013. +[5] W. Peppler, B. Kudva, J. Dobbins III, C. Lee, M. Van Lysel, B. Hasegawa, +and C. Mistretta, “Digitally controlled beam attenuator,” in Application of +Optical Instrumentation in Medicine X, vol. 347, pp. 106–111, SPIE, 1982. +[6] T. P. Szczykutowicz and J. Hermus, “Fluid dynamic bowtie attenuators,” +in Medical Imaging 2015: Physics of Medical Imaging, vol. 9412, pp. 219– +225, SPIE, 2015. +[7] P. Shunhavanich, S. S. Hsieh, and N. J. Pelc, “Fluid-filled dynamic bowtie +filter: a feasibility study,” in Medical Imaging 2015: Physics of Medical +Imaging, vol. 9412, pp. 399–406, SPIE, 2015. +[8] F. Liu, Q. Yang, W. Cong, and G. Wang, “Dynamic bowtie filter for cone- +beam/multi-slice ct,” PloS one, vol. 9, no. 7, p. e103054, 2014. +[9] J. W. Stayman, A. Mathews, W. Zbijewski, G. Gang, J. Siewerdsen, +S. Kawamoto, I. Blevis, and R. Levinson, “Fluence-field modulated x-ray +ct using multiple aperture devices,” in Medical Imaging 2016: Physics of +Medical Imaging, vol. 9783, pp. 232–237, SPIE, 2016. +[10] A. Mathews, G. Gang, R. Levinson, W. Zbijewski, S. Kawamoto, J. Siew- +erdsen, and J. Stayman, “Experimental evaluation of dual multiple aper- +ture devices for fluence field modulated x-ray computed tomography,” in +Medical Imaging 2017: Physics of Medical Imaging, vol. 10132, pp. 690– +695, SPIE, 2017. +[11] A. Ben-Tal and A. Nemirovski, “Lectures on modern convex optimization +(2020),” SIAM, Philadelphia, PA. Google Scholar Google Scholar Digital +Library Digital Library, 2021. +[12] Y. Nesterov, Lectures on convex optimization, vol. 137. Springer, 2018. +[13] K. Srinivasan, M. Mohammadi, and J. Shepherd, “Applications of linac- +mounted kilovoltage cone-beam computed tomography in modern radiation +therapy: A review,” Polish journal of radiology, vol. 79, p. 181, 2014. +[14] J. Wang, T. Li, Z. Liang, and L. Xing, “Dose reduction for kilovotage cone- +beam computed tomography in radiation therapy,” Physics in Medicine & +Biology, vol. 53, no. 11, p. 2897, 2008. +[15] A. Juditsky and A. Nemirovski, Statistical Inference via Convex Optimiza- +tion. Princeton University Press, 2020. +11 + diff --git a/NtE1T4oBgHgl3EQftgX6/content/tmp_files/load_file.txt b/NtE1T4oBgHgl3EQftgX6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..51df1e8e9c3f74741305542a839bd0f8fb7f72ed --- /dev/null +++ b/NtE1T4oBgHgl3EQftgX6/content/tmp_files/load_file.txt @@ -0,0 +1,322 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf,len=321 +page_content='Radiation design in computed tomography via convex optimization Anatoli Juditsky ∗ Arkadi Nemirovski † Michael Zibulevsky‡ January 10, 2023 Abstract Proper X-ray radiation design (via dynamic fluence field modulation, FFM) allows to reduce effective radiation dose in computed tomogra- phy without compromising image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' It takes into account patient anatomy, radiation sensitivity of different organs and tissues, and location of regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' We account all these factors within a general convex optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 1 Introduction Recently, there has been significant research interest in dynamic fluence field modulation (FFM), which consists in varying the beam shape throughout the scan allowing for the adaptation of the spatial x-ray distribution to conform to the patient anatomy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', [1] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' A range of hardware solutions have been proposed, including dynamic wedges [2, 3, 4], fluid-filled chambers [5, 6, 7, 8], and slitbased multiple aperture devices [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Until now, to the best of our knowledge, the problem of FFM radiation planning has been treated within non-convex optimization framework, and only rough distribution of radiation with relatively small number of parameters (co- efficients of basis functions) was modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In this work we utilize convex opti- mization techniques to optimize the fine structure of the fluence, namely, the amount of radiation sent into each projection bin (towards the corresponding detector pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' ∗LJK, Universit´e Grenoble Alpes, 700 Avenue Centrale, 38401 Domaine Universitaire de Saint-Martin-d’H`eres, France anatoli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='juditsky@univ-grenoble-alpes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='fr †Georgia Institute of Technology, Atlanta, Georgia 30332, USA, nemirovs@isye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='edu ‡Department of Computer Science, Technion—Israel Institute of Technology, Haifa 32000, Israel, mzib@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='technion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='il This work was supported by Multidisciplinary Institute in Artificial intelligence MIAI @ Grenoble Alpes (ANR-19-P3IA-0003) and by the Israel Council For Higher Education— Planning & Budgeting Committee 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='03379v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='med-ph] 6 Jan 2023 The main body of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In Section 2 we present the CT observation model underlying our derivations, introduce and motivate the loss index of radiation design responsible for the asymptotical, as the total amount of radiation grows, quality of the Maximum Likelihood recovery of the image of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' We then pose the problem of optimal radiation design as the problem of optimizing the loss index under general convex constraints on the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' The resulting optimization problem is convex and thus efficiently solvable (at least in theory, and to some extent, also in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In Section 3, we report on the (in our appreciation, encouraging) results of a “proof of concept” numerical experiment implementing the radiation design proposed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Section 4 contains some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 2 CT Radiation Design 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 CT observation scheme The CT observation scheme we intend to consider is as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' x ∈ Rn + is the “signal” – the discretized body attenuation density, so that n is the number of voxels in the field of view;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' w ∈ Rm is the vector of observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' observations are indexed by bins, m being the total number of bins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' A ∈ Rm×n is the “projection matrix”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' q ∈ Rn + is the vector with entries qi which are the expected numbers of photons sent to bin i during the stude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' We assume that the i-th observation stemming from signal x is ωi ∼ Poisson(qi exp{−[Ax]i}, (1) where Poisson(µ) is the Poisson distribution with parameter µ ≥ 0: Probι∼Poisson(µ){ι = k} = µk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' eµ, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' We assume also that entries ωi in the vector of observations ω are inde- pendent across i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Given observation ω stemming from the unknown signal x, we want to recover the vector Bx, where B ∈ Rν×n is a given matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 We measure the recovery error in the standard Euclidean norm ∥ · ∥2 on Rν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 1In our experiments Bx is a truncation of image x towards the region of interest (ROI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 Radiation sensitivity - from voxels to bins Informally, our goal is to find radiation design q, which provides best image quality given the whole-body effective dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Various organs and tissues have different sensitivity to radiation—effective dose caused by unitary radiation ex- posure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' This can be reflected in 3D voxel-wise sensitivity map s, which is standardly used in radiation therapy planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' We assume that, in addition, an (approximate) attenuation map of the body is available, and we can compute an auxiliary bin-sensitivity vector c with entries ci reflecting effective dose obtained from a unit of radiation sent into bin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Constant whole body effective dose is provided by the condition cT q = const, (2) Thus our goal is to find radiation design q, which provides the best image quality under constraint (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 Maximum Likelihood estimate With A and q given, we intend to recover Bx as B�x(ω), where �x(ω) is the Maximum Likelihood (ML) estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Denoting by aT i , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', m, the rows of A, the log-likelihood of observing ω, the signal being x, is Lq(ω, x) = �m i=1[ωi ln(qi exp{−aT i x}) − qi exp{−aT i x} − ln(ωi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=')] = �m i=1[−ωiaT i x − qi exp{−aT i x}] + R(q, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Consequently, maximizing Lq(ω, x) in x is the same as maximizing in x the concave in x function Lq(x) − [ � i ωiai]T x, Lq(x) = − � i qi exp{−aT i x} (3) and is therefore an efficiently solvable problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' the resulting estimate �x(ω) ∈ Argmax x � Lq(x) − [ � i ωiai]T x � of x is a function of ω depending on q as a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='4 Radiation Design via Loss index minimization Our goal is “presumably good,” in terms of the performance of the resulting estimate, selection of q in a given compact convex set Q ⊂ Rm +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' This informal goal can be stated formally in different ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' the formalization we propose is based on the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In the “noiseless case” where the observations are ωi = µi, µi = qi exp{−aT i x∗}, x∗ being the true signal (instead of being Poisson random variables with pa- rameters µi), the ML estimate exactly recovers the true signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In other words, 3 setting γ∗ = � i µiai, we get x+ ∈ Argmaxx[Lq(x) − γT ∗ x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Indeed, since Lq is concave and smooth in x, it suffices to verify that ∇xLq(x∗) = γ∗, which is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Now assume that the positive semidefinite Hessian matrix H := −∇2Lq(x∗) = � i µiaiaT i is positive definite, or, which is the same when q > 0, that A is of rank m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Let Lq(·) be the second order Taylor approximation of Lq(·) around x = x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Were we specifying our estimate of x as �x = �x(ω) := argmax x [Lq(x) − [ � i ωiai]T x], we would have ∇xLq(�x) = � i ωiai, ∇xL(x∗) = � i µiai, that is, �x−x∗ = −H−1 � i [ωi − µi]ai � �� � δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Hence, taking into account the immediate relation E{δδT } = H (recall that ωi ∼ Poisson(µi) are independent across i), E{∥B�x − Bx∗∥2 2} = E � δT H−1BT BH−1δ � = E � Tr � H−1BT BH−1δδT �� = Tr � H−1BT BH−1E � δδT �� = Tr � H−1BT B � = Tr(BH−1BT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' This computation suggests, as a meaningful “loss index” of q (the smaller it is, the better for our purposes) the function I∗(q) = Tr � �B �� i qi exp{−aT i x∗}aiaT i �−1 BT � � (4) Needless to recall, the above reasoning is informal—our estimate of x is not �x, it is �x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Nevertheless, it can be proved that when A is of rank m and q becomes large, specifically, q = t¯q with ¯q > 0, when t → ∞, our informal reasoning yields correct asymptotics of the expected squared ∥ · ∥2-error of the recovery of Bx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' We, however, neither present these asymptotic results, nor need them—our goal at this point is to find a convenient criterion to be optimized over q ∈ Q in order to get a presumably good measurement design, and to this end no rigorous justification of our choice is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' What indeed is important for us, is that I∗ is a convex function of q > 0, and thus is well suited for minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' A bad news is that strictly speaking, we cannot use I∗ as a criterion to be minimized—this quantity depends, as on a parameter, on the unknown to 2Namely, with q = t¯q, ¯q > 0, one has E{∥B�x − Bx∗∥2 2} = (1 + o(t))t−1I∗(¯q) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 4 us true signal x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' On the other hand, when replacing in the right hand side of (4) the unknown quantities ρ∗ i := exp{−aT i x∗} with their approximations ρi which are within factor c from ρ∗ i , that is c−1ρ∗ i ≤ ρi ≤ cρ∗ i , the resulting right hand side in (4) is within the same factor from the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Therefore, assuming that an approximate attenuation map ρ is available a priori, or that it can be estimated using observation ω from a pilot run with, say, equal to each other “moderate” qi = q, we may select q by minimizing over Q the observable criterion I(q) = Tr � �B �� i qiρiaiaT i �−1 BT � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' (5) This is, essentially, the course of actions we propose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 Regularization The problem of minimizing I(q) over q ∈ Q is convex and as such can, the- oretically, be solved to whatever high accuracy in a computationally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' At the same time, numerical experimentation shows that this prob- lem may happen to be ill-conditioned, resulting in difficulties when solving it numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' To overcome, to some extent, these difficulties, we can replace minimizing over q ∈ Q the objective I(q) given by (5) with minimizing over the same domain the penalized objective Iλ(q) = Tr � �B �� i qiρiaiaT i + λIn �−1 BT � � (6) where regularizing coefficient λ is nonnegative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' a good value of this coefficient could be found experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Thus, in the sequel we intend to optimize mea- surement design by solving convex optimization problem min q∈Q � � �Iλ(q) := Tr � �B �� i qiρiaiaT i + λIn �−1 BT � � � � � (7) which can be solved by standard Convex Optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' It turns out, however, that one can utilize problem’s structure to design a dedicated algorithm which, as our experiments show, is much faster than the “general purpose” algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' The following underlying observation is essentially well known: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Let P be an m×n matrix of rank n, B be a k×n matrix, and q, ρ be positive m-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Then, setting W = W[q] := Diag{qiρi, i ≤ m}, Iλ(q) := Tr � �B �� i qiρiaiaT i + λIn �−1 BT � � 5 = � � � min G∈Rk×m � λ−1∥GA − B∥2 F + Tr(GW −1GT ) � , λ > 0 (8a) min G∈Rk×m � Tr(GW −1GT ) : GA = B � , λ = 0 (8b) where ∥ · ∥F is the Frobenius norm We provide the proof of Proposition 1 in Appendix A for the sake of com- pleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Alternating minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Proposition 1 suggests the alternating mini- mization algorithm for minimizing Iλ(q) over q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Assume for the sake of definiteness that λ > 0 (modification for the case of λ = 0 is immediate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' By (8a), the problem of interest (7) can be rewritten equivalently as min G∈Rk×n,q∈Q Φ(G, q) := � λ−1∥GA − B∥2 F + ∥GW −1(q)GT ∥2 � To solve the latter problem, we start with some positive q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' At a step t of the algorithm, given positive qt−1 ∈ Q, we minimize Φ(G, qt−1) over G ∈ Rk×m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' this is an unconstrained mini- mization problem with strongly convex quadratic objective, and its solu- tion Gt is given by explicit formula: Gt = B[AT W[qt−1]A + λIn]−1AT W[qt−1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' (9) after Gt is computed, we specify qt by minimizing Φ(Gt, q) over q ∈ Q, which boils down to finding qt ∈ Argmin q∈Q �m j=1 ∥Colj(Gt)∥2 2 qjρj � �� � =Tr(GtW −1[q]GT t ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' When Q is simple, the latter problem is simple as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' For instance, in the case of Q = {q ∈ Rm + : cT q ≤ 1} with c > 0, assuming that Gt has no zero columns (which can be ensured by an arbitrarily small perturbation of Gt), the solution is given by [qt]j = rj/cj cT r , rj = ∥Colj[Gt]∥2 � cj/ρj, 1 ≤ j ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' (10) After Gt, qt are computed, we pass to step t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 3 Computational experiment Here we report on a toy “proof of concept” numerical experiment on recovering n = 32 × 32-pixel Shepp-Logan type phantom with 68 projection angles spread uniformly over 360 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 6 Figure 1: Top row: phantom, sensitivity image and region of interest (ROI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Bottom row: phantom sinogram, sensitivity of the bins and sinogram of the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 Experimental setup Unlike conventional tomography, where 180 degree angle span is sufficient, we consider 360 degree span to account for the fact that effective radiation dose is generally different when photons are sent along the same line from opposite directions due to different attenuation between the source and a particular body point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In our experiments, the number of bins is m = 2828;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' we utilize the baseline (uniform) radiation design qi ≡ 2e5 photons/bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Figure 1, top row, shows the attenuation map, radiation sensitivity, and region of interest (ROI) which we aim to reconstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Bottom row shows the corresponding maps in the sinogram (projection) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In particular, the middle picture shows radiation sensitivity in projection space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' effective dose caused by a unit of radiation sent into each projection bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' One can see a bright curve strip of high sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' As we will see later, our algorithm will reduce amount of radiation sent into this strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Building the field design vector q in the model presented in the previous section requires, along with the matrices A and B readily given by the geometry of bins and the ROI, the knowledge of vectors c and ρ with entries indexed by bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' The entries in c are proportional to our guess on effective dose from a single photon sent into the corresponding bin, and entries in ρ are estimations of the quantities exp{−aT i x∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Both these vectors depend on the actual image x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' “In real life” they may be either considered available a priori or should be obtained from a pilot recovery of x∗ in which the radiation is a small fraction of R split, say, equally between m bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In our “proof of concept” experiment where the goal is to understand potential of radiation design we use “ideal” values of ρ and c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' in particular, we set ρi = ρ∗ i = exp{−aT i x∗} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' To specify c 7 Phantom Sensitivityimage ROlimage 250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 10 10 200 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 20 20 100 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 50 30 30 30 0 10 20 30 10 20 30 10 20 30 Projections (sinogram) Log10sensitivityofbins ProjectionsROl 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 20 3 20 3 20 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 40 2 40 4 40 5 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 60 60 60 0 A 0 10 30 40 10 203040 10203040we act as follows: first, we utilize x∗ and pixel sensitivities image s depicted in Figure 1 to compute the effective dose ci caused by a single photon sent through the bin i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Then, to get c, we scale c to ensure that cT q = 1, where q is the uniform (“baseline”) design such that the radiation in every bin is R/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 Computations Design optimization was carried out using alternating minimization (9)–(10) described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 5-6 iterations of the process turned out to be sufficient to get high accuracy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' An alternative, somehow more time consuming solution is provided by the first order algorithm—Mirror Descent with simplex setup, see [11, Lecture 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Maximum Likelihood estimate was computed utilizing Nesterov’s Fast Gradients [12] for minimizing smooth convex functions over the nonnegative orthant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 Numerical results We present results of two experiments for the phantom image presented in Fig- ure 1 and two values of the regularization parameter, λ = 0 and λ = 1e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Fig- ure 2 shows the obtained optimized radiation design in the penalized problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' The optimization results in significant reduction of radiation in the sensitive area and its increase in the ROI-related part of the sinogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' In each experi- Figure 2: Radiation design for regularization λ=1e3 (right image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=') For com- parison, sensitivity and ROI sinograms are presented in the left and the middle plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' ment we compute N = 100 recoveries utilizing baseline and optimized designs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' in Figure 3 we present the histograms of the mean square error (MSE—squared recovery error per ROI pixel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Figure 4 shows typical recoveries of the ROI image in the case of λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' One can observe a clear image quality improvement under the optimized design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' 4 Concluding remarks We present the first attempt to treat CT radiation design within a convex optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Unlike conventional design, it takes into account the 8 Log10sensitivityofbins ROlsinogram Optimal radiation design 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 10 10 12 10 15 20 3 20 10 20 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 30 8 30 10 40 4 40 6 40 50 50 4 50 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='5 60 60 N 60 5 0 0 1020 3040 10203040 10 20304010-4 10-3 10-2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='25 4 5 6 7 8 9 10 10-5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='25 Figure 3: Distribution of the MSE in experiments with λ = 0 (left plot) and λ = 1e3 (right plot): blue histogram—baseline design, red histogram—optimized design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Figure 4: Original and reconstructed ROI (baseline and optimized designs), λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' radiation sensitivity map of the body, therefore minimizing the total effective dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Further challenges and opportunities 3D reconstruction In the 3D case the number of voxels scales as n3 with single-dimension grid size, while the number of bins grows as n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' This gives much more degrees of freedom for the optimal design than in the 2D case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' On the other hand, a faster algorithm is needed to implement radiation design in a reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Our approach is especially attractive for use in radiation therapy rooms, where a preliminary radiation sensitivity map of a patient is readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' It may be used e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' with kilo-voltage cone-beam CT (kV-CBCT) systems inte- grated into the gantry of linear accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' One more point to note: usually cone beam CT gantry moves with max- imal axial speed so that each body point is exposed to x-ray only once for a given transaxial angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' It may be better to slow this movement down, allowing multiple axial x-ray angles for each tranaxial angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' This would improve the re- construction noise-to-dose ratio in CT systems in both cases, with and without 9 Original Reconstructedbasic Reconstructedoptimal 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='3 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='1 10 10 0 10 2 4 6 8 10 2 6 8 10 6 10control of radiation pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Radiation design under constraints In this work the design is optimized using an asymptotic maximum likelihood reconstruction model under unique effective dose constraint (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' One can easily impose extra convex constraints on the radiation vector q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', total radiation dose, maximal dose per pixel, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Furthermore, the proposed design optimization framework can be adapted to account for the available a priori information about the body image when it is formulated in the form of convex constraints on the image space (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=', [15, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' A Proof of Proposition 1 Given m × n matrix P of rank n and λ > 0, consider the optimization problem min H∈Rk×m � λ−1∥HP − B∥2 F + ∥H∥2 F � The objective in this unconstrained minimization problem is a strongly convex quadratic function, so that optimal H is given the unique solution to the Fermat equation λ−1P[P T HT − BT ] + HT = 0, resulting in H = B[λ−1P][λ−1PP T + Im]−1 = B[P T P + λIn]−1P T , where the second equality is due to Sherman-Morrison formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Direct compu- tation shows that for this H one has λ−1∥HP − B∥2 F + ∥H∥2 F = Tr(B[P T P + λIn]−1BT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Setting P = W 1/2A and substituting the optimization variable H with GW −1/2, we arrive at (8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Passing in the latter relation to limit as λ → +0 and taking into account that A is of rank n, we arrive at (8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE1T4oBgHgl3EQftgX6/content/2301.03379v1.pdf'} +page_content=' Gang, J.' 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MAIN +©ESO 2023 +January 9, 2023 +The CARMENES search for exoplanets around M dwarfs +Wolf 1069 b: Earth-mass planet in the +habitable zone of a nearby, very low-mass star⋆ +D. Kossakowski1, M. Kürster1, T. Trifonov1,2, Th. Henning1, J. Kemmer3, J. A. Caballero4, R. Burn1, S. Sabotta3, +J. S. Crouse5,6,7,8, T. J. Fauchez5,9,6, E. Nagel10,11, A. Kaminski3, E. Herrero12,13, E. Rodríguez14, +E. González-Álvarez15, A. Quirrenbach3, P. J. Amado14, I. Ribas12,13, A. Reiners16, J. Aceituno17,14, V. J. S. Béjar18,19, +D. Baroch12,13, S. T. Bastelberger7,5,8,6, P. Chaturvedi11, C. Cifuentes4, S. Dreizler16, S. V. Jeffers20, R. Kopparapu5,6, +M. Lafarga21,12,13, M. J. López-González14, S. Martín-Ruiz14, D. Montes22, J. C. Morales12,13, E. Pallé18,19, A. Pavlov1, +S. Pedraz17, V. Perdelwitz23,10, M. Pérez-Torres14,24, M. Perger12,13, S. Reffert3, C. Rodríguez López14, +M. Schlecker25,1, P. Schöfer14,16, A. Schweitzer10, Y. Shan26,16, A. Shields27, S. Stock3, E. Wolf28,5,29, +M. R. Zapatero Osorio15, and M. Zechmeister16 +(Affiliations can be found after the references) +Received 28 October 2022 / accepted 21 December 2022 +ABSTRACT +We present the discovery of an Earth-mass planet (Mb sin i = 1.36 ± 0.21 M⊕) on a 15.6 d orbit of a relatively nearby (d ∼ 9.6 pc) and low-mass +(0.167 ± 0.011 M⊙) M5.0 V star, Wolf 1069. Sitting at a separation of 0.0672 ± 0.0014 au away from the host star puts Wolf 1069 b in the habitable +zone (HZ), receiving an incident flux of S = 0.652 ± 0.029 S ⊕. The planetary signal was detected using telluric-corrected radial-velocity (RV) +data from the CARMENES spectrograph, amounting to a total of 262 spectroscopic observations covering almost four years. There are additional +long-period signals in the RVs, one of which we attribute to the stellar rotation period. This is possible thanks to our photometric analysis including +new, well-sampled monitoring campaigns undergone with the OSN and TJO facilities that supplement archival photometry (i.e., from MEarth and +SuperWASP), and this yielded an updated rotational period range of Prot = 150 − 170 d, with a likely value at 169.3+3.7 +−3.6 d. The stellar activity +indicators provided by the CARMENES spectra likewise demonstrate evidence for the slow rotation period, though not as accurately due to +possible factors such as signal aliasing or spot evolution. Our detectability limits indicate that additional planets more massive than one Earth +mass with orbital periods of less than 10 days can be ruled out, suggesting that perhaps Wolf 1069 b had a violent formation history. This planet is +also the sixth closest Earth-mass planet situated in the conservative HZ, after Proxima Centauri b, GJ 1061 d, Teegarden’s Star c, and GJ 1002 b +and c. Despite not transiting, Wolf 1069 b is nonetheless a very promising target for future three-dimensional climate models to investigate various +habitability cases as well as for sub-m s−1 RV campaigns to search for potential inner sub-Earth-mass planets in order to test planet formation +theories. +Key words. methods: data analysis – planetary systems – stars: individual: Wolf 1069 – stars: low-mass – techniques: radial velocities +1. Introduction +An impressive 5000 exoplanets and counting have been detected +thus far1, largely thanks to the past and ongoing radial-velocity +(RV) and transit surveys. On the hunt for an Earth analog, out +of these thousands of planets, only ∼50 have been found to sit +in the so-called habitable zone (HZ) of their stellar host2, which +is defined to be the circumstellar region in which liquid water +could potentially exist on the surface of the planet (Kasting et al. +1993; Kopparapu et al. 2013). Only 20 of these are considered +to be Earth-sized, defined by radii of 0.8 R⊕ < Rp < 1.6 R⊕ or +by masses of 0.5 M⊕ < M sin ip < 3 M⊕. Moreover, a major- +ity of them have been discovered around M-dwarf stellar hosts, +⋆ RVs and additional data (i.e., stellar activity indicators as shown in +Fig. 5 are available in electronic form at the CDS via anonymous ftp to +cdsarc.u-strasbg.fr (TBD)). +1 https://exoplanetarchive.ipac.caltech.edu/, accessed on +16 September 2022. +2 The +Habitable +Exoplanet +Catalog: +http://phl.upr.edu/ +projects/habitable-exoplanets-catalog, last updated on 6 +December 2021 and further discussed in Appendix B. +most likely due to the ease in detectability considering the higher +planet-to-star mass and radius ratios (see e.g., Zechmeister et al. +2009; Seager 2010; Bonfils et al. 2013; Shields et al. 2016; Per- +ryman 2018). +The definition of the HZ is not a definite implication for +a life-hosting world, but rather acts as a good indicator for a +planet to show further promising potential for markers of sur- +face habitability. There are in fact a variety of factors that affect +its habitability potential, such as the X-ray/UV emission or age +in regards to the stellar host, or processes due to the planet it- +self including its atmospheric composition or its ability to retain +certain elements in its atmosphere (e.g., Dong et al. 2018; Kop- +parapu et al. 2020). For this reason, it is not only crucial to first +gather planets that are situated within the HZ, but also, it is nec- +essary to build a better understanding of the stellar effects on the +planet’s habitability (e.g., Segura et al. 2003, 2010; Hilton 2011; +Cohen et al. 2014; Chadney et al. 2016), and to also characterize +its atmosphere observability, with instruments such as the James +Webb Space Telescope (JWST; Gardner et al. 2009). +Article number, page 1 of 26 +arXiv:2301.02477v1 [astro-ph.EP] 6 Jan 2023 + +A&A proofs: manuscript no. MAIN +Even though most HZ planets around low-mass M dwarfs +are RV-only detections that do not transit, we can nonetheless +generate useful indicators to investigate their habitability further. +Three-dimensional (3D) general circulation model (GCM) cli- +mate simulations (e.g., Way et al. 2017; Wolf et al. 2022) can +produce predictions to investigate various atmosphere composi- +tions to test how durably habitable the planet is. These analyses, +along with a push for improvements in thermal emission and re- +flected light phase curve observations, are crucial given the rise +of nontransiting planets found in the HZ of stellar hosts within +the solar neighborhood (e.g., Anglada-Escudé et al. 2016; Drei- +zler et al. 2020). +In this paper, we turn our attention to the discovery of +Wolf 1069 b, an Earth-mass planetary companion orbiting a mid- +type M dwarf within the conservative HZ limits, as defined by +Kopparapu et al. (2013). This planet could very well possess the +key factors in making it indeed a habitable world according to +preliminary 3D GCM models. Also, in contrast to other habit- +able worlds in the conservative HZ with similar host stars (i.e., +Kepler-1649, Proxima Centauri, GJ 1061, Teegarden’s Star, and +GJ 1002), Wolf 1069 b is the only one within the conservative +HZ without an inner planet, based on our current detection lim- +its. This notion is supported by the works of Burn et al. (2021), +Mulders et al. (2021), and Schlecker et al. (2021), where we ex- +pect a lower planet occurrence rate for stars with M⋆ < 0.2 M⊙ +than for stars with 0.2 M⊙ < M⋆ < 0.5 M⊙ for both the peb- +ble and core accretion scenarios. Granted, these are theoretical +predictions as more observation-based evidence is required to +confirm this, and Wolf 1069 b could still be accompanied by +closer-in and outer planets. Nevertheless, the concept that only +one planet survives is predicted by formation models if there +were at least one giant impact at the late stage. This would en- +hance the chance of having a massive moon similar to the Earth +and might also stir up the interior of the planet to prevent strati- +fication and sustain a magnetic field (e.g., Jacobson et al. 2017). +As remote as this appears, the search for exo-moons is no longer +so far-fetched in recent times (e.g., Martínez-Rodríguez et al. +2019; Dobos et al. 2022). +The paper is outlined as follows. Section 2 first presents the +comprehensive spectroscopic and photometric data collected for +this work. Then, the host star and its properties are introduced in +Sect. 3, in which we determine and update its rotational period +using newly taken photometry from our facilities. The various +signals in the RVs for this system are investigated and modeled +in Sect. 4, where the results and prospects for Wolf 1069 b are +then discussed in Sect. 5. We finally display our conclusions in +Sect. 6. +2. Observational data +2.1. CARMENES high-resolution spectroscopy +The CARMENES3 instrument is located at the 3.5 m telescope +at the Calar Alto Observatory in Spain and consists of two sep- +arate spectrographs residing in two channels: the visual (VIS), +which covers the spectral range 520–960 nm (spectral resolution +of R = 94 600), and the near-infrared (near-IR), which covers +the 960–1710 nm range (R = 80 400) (Quirrenbach et al. 2014, +2018). Wolf 1069 (Karmn J20260+585) was one of the ∼300 +stars initially chosen as part of the CARMENES Guaranteed +3 Calar Alto high-Resolution search for M dwarfs with Exo-earths with +Near-infrared and optical Échelle Spectrographs, http://carmenes. +caha.es +Time Observation (GTO) program (Reiners et al. 2018), and +276 observations were since accumulated spanning 1450 days +(June 2016 – June 2020). There were six measurements that +were missing a drift correction as well as an additional eight +that had low signal-to-noise ratio, and were, for this reason, +discarded, resulting in 262 usable RV measurements. Further- +more, the spectra were notably affected by telluric absorption +(e.g., Reiners et al. 2018). We corrected them by employing the +template division telluric modeling methodology, a technique to +remove telluric absorption lines from stellar spectra with numer- +ous intrinsic lines (Nagel et al. 2022). This technique is suit- +able for separating telluric and spectral components based on +the Earth’s barycentric motion throughout the year. The telluric- +free spectra of Wolf 1069 are then produced by fitting a syn- +thetic transmission model of the Earth’s atmosphere to each in- +dividually extracted telluric spectrum with molecfit (Smette +et al. 2015). The weighted root mean square (wrms) and the me- +dian uncertainty of the remaining 262 data points are 2.66 m s−1 +and 1.67 m s−1, respectively. The simultaneously taken near-IR +measurements were not considered as part of the analysis given +the notably higher wrms of 7.0 m s−1 along with the significantly +higher mean uncertainty of each observation. Therefore, we con- +tinue the analysis with the 262 telluric-absorption-corrected VIS +spectra. The CARMENES RV data with their uncertainties are +displayed in the top panel of Fig. 1. +The raw data are first pipelined through the standard guar- +anteed time observations via caracal (Caballero et al. 2016b). +Then, the RVs are determined using serval4 (Zechmeister et al. +2018), where the spectra are corrected for barycentric motion, +secular acceleration, and instrumental drift, and then nightly +zero-points were calculated and applied (Trifonov et al. 2020). +In addition, serval produces various stellar activity indica- +tors such as the chromatic index (CRX), the differential line +width (dLW), the Hα index, the Ca ii IR triplet (IRT) lines, and +the Na i D doublet lines. We also obtained the photospheric +absorption band indices TiO λ 7050 Å, TiO λ 8430 Å, and +TiO λ 8860 Å from the nontelluric-corrected spectra using spec- +tral ranges without notable telluric contamination, as defined by +Schöfer et al. (2019). Lastly, we computed the cross-correlation +function (CCF) and its full-width half-maximum (FWHM), con- +trast (CTR), and bisector velocity span (BVS) values computed +with the raccoon5 code, adopting the approach of using bi- +nary masks as explained in Lafarga et al. (2020). These indi- +cators were investigated for the rotation period of the stellar host +(Sect. 3.2). +2.2. Our photometric campaigns +We carried out simultaneous, continuous photometric follow-up +of Wolf 1069 from 2017 to 2020 with the photometric facilities +as listed below. A summary of the various photometric data sets +is found in Table 1. They are also displayed as a time series in +Fig. 2. +Observatorio de Sierra Nevada (OSN). +The Observatorio de +Sierra Nevada (OSN)6, currently maintained by the Instituto de +Astrofísica de Andalucía (IAA) and situated at Loma de Dilar +in Granda, Spain, hosts two Nasmyth optical telescopes with +apertures of 90 cm (T90) and 150 cm (T150). Both telescopes +4 https://github.com/mzechmeister/serval +5 https://github.com/mlafarga/raccoon +6 https://www.osn.iaa.csic.es/en +Article number, page 2 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +10 +5 +0 +5 +10 +RV (m/s) +GP component +CARMENES +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Time (BJD - 2457000) +10 +0 +10 +RV resid. (m/s) +10 +5 +0 +5 +10 +RV (m/s) +P = 15.56 d t0 = 2458511.63 +0.4 +0.2 +0.0 +0.2 +0.4 +Phase +10 +0 +10 +RV resid. (m/s) +Fig. 1. RV time series and phase-folded plots for Wolf 1069 b. Top panel: CARMENES VIS RV measurements for Wolf 1069 along with the +best-fit model (dark gray line) and the stellar rotation period modeled by a dSHO-GP (orange). The light gray band indicates the 68% confidence +interval of the model. Bottom panel: RVs phase-folded to the period of Wolf 1069 b at 15.6 d (Kb = 1.07 ± 0.17 m s−1) and with the GP component +subtracted out. The black circles represent the data points binned to 0.1 in phase space for visualization purposes. The bottom panel for each plot +represents the residuals after subtracting out the model. There are two data points that did not fit within the boundary for visual reasons. +are equipped with similar VersArray 2k × 2k CCD cameras, +which deliver images with fields of view (FOV) of 13.2 arcmin +× 13.2 arcmin (T90; Amado et al. 2021) or 7.92 arcmin × +7.92 arcmin (T150; Rodríguez et al. 2010). Each camera is based +on a high quantum efficiency back-illuminated CCD chip, type +Marconi-EEV CCD42-4, with optimized response in the ultravi- +olet. We monitored Wolf 1069 using both telescopes and various +observing runs between the years 2017 and 2020. The observa- +tions with the T90 telescope were collected using both Johnson +V and R filters during three observing runs as tabulated in Ta- +ble 1. Each epoch typically consisted of 20 exposures of 80 s in +each filter per night. The resulting light curves were obtained by +the method of synthetic aperture photometry. Each CCD frame +was corrected in a standard way for bias and flat-fielding. Dif- +ferent aperture sizes were tested in order to choose the best one +for our observations. A number of nearby and relatively bright +stars within the frames were selected as reference stars to pro- +duce differential photometry of Wolf 1069. Outliers due to poor +observing conditions or very high airmass were removed. +The observations with the T150 telescope were collected +during a single observing run and were reduced in the same way. +In this case, each epoch typically consisted of 30 exposures of +80 s, 35 s, and 10 s in each V, R, and I filter, respectively, per +night. +Telescopi Joan Oró (TJO). +Simultaneous to the OSN pho- +tometry, observations for Wolf 1069 were carried out with the +80 cm Telescopi Joan Oró (TJO)7 at the Observatori del Montsec +in Lleida, Spain. The images were obtained with an exposure +time of 60 seconds using the Johnson R filter of the LAIA im- +ager, a 4k × 4k CCD with a FOV of 30 arcmin and a plate +scale of 4 arcsec/pixel. The images were calibrated with darks, +bias, and flat fields with the ICAT pipeline (Colome & Ribas +2006) of the TJO. The differential photometry was extracted with +AstroImageJ (Collins et al. 2017) using the aperture size that +minimized the rms of the resulting relative fluxes, and a selection +7 http://www.ieec.cat/content/206/what-s-the-oadm/ +Article number, page 3 of 26 + +A&A proofs: manuscript no. MAIN +of the 20 brightest comparison stars in the field, which did not +show variability. Then, data points with a low S/N were filtered +from the light curve, and any points with relative fluxes greater +than 1.08 or less than 0.96 were removed before nightly binning. +2.3. Photometric monitoring surveys +Additionally, we compiled a collection of archival long-term +photometric data taken by monitoring surveys as described be- +low. They are also listed in Table 1 and illustrated in Fig. 2. +SuperWASP. The SuperWASP8 project is led by a chiefly UK- +based consortium. Using two arrays of robotic telescopes oper- +ating in the northern and southern hemispheres, the survey ob- +tains light curves for millions of objects at high cadence to look +for transiting planets and study other astrophysical phenomena +across the entire sky (Pollacco et al. 2006; Butters et al. 2010). +SuperWASP-North is located at the Roque de los Muchachos +Observatory in La Palma, whereas SuperWASP-South is at the +South African Astronomical Observatory near Sutherland, South +Africa. Each observatory consists of eight wide-angle cameras +with Canon 200 mm, f/1.8 lenses that feed into 2048 × 2048 +CCDs. The pixel scale is 13.7 arcsec. +Wolf +1069 +was +monitored +for +four +seasons +with +SuperWASP-North from 2007 to 2010, though only sparsely in +the last two seasons. The usable data spans from June 2007 to +August 2008 with a ∼6 month gap. We received the complete +light curve, corrected for instrumental and atmospheric system- +atics, from the SuperWASP team. The detrending procedure +is nonaggressive and expected to preserve true astrophysical +signals (including rotational modulation), as documented by +Tamuz et al. (2005). After clipping the final two seasons and +iteratively rejecting outliers commensurate with the size of the +data set in each season, we binned the data nightly such that the +weighted mean and error of all the data points that go into each +bin constitutes the flux and error of that bin. +MEarth-North. +Wolf 1069 was observed by the MEarth9 +project (Irwin et al. 2015), specifically with the MEarth-North +array composed of eight 40 cm telescopes, each equipped with +a 25.6 arcmin × 25.6 arcmin FOV Apogee U42 camera, and +located at the Fred Lawrence Whipple Observatory on Mount +Hopkins nearby Tucson, Arizona, USA. Using the light curves +from the latest data release10, the target was observed for more +than six years with telescope one (“MEarth-tel01”) and tele- +scope five (“MEarth-tel05”). The MEarth project generally uses +a RG715 long-pass filter, except for the 2010–2011 season +when an I715−895 interference filter was chosen. In the case of +Wolf 1069, with observations collected from both telescopes +later than October 2011, the RG715 filter was always used, and +thus, we consider each photometric light curve as its own. The +data were nightly binned following the same procedure as for +the SuperWASP data. Particularly for first season with MEarth- +tel05, we excluded certain nightly measurements where only one +observation was taken to ensure accurate data quality. This con- +stituted ∼15 nights out of the final 228 (Table 1), which were in +the end not considered for the final rotational period determina- +tion due to large noisiness (see Sect. 3.2). +8 Super-Wide Angle Search for Planets. +9 https://www.cfa.harvard.edu/MEarth/Welcome.html +10 DR10: +https://lweb.cfa.harvard.edu/MEarth/DR10/ +north2011-2020/ +TESS. Wolf 1069 was thus far observed in three of the north- +ern sectors (15 – camera #2 CCD #4; 16 – camera #2 CCD #3; +17 – camera #3 CCD #4, 15 August – 2 November 2019) during +the nominal mission of the Transiting Exoplanet Survey Satellite +(TESS; Ricker et al. 2015) with the short 2-minute cadence pho- +tometry, as well as in three sectors (41 – camera #2 CCD #1; 23 +July 2021 – 20 August 2021, 55 – camera #3 CCD #3; 05 August +2022 – 01 September 2022; 56 – camera#3 CCD #3; 01 Septem- +ber 2022 – 30 September 2022) during the extended mission with +2-minute and 20-second cadence11. The target is being currently +observed in one sector (57 – camera #3 30 September 2022 – +29 October 2022). The publicly available data from all sectors +were downloaded from the Mikulski Archive for Space Tele- +scopes12. Following the typical procedure, these data are cor- +rected for artifacts and systematic trends (Presearch Data Con- +ditioning, PDC_SAP flux – Smith et al. 2012; Stumpe et al. 2012, +2014), provided by the Science Processing Operations Center +(SPOC; Jenkins et al. 2016). We use these data for our analysis. +3. Stellar properties +3.1. Basic astrophysical properties +Wolf 1069 (GJ 1253, Karmn J20260+585) is a slowly rotating, +high-proper motion M5.0 V star at less than 10 pc discovered +by Wolf (1920). For decades, it was the subject only of astro- +photometric analysis of stars in the solar neighborhood (Luyten +1955; Gliese & Jahreiß 1979; Probst 1983; Weis 1984), with the +first spectral analysis by Bidelman (1985). Since the end of the +20th century, Wolf 1069 was more investigated in X-rays (Wood +et al. 1994; Stelzer et al. 2013), with high-resolution imaging +(Dieterich et al. 2012; Jódar et al. 2013; Janson et al. 2014; Lam- +man et al. 2020), and especially for determining its astrophysical +stellar parameters (Rojas-Ayala et al. 2012; Mann et al. 2015; +Rajpurohit et al. 2018; Marfil et al. 2021). +We summarize the stellar properties of Wolf 1069 in Table 2. +Following Cifuentes et al. (2020), we integrated the spectral en- +ergy distribution and computed the bolometric luminosity using +broadband photometry and the latest parallactic distance from +Gaia Data Release 3 (DR3; Gaia Collaboration et al. 2022). +From this value and the effective temperature of the star, deter- +mined spectroscopically by Passegger et al. (2019), we derived +the radius from the Stefan-Boltzmann law, and the mass from +the mass-radius relation by Schweitzer et al. (2019). All tabu- +lated parameters agree within uncertainties with published val- +ues (e.g., Jenkins et al. 2009; Terrien et al. 2015; Lafarga et al. +2020), except for the rotation period, which we concluded to +be 150–170 d. Our Prot determination is explained in detail and +compared with the literature in Sect. 3.2. Although Wolf 1069 +kinematically belongs to the Galactic thin disk (i.e., younger +ages of ∼1-5 Gyr) according to the galactocentric velocities cal- +culated as in Montes et al. (2001) with a custom made python +code (Cortés-Contreras et al. in prep.), the very long Prot de- +termined by us points toward older ages (∼7-11 Gyr; Newton +et al. 2016). The low activity indicators (Hα and X-ray emis- +sion) and slow rotational velocity, v sin i, are also in line with a +long Prot. As a result, Wolf 1069 is a very weakly active star, +which facilitates the identification of RV signals with very low +semi-amplitudes. +11 https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/ +wtv.py +12 https://mast.stsci.edu +Article number, page 4 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +−3000 +−2800 +−2600 +−2400 +−2200 +−2000 +0.975 +1.000 +1.025 +−800 +−600 +−400 +−200 +0 +200 +0.975 +1.000 +1.025 +1000 +1200 +1400 +1600 +1800 +2000 +0.975 +1.000 +1.025 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Time - 2457000 (days) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized flux +SuperWASP +MEarth-tel01-s1 +MEarth-tel05-s1 +OSN-V-T150 +OSN-R-T150 +OSN-I-T150 +OSN-V-T90 +OSN-R-T90 +TJO-R +MEarth-tel01-s2 +MEarth-tel05-s2 +Fig. 2. Time series of the long-term photometry for Wolf 1069 color coded by instrument and filter. The time range for each panel is consistent +among all panels. The MEarth-tel05-s1 data were not included in the final rotational period determination but are faintly shown for illustrative +purposes. Given that the GP model is unique to each instrument with its own amplitude hyperparamter (see for example Fig. 8 in Kemmer et al. +2020), the extrapolated GP models of two instruments (MEarth-tel-05 and OSN-R-T90) are overplotted with the same color as their respective +data sets for illustrative purposes. +Table 1. Observing log of ground-based long-term photometric observations acquired for Wolf 1069. +Instrument +Date +Filter +∆ta +Nobs +Nnightsb +rmsc +Begin +End +(d) +(ppt) +SuperWASP +13 June 2007 +12 August 2008 +400–700 nm +426 +10 436 +172 +10.05 +MEarth-tel01 +� +29 September 2012 +08 July 2015 +������������� +RG715 +1 012 +2 595 +183 +6.02 +13 September 2017 +19 November 2018 +431 +6 237 +177 +6.85 +MEarth-tel05 +� +28 May 2012† +10 November 2015† +1 260 +2 716 +228 +5.44 +13 September 2017 +19 November 2018 +431 +5 665 +175 +6.52 +OSN-T150 +��������� +31 August 2017 +06 December 2017 +V +97 +1 322 +41 +8.65 +31 August 2017 +06 December 2017 +R +97 +1 276 +39 +5.90 +14 September 2017 +06 December 2017 +I +84 +1 078 +37 +6.38 +OSN-T90 +��������������������� +29 June 2017 +04 September 2017 +��������� +V +67 +439 +24 +6.07 +21 June 2019 +29 October 2019 +130 +719 +45 +7.42 +26 June 2020 +02 November 2020 +129 +1187 +64 +8.73 +29 June 2017 +04 September 2017 +��������� +R +67 +442 +24 +5.68 +21 June 2019 +29 October 2019 +130 +716 +45 +8.10 +26 June 2020 +02 November 2020 +129 +1207 +64 +6.40 +TJO-T80 +19 December 2018 +12 January 2020 +R +388 +1 345 +79 +10.02 +Notes. (a) Time span of the observation. (b) Number of nightly binned observations. (c) Root mean square in parts-per-thousand. +Data sets that were not used for the photometric rotational period determination are indicated by a dagger †. +Article number, page 5 of 26 + +A&A proofs: manuscript no. MAIN +0.00 +0.08 +SuperWASP +SuperWASP +0.00 +0.08 +MEarth-tel01-s1 +MEarth-tel01-s1 +0.0 +0.2 +MEarth-tel01-s2, MEarth-tel05-s2 +MEarth-tel01-s2, MEarth-tel05-s2 +0.0 +0.4 +OSN-T150-I +OSN-T150-I +0.0 +0.4 +OSN-T150-V, OSN-T90-V +OSN-T150-V, OSN-T90-V +0.0 +0.2 +OSN-T150-R, OSN-T90-R, TJO-T80-R +OSN-T150-R, OSN-T90-R, TJO-T80-R +0.0 +0.5 +1 +0.00 +0.15 +All instruments +All instruments +0.0 +0.005 +0.01 +10.0 +5.0 +2.0 +100 +100 +150 +200 +300 +400 +f [1/d] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +GLS Power (ZK) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Period [d] +Fig. 3. GLS periodograms for the long-term photometric data of Wolf 1069. The right panels are zoomed in to the longer-period regime. Each +horizontal panel represents an effective instrument that was considered for the determination of the stellar rotation period (Sect. 3.2). The normal- +ized GLS power of the sampling of the data for each row is shown in gray. The range for the photometric rotation period of 150–170 d is shaded +in orange. Some significant alias signals due to the sampling of each respective data set are illustrated with a vertical dashed line, whereas the true +signal is marked with a solid line. +3.2. Stellar rotation period +Using MEarth light curves from 2011 to 2014, Díez Alonso et al. +(2019) determined a photometric rotation period of 57.7 d for +Wolf 1069. However, the same data were analyzed earlier by +Newton et al. (2016), who reported only a tentative signal at +142.1 d and noted that it did not meet their criteria for a con- +fident detection. To further establish the stellar rotation period, +we examined all available photometric measurements, including +the newer-taken observations from the OSN and TJO telescopes, +along with various stellar activity indicators available from the +CARMENES spectra. +Article number, page 6 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +Table 2. Stellar parameters of Wolf 1069. +Parameter +Value +Reference +Basic identifiers and data +Name +Wolf 1069 +Wolf1920 +GJ +1253 +Gli57 +Karmn +J20260+585 +Cab16 +Gaia DR3 +2188318517720321664 +Gaia DR3 +G (mag) +12.352 ± 0.003 +Gaia DR3a +Coordinates and spectral classification +α (ICRS) +20:26:05.80 +Gaia DR3 +δ (ICRS) ++58:34:31.4 +Gaia DR3 +Sp. type +M5.0 V +Reid95 +Parallax and kinematics +π (mas) +104.441 ± 0.026 +Gaia DR3 +d (pc) +9.5747 ± 0.0024 +Gaia DR3 +µα cos δ (mas yr−1) +261.038 ± 0.032 +Gaia DR3 +µδ (mas yr−1) +542.906 ± 0.031 +Gaia DR3 +γ (km s−1) +–60.047 ± 0.020 +Lafa20 +U (km s−1) +–23.1335 ± 0.0057 +This work +V (km s−1) +–61.2071 ± 0.0200 +This work +W (km s−1) +–8.4689 ± 0.0060 +This work +Galactic population +Thin disk +This work +Photospheric parameters +Teff (K) +3158 ± 54 +Pass19 +log g⋆ (cgs) +4.93 ± 0.06 +Pass19 +[Fe/H] (dex) +0.07 ± 0.19 +Pass19 +v sin i⋆ (km s−1) +<2 +Rein18 +pEW(Hα) +−0.045 ± 0.082 +Fuhr22 +⟨B⟩ (G) +<620 +Rein22 +log FX (mW m−2) +−13.27 +Stel13 +Prot (d) +150–170 +This workb +Physical parameters +L⋆ (10−4 L⊙) +29.44 ± 0.28 +This work +R⋆ (R⊙) +0.1813 ± 0.0063 +This work +M⋆ (M⊙) +0.167 ± 0.011 +This work +Conservative HZ (au)c +[0.056,0.111] +This work +Notes. (a) See Table D.1 for multiband photometry different from Gaia +DR3 G band. (b) See Sect. 3.2 for the Prot determination. (c) For planets +with Mp = 1M⊕ +References. Cab16: Caballero et al. (2016a); Cif20: Cifuentes et al. +(2020); Gaia DR3: Gaia Collaboration et al. (2022); Fuhr20: Fuhrmeis- +ter et al. (2020); Fuhr22: Fuhrmeister et al. (2022); Gli57: Gliese +(1957); Lafa20: Lafarga et al. (2020); Mar21: Marfil et al. (2021). +Pas19: Passegger et al. (2019); Reid95: Reid et al. (1995); Rein22: Rein- +ers et al. (2022); Schw19: Schweitzer et al. (2019); Stel13: Stelzer et al. +(2013); Wolf1920: Wolf (1920). +Long-term photometry. Considering each photometric data set +alone, namely, OSN, TJO, SuperWASP, and MEarth, all indi- +cate hints toward a prominent peak of ∼150–165 d when taking +a look at the generalized Lomb-Scargle (GLS; Zechmeister & +Kürster 2009) periodograms, along with strong peaks present at +the respective alias frequencies (Fig. 3). Interestingly, while this +periodicity does present itself significantly in most photometric +data sets, the most prominent peak in some is rather at a longer +period (∼200–300 d), which can also be connected to an alias +signal of the presumed rotational period due to the sampling for +each of the respective data sets. Focusing on the first season of +the photometry from MEarth, solely from tel-01 as the data from +tel-05 are rather noisy, the highest peak is around 140 d, similar +to Newton et al. (2016). On the other hand, the second observ- +ing block of the MEarth data, considering now data from both +the tel-01 and tel-05 instruments, peaks clearly at ∼158 d, with +present alias signals (i.e., ∼110 d and ∼300 d) due to the sam- +pling of 320 d. +To determine the rotational period of Wolf 1069, we per- +formed a fit with juliet13 (Espinoza et al. 2019) using the +sum of two stochastically driven, damped harmonic oscillator +(SHO) terms, or a double SHO Gaussian process (dSHO-GP), +as done so in previous works such as, David et al. (2019), Gillen +et al. (2020), and Kossakowski et al. (2021) first considering +all the data sets. A summary of the priors used for the analy- +sis is found in Table C.1. To set this up, we treated the OSN +data as effectively different instruments arranged together only if +the telescope size (i.e., T90 and T150) and filter (i.e., V, R, and +I) were the same across multiple observational seasons. Each +telescope of the MEarth data were separated into two tempo- +ral subsets, namely MEarth-tel05-s1 and MEarth-tel05-s2, and +the same for MEarth-tel01 though without tel05, to account for +possible changes in stellar activity behavior over long periods of +time (i.e., ∼800 d). The SuperWASP data were kept as is. We im- +posed a log-uniform prior for the rotational period, Prot, shared +among all instruments ranging from 10 d to 200 d to avoid sam- +ples from populating near the longer-period alias signal. Like- +wise, the quality factor for the secondary oscillation, Q0, and the +difference between the quality factors of the first and second os- +cillations, δQ, were also shared. As for the fractional amplitude +of the secondary oscillation with respect to the primary one, f, +this parameter was shared among instruments with the same fil- +ter, that is, wavelength. The amplitude of the dSHO-GP, σGP, +followed suit as the dSHO-GP is trying to model the underlying +physical behavior of the star that is wavelength dependent. To +then account for any individual instrumental systematics, each +respective instrument had its own offset value and jitter term14 +(Baluev 2009). +From the posterior results of this fit, we obtain a rotational +period of 169.3+3.7 +−3.6 d, compatible with the peaks in the peri- +odograms and their widths (Fig. 3). To further experiment, we +also tested out the same model setup, but this time solely on +the photometry contemporaneously taken with the RVs, mean- +ing those data comprising the last panel of Fig. 2. The reason for +considering just these data is that, it could be possible that the +star underwent some changes in activity on the long scale due to +spot migration or differential rotation. The results of this run give +a rotational period of 170+15 +−11 d, which is 1-σ well in agreement +when considering all photometry, ensuring that the behavior of +the star is still applicable to today. Therefore, we determine the +photometric stellar rotation period to be 169.3+3.7 +−3.6 d. +Spectroscopy. We furthermore explored the various stellar ac- +tivity indicators provided by the CARMENES spectra as well +as the RVs themselves to look for agreement with the pho- +tometric rotational period. Wolf 1069 is an Hα-inactive star +(pEW’(Hα) Å > −0.3; Schöfer et al. 2019). +While there are no strong or moderate correlations found +between the RVs and the stellar activity indicators using the +Pearson’s p-probability, the GLS periodograms of the indica- +tors nonetheless consist of a variety of prominent peaks (see +Fig. 5). While some peaks found in the periodograms do coin- +cide with the rotation period derived from the photometry, that +13 https://juliet.readthedocs.io/en/latest/index.html +14 The jitter term is an additional noise count that is added in quadrature +to the nominal uncertainty values. +Article number, page 7 of 26 + +A&A proofs: manuscript no. MAIN +is, ∼169 d, there are nonetheless other existing signals with even +higher significance ranging in periods from ∼200–260 d, or even +in the longer-period regime of up to a few hundred days. Par- +ticularly, these signals are present in the CRX, dLW, Hα index, +TiO λ 7050 Å, TiO λ 8430 Å, TiO λ 8860 Å, CTR, and FWHM. +There are still instances where the significant peaks coincide +with the expected alias frequencies, however, the power at the +rotational period is sometimes consistent with zero, as for in- +stance in the Ca ii IRT3 indicator. This could be pure coinci- +dental, we cannot exclude the possibility of it being a chance +alignment. There are also peaks at even longer-period regimes +(i.e., >1000 d), which could be due to a stellar magnetic cycle +not related to the stellar rotation itself. This behavior appears +similar to EV Lac (see Jeffers et al. 2022, for a detailed analy- +sis of the various periodicites), where it was shown that different +indicators can respond with a phase lag, or nonuniformly with +the rotational variation of the star. In this work, we focus only on +signals pertaining to the stellar rotation period, as well as those +connected to the other RV signals (see Sect. 4.1). +Wolf 1069 is considered to be a low-activity, low-mass star +following the categorizations in Lafarga et al. (2021). Hence, ap- +plying the findings made by Lafarga et al. (2021), the most effec- +tive indicators for identifying the stellar rotation period comprise +the dLW, CTR, and FWHM, which are tracing the varying width +of the absorption lines in the spectra, as well as the indices of +the chromospheric lines, Hα and Ca ii IRT, and sometimes the +RVs themselves. The CRX and BVS in this instance are not as +beneficial. Taking a closer look at these, we see that in fact, the +dLW does peak at the expected rotation period and fits quite well +to it. Likewise, when fitting for 169 d, the CTR and FWHM also +match quite well in the phase-folded plots (not shown). The Hα +index and Ca ii IRT, however, do not agree with this periodicity, +but rather show long-term trends, that may be related to a longer +magnetic cycle as these indicators are also associated with mag- +netic cycles. Another significant signal to mention in the dLW +and the CTR appears at ∼29.5 d, which is close to twice the or- +bital period of the planetary signal of interest (i.e., 15.6 d, see +Sect. 4.2). However, this periodicity is most likely due to instru- +mental effects, namely, the contamination from the light of the +Moon. Within the RVs, there is a peak at 165 d, though not sig- +nificant, after subtracting out the other more prominent peaks +(see Fig. 4 d). When performing the final fit on the RVs (see +Sect. 4.2), we apply a dSHO-GP on this signal arriving at a value +of Prot, RV = 165.6+3.3 +−3.4 d. +To bring all this evidence together, the photometric data point +to a rotational period of ∼169 d, the RVs to ∼165 d, and a por- +tion of stellar activity indicators similarly hint toward this value, +though sometimes exhibit higher peaks at longer periods. Based +on the photometry, it is evident that the variability of the star +is changing from season to season (see Fig. 2), so the period- +icities detected in the indices skewing away from the predicted +rotational period may be exhibiting the evolution of the spot con- +figuration. For example, some activity indices are evidently still +consistent (i.e., same period or aliasing due to the sampling), +although others are not as consistent, which could be a result +of other issues, stellar cycles, spot evolution, data precision and +sampling, that make the interpretation nontrivial. We therefore +adopt the rotational period of Wolf 1069 to be within the range +of 150−170 d, with indications toward Prot = 169.3+3.7 +−3.6 d, though +values outside this constrained range cannot be easily excluded. +This is consistent with the predictions for a low-mass, inactive +M dwarf (Newton et al. 2017, their Fig. 5). +4. Analysis and results +4.1. Signal detection +We first computed a GLS periodogram on the RVs using the +Exo-Striker15 tool (Trifonov 2019) to initially identify poten- +tially interesting signals. A series of GLS periodograms after se- +quentially subtracting out the most prominent sinusoidal signals, +along with the discrete Fourier transform of the window func- +tion, can be found in Fig. 4. The sampling of the data showed +three notable peaks at 899 d, 365 d (the yearly sampling), and +270 d. To quantify the significance of a signal’s peak, we com- +puted the false alarm probability (FAP) levels of 10, 1, and 0.1 % +using 10 000 randomizations of the input data where the fre- +quency ranged from 1/tbaseline to 1, with a frequency resolution +of 1/tbaseline and oversampling factor of ten. +Already, in the original RV data set there are three prominent +peaks at 15.6 d, ∼90–100 d and ∼400 d, all hovering the 0.1 % +FAP level (Fig. 4 b). To also mention, there are equally promi- +nent peaks at 0.94 d and 1.066 d, both aliases of the 15.6 d signal +due to the daily sampling. Furthermore, there are strong yearly +aliases for the 15.6 d signal, particularly at 14.9 d and 16.2 d. Per- +forming tests with the AliasFinder (Stock & Kemmer 2020), +we find that the true periodicity is the one at 15.6 d (see Fig. A.1), +which we adopt when considering this signal. Continuing on, +there are an additional three peaks around the 10 % FAP level, +namely at 165 d, ∼190 d, and ∼145 d, in order of significance, +where the first one corresponding to the rotation period. The +other two can be explained as aliases due to the 365-d and 270- +d sampling of the ∼400 d and ∼90–100 d signal, respectively. A +full outline of all the aliases is tabulated in Table 3. +As it does not matter which of the peaks of equal signif- +icance is chosen first to subtract out for the prewhitening, we +opt for the 15.6 d considering its aliases are also those that are +most prominent. After subtracting the 15.6 d signal (Fig. 4 c), +all corresponding aliases disappear, as expected. Both the ∼90– +100 d and ∼400 d signal reach above the 0.1 % FAP level, where +the 165 d, ∼190 d, and ∼145 d signals also become more promi- +nent. In the next succession of simultaneously modeling a two- +sinusoidal model with the next highest peak (15.6 d, 90.3 d), +there are no longer any signals that have an FAP level less than +0.1 %. The highest peak at ∼400 d is above the 10 % FAP level, +and its alias due to the yearly sampling (i.e., ∼190 d) is appar- +ent but no longer significant. The 165 d is still above 10 % FAP +level. +The next signal to subtract for would be the one at ∼400 d. +However, we have evidence that this signal is related to tel- +luric contamination. We specifically compared the nontelluric- +corrected RVs with those corrected for tellurics, finding that +there is a prominent peak at 388 d in the telluric component +(Fig. 6, top panel). We conclude that even though we use the +telluric-corrected spectra, the contamination still likely man- +ifests as this process of telluric removal is nontrivial and, +nonetheless, introduces residuals in the corrected spectra (Nagel +et al. 2022). Moreover, the peak in the RVs is more so at 404 d, +whereas the peak in the periodogram of the telluric component +is actually at 388 d. We presume that this could be due to the +fact that the alias due to the 270-d sampling of the 165 d signal +is 420 d, thus, adding some power there, in turn broadening the +peak and veering the periodicity away from 388 d to a value in +between the two. With this in mind, and considering the photo- +metrically determined rotation period, we proceed with simul- +taneously subtracting the 165 d signal next. Finally, there are no +15 https://github.com/3fon3fonov/exostriker +Article number, page 8 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +Sampling periods (d) +Alias order +Present periods (d) +15.6 +90.3 +165 +388 +( f = 0.06410 d−1) +(f = 0.01107 d−1) +( f = 0.00602 d−1) +(f = 0.00258 d−1) +m = 1 +14.8 † +67.7 † +102.8 +159.2∗ +270 +m = −1 +16.6 +135.7 +431.0∗ +887.8 +( fs = 0.00370 d−1) +m = 2 +14.0 † +54.1 † +74.5 +100.2 +m = −2 +17.6 +272.7 +722.9 +207.0 +m = 1 +15.0 +72.4 † +114.1 +† +188.1 +365 +m = −1 +16.3 +120.0 +304.5 +† +6157.4 +( fs = 0.00274 d−1) +m = 2 +14.4 †∗ +60.4 ∗ +86.9 +124.1 +m = −2 +17.1 +178.7 +1836.1 +344.6 +m = 1 +15.3 † +82.1 † +140.1 +271.0 +899 +m = −1 +15.9 † +100.4 +203.6 +682.6 +( fs = 0.00111 d−1) +m = 2 +15.1 +75.2 †∗ +121.2 +208.3 +m = −2 +16.2 +113.0 +263.2 +2835.9 +Table 3. Aliases of the significant peaks found in the RVs given the sampling period tabulated. The periods of the aliases are computed using +Palias = 1/ falias, where falias = f + m · fs following f as the frequency of the true signal, m as the alias order, and fs as the sampling frequency. +Aliases that are significant (FAP > 10%) are bolded, whereas those that are present but not significant are indicated with a dagger †. Ones marked +with an asterisk signify peaks that are close to, but not exactly, the center of the expected alias. +more signals above 10 % FAP after removing three simultaneous +sinusoidal signals (15.6 d, 90.3 d, 165 d). +4.2. RV model comparison +Out of the three prominent signals, only the one at 165 d does not +reach our significance criterium (FAP < 0.1 %, though greater +than 10 %), but is however related to stellar activity and thus +might be important to model. We therefore continue the analysis +by testing out “three-signal models” while properly considering +the activity, with comparison to “two-signal models” by ignor- +ing it. For reasons explained in Sect. 4.3, the origin of the 90.3 d +is ambiguous. Thus, we model it solely with a sinusoid when +including this signal in a fit. For the 15.6 d signal, there is no +evidence against it to not be planetary, therefore, we apply cir- +cular or eccentric Keplerian orbits. For eccentric fits, the eccen- +tricity was parametrized with S1 = √e sin ω and S2 = √e cos ω +with uniform priors, U(−1, 1) as recommended by Eastman et al. +(2013). +The modeling and comparison of models is all performed +with juliet, a versatile python package for simultaneous transit +and RV fitting, as already described in depth in other works such +as Kemmer et al. (2020), Stock et al. (2020), Bluhm et al. (2020), +and Kossakowski et al. (2021). Following the same recipe as +in the mentioned references, we use the computation of the +Bayesian log evidence (ln Z) to compare models. Models with +a ∆ ln Z ≳ 2.5 in comparison to another one show moderate ev- +idence in favor of the winning model (e.g., Trotta 2008; Feroz +et al. 2011). Any value greater than 5 signifies strong evidence +toward the winning model and anything below 2.5 indicates that +neither of the two models are favored. +To begin, we first ran a flat model with a white-noise term +(i.e., jitter term, σCARMENES-VIS) to act as a basis. Likewise, we +ran a red-noise model using a dSHO-GP centered around the +stellar rotation period (see below for setup details). For the 15.6 d +signal, we imposed an appropriately sized uniform prior for the +period, U(15.4 d, 15.7 d) in order to cover the width of the peak +as in the periodogram and to ensure that this true signal would +be picked up rather than its nearby aliases. The time-of-transit +center, which is used in juliet to parametrize the phase of the +orbit, was chosen to be uniform and set to cover one cycle during +the most-sampled time epoch of the RV data set, U(2458502 d, +2458515 d). Likewise, for the ∼90 d signal, the prior on the pe- +riod was kept wide enough to capture the tails of the posterior +sample distribution, U(85 d, 95 d). To account for the stellar +rotation period, we experimented modeling it with a sinusoid +and with a dSHO-GP. The photometrically determined rotational +period of 150–170 d (Sect. 3.2) was taken into consideration +for the prior setup on the period. Within the RV periodograms +(Sect. 4.1), the periodicity shows up as closer to 165 d and en- +counters various neighboring signals due to aliasing of the other +signals. For this reason, the prior on the period was kept rela- +tively narrow and uniform from 155 d to 175 d. Table C.2 show- +cases a full overview of the employed priors, including those for +the instrument. +Results. A table showcasing the Bayesian log evidence for the +assortment of the tested models tested can be found in Table 4. +The winning model comprises one Keplerian for the 15.6 d sig- +nal alongside a dSHO-GP centered on 165 d to describe the be- +havior of the stellar rotation period. We note that including an +extra sinusoid term for the 90.3 d signal to this model is equiv- +alent in terms of the Bayesian log evidence, even though this +signal is evidently prominent in the GLS periodogram (Fig. 4). +Given the ambiguity of the nature of this signal (Sect. 4.3), we +find it appropriate to omit it from the final model and allow the +dSHO-GP to moderately absorb it for the time being. The crucial +aspect is that the interpretation of this signal and, thus, how we +consider it in our models, which we acknowledge may adjust in +the future, does not drastically alter the planetary parameters of +the 15.6 d signal (Fig. 7). +For the 15.6 d signal, an eccentric Keplerian orbit was con- +sistent with a circular one. The distribution of the eccentricity +was consistent with zero. Focusing on the stellar rotation period, +the dSHO-GP was preferred over a sinusoid (∆ ln Z > 10) as it +is most likely better suited in describing the quasi-periodic be- +Article number, page 9 of 26 + +A&A proofs: manuscript no. MAIN +0.00 +0.05 +0.10 +0.2 +(a) window function +0.00 +0.01 +899d +365d +270d +0.00 +0.05 +0.10 +0.1 +(b) original +0.00 +0.01 +0.00 +0.05 +0.10 +0.1 +(c) 1 sinusoid (15.6 d) +0.00 +0.01 +0.00 +0.05 +0.10 +0.1 +(d) 2 sinusoids (15.6 d, 90.3 d) +0.00 +0.01 +0.00 +0.05 +0.10 +0.1 +(e) 3 sinusoids (15.6 d, 90.3 d, 165 d) +0.00 +0.01 +0.0 +0.05 +0.1 +0.06 +(f) 1 Kep (15.6 d) + dSHO-GP165d +0.0 +0.01 +15.6 +388 +165 +90.3 +10.0 +388 +165 +90.3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f [1/d] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +GLS Power (ZK) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Period [d] +Fig. 4. GLS periodograms for the RVs of Wolf 1069 after sequentially subtracting out the most prominent signals. The horizontal dashed, dot- +dashed, and dotted lines represent the 10 %, 1 %, and 0.1 % FAP levels (from bottom to top). There were no significant signals with periods shorter +than 10 d other than the aliases due to the daily sampling. The right panel is a closer zoom-in of the left panel to highlight the longer-period signals. +The 15.6 d planetary signal is illustrated with a vertical blue solid line, and the 90.3 d signal and its alias due to the 270-d sampling with a green +solid and dashed line, respectively. The range for the photometric rotation period is shaded in orange, where the stellar rotation period within +the RVs is marked with an orange solid line. The component of residual telluric contamination at 388 d and its alias due to the 365-d sampling +period are also represented with a vertical magenta solid and dashed line, respectively. Panel (a): the window function of the data set. Panel (b): no +signal fitted, solely the original RVs with an offset and jitter term. Panel (c): residuals after subtracting the 15.6 d signal. Panel (d): residuals after +subtracting a simultaneous model fit of two sinusoids at 15.6 d and 90.3 d. Panel (e): residuals after subtracting a simultaneous model fit of three +sinusoids at 15.6 d, 90.3 d, and 165 d. Panel (f): residuals after subtracting the final model choice including 1 Keplerian at 15.6 d (further described +in Sect. 4.2). +havior of the stellar activity. Furthermore, this corresponds well +to the fact that there seems to be a high level of spot evolution as +we have encountered in the stellar activity indicators (Sect. 3.2) +such that this is also demonstrated as an effect in the RVs. +To conclude, the final model consists of ten free parameters +applied on the 262 RV data points. The 15.6 d signal is best de- +scribed by a circular Keplerian model and the 165 d signal by a +dSHO-GP to account for the stellar rotation period. The best RV +model from the posteriors is shown in Fig. 1, where values for +the derived planetary parameters can be found in Table 5. The +full posterior overview for all model parameters is located in Ta- +ble C.3. A visual inspection of the posterior probability densities +for key parameters are displayed in Figs. C.1 and C.2. +4.3. Investigating the low-amplitude 90.3 d signal +The 90.3 d signal is significant with an FAP level near 0.1 % in +the GLS periodograms (Fig. 4), yet it does not statistically im- +prove the fit when including it in the RV models including also +the rotation period (Table 4). This could be an artifact that the +semi-amplitude hovers around the 1 m s−1 limit (K90.3 d = 91 ± +38 cm s−1), thus, making it possibly difficult to justify including +such a low-amplitude signal with a relatively large uncertainty +in the models. Likewise, the dSHO-GP kernel used in the model +is equipped to pick up the rotational period, Prot, and half of the +rotational period, Prot/2. It could be that 90.3 d is decently close +to 83 d (i.e., Prot/2) and, thus, the dSHO-GP is performing its job +of absorbing this signal. In fact, this is demonstrated in the GP +component of the RV model shown in Fig. 1. It is nonetheless +evident that this periodicity is present in the RV data as can be +seen in the residuals of the RV model. Its nature, however, ap- +pears to be quite dubious. Below, we explore the signal further +to better understand its origin. +Its periodicity is near a quarter of one year. A suspicion +could be that this signal should be present in the telluric- +contamination-only component of the RVs, though, it is not +Article number, page 10 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +0.08 +CRX +0.15 +dLW +0.08 +Caii-IRT1 +0.08 +Caii-IRT2 +0.08 +Caii-IRT3 +0.15 +Hα +0.1 +NaD1 5890˚A +0.1 +NaD2 5896˚A +0.3 +TiO 7050˚A +0.25 +TiO 8430˚A +0.2 +TiO 8860˚A +0.08 +BVS +0.1 +CTR +0.0 +0.05 +0.1 +0.15 +FWHM +0.0 +0.01 +15.6 +388 +165 +90.3 +10.0 +388 +165 +90.3 +f [1/d] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +GLS Power (ZK) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Period [d] +Fig. 5. GLS periodograms for various known stellar activity indicators from the CARMENES spectroscopy. The window function of the data +sampling can be found in Fig. 4 a. For consistency, the colored vertical lines and the frequency width of the panels on the left correspond exactly +to those in the RV GLS periodograms (Fig. 4). The horizontal dashed, dot-dashed, and dotted blue lines represent the 10 %, 1 %, and 0.1 % FAP +levels (from bottom to top). +Article number, page 11 of 26 + +A&A proofs: manuscript no. MAIN +Table 4. Model comparison using the Bayesian log evidences on the +RVs for Wolf 1069. +Model +ln Z +∆ ln Z +Base models +Flat +−640.00 +−33.14 +dSHO-GP165 d +−618.51 +−11.65 +One-signal model +1 Kep15.6 d +−631.49 +−24.63 +Two-signal models +1 Kep15.6 d + 1 Sin90.3 d +−620.62 +−13.76 +1 Kep15.6 d + dSHO-GP165 d +−606.86 +0.0 +Three-signal models +1 Kep15.6 d + 2 Sin90.3 d, 165 d +−616.83 +−9.96 +1 Kep15.6 d + 1 Sin90.3 d + dSHO-GP165 d +−606.94 +−0.08 +Notes. The chosen model was the 1 Kep15.6 d + dSHO-GP165 d, as +marked as the bold-faced row. A better model would have a larger, more +positive ∆ ln Z. Regarding the model names, “Kep” refers to a circular +Keplerian orbit and “Sin” to a sinusoidal signal. The period values are +quoted as the median of the posterior distribution and can vary slightly +depending on the model choice. +Table 5. Derived posterior parameters for Wolf 1069 b. +Parameter name +Posterior(a) +Unit +b +Pp +15.564+0.015 +−0.015 +d +t0,p (BJD) +2458511.63+0.45 +−0.46 +d +Kp +1.07+0.17 +−0.17 +m s−1 +S 1,p = √ep sin ωp +0.0 (fixed) +. . . +S 2,p = √ep cos ωp +0.0 (fixed) +. . . +M sin ip +1.26+0.21 +−0.21 +M⊕ +ap +0.0672+0.0014 +−0.0014 +au +Teq(b) +250.1+6.6 +−6.5 +K +S p +0.652+0.029 +−0.027 +S ⊕ +Notes. (a) Error bars denote the 68% posterior credibility intervals. +(b) The equilibrium temperature of the planet assuming zero Bond +Albedo and one emissivity. +(Fig. 6, top panel). We additionally tested breaking up both the +telluric-corrected (TC) and nontelluric-corrected (nonTC) RVs +into “blue” and “red” subsets. To do this, we can recompute the +RVs by selecting certain orders. The CARMENES VIS channel +consists of 55 RV orders, 42 of which are used to compute the +RV measurement via a weighted mean through serval (Zech- +meister et al. 2018). For the blue and red subset, we consid- +ered the first 21 and last 21 orders, respectively. Thus, the blue +and red subsets span roughly 570 nm to 700 nm and 700 nm to +910 nm, respectively. A GLS periodogram for both subsets and +for both data sets is shown in Fig. 6. Taking a look at the nonTC +spectra, the 388 d (telluric-attributed) signal, its yearly alias at +188 d, and the 90.3 d, as well as a neighboring signal at 97 d, +are substantially stronger in the red than in the blue. This is in +agreement that the telluric contamination is stronger in the red +than in the blue, given that there are sharper, deeper telluric- +absorption features in the redder part of the VIS channel (Fig. 1 +0.08 +0.08 +0.0 +0.01 +0.08 +0.065 +388 +165 +90.3 +15.6 +f [1/d] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +GLS Power (ZK) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Period [d] +Fig. 6. GLS periodograms of the telluric-only (nonTC subtracted from +TC; top), nonTC (middle) and TC (bottom) RVs. The red and blue colors +represent the “blue” and “red” subsets within the VIS channel of the +CARMENES instrument (Sect. 4.3). The vertical lines match those in +Fig. 4 for consistency and the horizontal lines are identical in the bottom +panels but differ slightly from the top panel. +in Reiners et al. 2018). Meanwhile, the power of the 15.6 d and +165 d signals is consistent within one another in both subsets. +After the telluric correction, some residual telluric contamina- +tion remains, though dampened, indicating that the correction +for tellurics was indeed effective but left some residual effect as +can be seen in the RV periodograms (Fig. 4). The power of the +15.6 d and 165 d signals is still compatible, which is most im- +portant. Therefore, even though the 90.3 d has no appearance in +the telluric-contamination-only component, it does exhibit chro- +maticity and behavior similar to other telluric signals, pointing +less in favor for a planetary signal and potentially more in favor +of telluric effects. It is, however, puzzling as to why the 90.3 d +signal does not peak in the telluric-only RVs (Fig. 6). +To summarize, we are not able to distinctly decipher the ori- +gin of the 90.3 d signal. Even if it were planetary, we currently +do not have enough evidence to support this claim. As our main +concern is the 15.6 d signal and we presented that its planetary +parameters are independent of whether the 90.3 d signal is con- +sidered or not (Fig. 7, particularly between 1 Kep15.6 d + 1 dSHO- +GP165 d and 1 Kep15.6 d + 1 Sin90.3 d + dSHO-GP165 d), we choose +not to include this signal in the final model as a precaution. Fur- +ther investigation or RV monitoring may be beneficial, however, +this lies beyond the scope of this paper. +4.4. Transit search within TESS +With a minimum mass of 1.26±0.21 M⊕ and assuming an Earth- +like core-mass fraction of 0.26, we obtained a radius estimate of +1.08 R⊕ for Wolf 1069 b using the mass-radius relation as given +by Zeng et al. (2016). This then translates to an expected transit +depth of ∼3.6 ppt, which should be easily detectable with TESS, +though not with the other available photometric facilities (i.e., +SuperWASP and MEarth). The transit probability is however +rather low at 1.2% (p ≈ R⋆/ap). Nonetheless, propagating the +orbital period and t0 from the RV fit with their 1-σ uncertainties +(taken from Table 5), we unfortunately did not find any hint of +a possible transit. We additionally checked to confirm that the +transits could not have happened during the data gaps, where +Article number, page 12 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +K (m s +1) +1 Kep15.6d + dSHO-GP165d +1 Kep15.6d + 1 Sin90.3d +1 Kep15.6d + 1 Sin90.3d + dSHO-GP165d +1 Kep15.6d + 2 Sin90.3d, 165d +TC +non-TC +Fig. 7. Box plot of the posteriors for the distributions of the minimum mass for the 15.6 d signal based on the model choice. The gray and blue +boxes represent the 25 % and 75 % quartiles of the posterior from the telluric-corrected (TC) and nontelluric-corrected (nonTC) RVs, respectively. +The red vertical line represents the median value of the minimum mass of the 15.6 d signal when applying the most favored model. The extending +gray lines depict the rest of the distribution and the dots are deemed as “outliers”. The models named here match those in Table 4. +only one would fall within an observational gap. Likewise, +we checked with the transit-least-squares16 method (Hippke & +Heller 2019), though no interesting signals popped up. Given +this information, we were able to obtain a maximum inclination +for Wolf 1069 b of imax = arccos +� +R⋆/ap +� += 89.35 deg. +5. Discussion +5.1. On the promising habitability of Wolf 1069 b +Plugging in the stellar luminosity and effective temperature (Ta- +ble 2), Wolf 1069 b, with a distance of 0.0672 ± 0.0014 au +to the star, sits comfortably within the conservative HZ lim- +its, namely, 0.056 au to 0.111 au given the runaway-greenhouse +and maximum-greenhouse limits, respectively (Kopparapu et al. +2013, 2014). Even more so, it is very likely that Wolf 1069 b is +indeed an Earth-like planet with Earth-like composition (32.5% +iron mass fraction and 67.5% silicates) and radius around one +Earth radii (following Fig. 1 in Luque & Pallé 2022), as we also +estimated in Sect. 4.4. Figure 9 puts Wolf 1069 b in context with +other planets around M-dwarf stars that are most likely to have +a rocky composition and maintain surface liquid water as listed +in the Habitable Exoplanet Catalog17 with some modifications +(Appendix B). To this effect, Wolf 1069 b resembles best Prox- +ima Centauri b, GJ 1061 d, Teegarden’s Star c, Kepler-1649 c, +and GJ 1002 b and c. With the exception of Kepler-1649 c, +all are RV-only detections. Furthermore, all 14 planetary sys- +tems illustrated contain more than one planetary companion, ex- +cluding Wolf 1069, Ross 128, and Kepler-1229, as discussed in +Sect. 5.2. When considering the occurrence rate for planets with +1–10 Earth masses on periods shorter than 10 d around later- +type M dwarfs, this value lies between ∼0.56–1.75 planets per +star (Ribas et al. 2022; Hardegree-Ullman et al. 2019). Regard- +ing the proximity of these systems, Dressing & Charbonneau +(2015) estimated that the nearest nontransiting HZ planet could +be 2.6±0.4 pc away, and is within 3.5 pc with 95% confidence for +potentially habitable 1–1.5 R⊕ planets. Soon after, Proxima Cen- +tauri b was discovered at a distance of 1.30 pc (Anglada-Escudé +et al. 2016), GJ 1061 d at 3.67 pc (Dreizler et al. 2020), Teegar- +16 https://github.com/hippke/tls +17 https://personal.ems.psu.edu/~ruk15/planets/ +den’s Star c at 3.83 pc (Zechmeister et al. 2019), and GJ 1002 b +and c at 4.85 pc (Suárez Mascareño et al. 2022). Wolf 1069 b is +located at a distance of 9.57 pc, making it the sixth closest, con- +servative HZ Earth-mass planet to us. Other closer contenders +included Ross 128 b (d = 3.38 pc) and GJ 273 b (d = 5.83 pc), +though these planets lie in the optimistic HZ. +Wolf 1069 b is in the slow rotator regime, and possibly in +tidal equilibirum rotation (e.g., Heller et al. 2011), that can lead +to unique atmospheric circulation pathways (e.g., Dole 1964; +Yang et al. 2019; Del Genio et al. 2019b). The impacts of this +slow rotation on both the potential habitability and impact on +observations have been discussed in detail by several 3D GCMs +(see e.g., Edson et al. 2012; Leconte et al. 2013). Preliminary +results from GCMs climate simulations using both the ExoCAM +model (Wolf et al. 2022) and the ROCKE-3D model (Way et al. +2017) suggest that Wolf 1069 b could maintain moderate temper- +atures and surface liquid water for a large range of atmospheric +compositions and surface types. Simulations explore a variety of +surface pressures, N2, CO2, CH4, and H2O abundances, along +with desert, solid rock, slab ocean, and dynamic ocean surfaces. +The comprehensive analysis of these 3D climate results and the +observational signals that could be used to differentiate between +climate states of Wolf 1069 b show the planet to be durably hab- +itable (Crouse et al. in prep.). Figure 8 shows the surface temper- +ature map produced with the ExoCAM GCM assuming a Mod- +ern Earth-like atmospheric composition. The red line delimiting +the open ocean shows that a significant fraction of the day side +surface could maintain liquid water, therefore day-side habitable +conditions. While the presence and nature of any atmosphere on +Wolf 1069 b (and existing M-dwarf planets in general) remains +theoretical, the support of habitable conditions over such a wide +range of possible atmospheric states puts Wolf 1069 b in the +same elevated class as Proxima Centauri b (Turbet et al. 2016; +Del Genio et al. 2019a), TRAPPIST-1 e (Wolf 2017; Turbet et al. +2018; Fauchez et al. 2019), and TOI-700 d (Suissa et al. 2020) +as a primary target to search for habitability and biosignature +markers. +Similar to Proxima Centauri b, Wolf 1069 b does not transit +its host star, meaning that observation and analysis of thermal +emission and reflected light phase curves will need to be em- +ployed to probe its atmosphere. Given the brightness of the host +Article number, page 13 of 26 + +A&A proofs: manuscript no. MAIN +Fig. 8. Surface temperature map of Wolf 1069 b produced by the Ex- +oCAM GCM, assuming a Modern Earth-like atmosphere. The map is +centered at the substellar point and the red line delimits the area where +water is at the liquid phase on the surface. +star and the distance to Earth, and assuming atmosphere mod- +els and albedo similar to the ones predicted for the TRAPPIST- +1 planets and Proxima Centauri b (Turbet et al. 2022), the at- +mospheric characterization of Wolf 1069 b might be within +the reach of the ELT18 instrumentation. ANDES19 (formerly +known as HIRES; Maiolino et al. 2013) will be the first instru- +ment theoretically capable of detecting the reflected light from +HZ rocky planet atmosphere in the early 2030’s. However, for +Wolf 1069 b, the small angular separation in the sky between +the planet and the host stars, 7.01 milliarcsec, makes these ob- +servations very challenging, even with the use of extreme adap- +tive optics systems. Further instrumental advances, such as the +proposed PCS20 instrument for the ELT (Kasper et al. 2021) +or space-based coronographic/interFerometric missions, might +be needed. While such observations are very challenging, many +of the nearest planets found in the conservative HZ around M +dwarfs are nontransiting, RV detections, indicating that perhaps +more time and investment into the development of such obser- +vations should be considered if we want to establish ground +statistics using all of the thus-far detected, potentially habitable +worlds. +5.2. The case of Wolf 1069 b as a lone, short-period planet +Our comprehensive analysis of the RV and photometric data sug- +gests that Wolf 1069 b is the only bonafide planet in the sensitive +domain of the planetary parameter space. We characterized this +domain with injection-and-retrieval tests by taking the residu- +als of the winning model without a GP (Sect. 4.2) and creating +simulated RV time series using Eqn. 2 in Sabotta et al. (2021). +This was repeated 50 times on 30 log-uniformly distributed grid +points in mass and 60 in period, allowing us to rule out additional +planets with at least one Earth mass and periods of less than +10 days (Fig. 10). Wolf 1069 b joins a sample of currently two +18 Extremely Large Telescope. +19 ArmazoNes high Dispersion Echelle Spectrograph, https://elt. +eso.org/instrument/ANDES/ +20 Planetary Camera and Spectrograph. +0.0 +0.5 +1.0 +1.5 +2.0 +Insolation (S⊕) +2400 +2600 +2800 +3000 +3200 +3400 +3600 +3800 +4000 +Teff (K) +Teegarden’s Star +b +Teegarden’s Star +c +Proxima Centauri +b +TOI-700 +d +GJ 1061 +c +GJ 1061 +d +Kepler-1649 +c +TRAPPIST-1 +d +TRAPPIST-1 +e +TRAPPIST-1 +f +TRAPPIST-1 +g +Ross 128 +b +Kepler-296A +e +Kepler-296A +f +Kepler-1229 b +K2-72 e +GJ 273 b +LP 890-9 +c +GJ 1002 +b +GJ 1002 +c +Wolf 1069b +Radius (R⊕) +0.5 +1 +1.5 +2 +Fig. 9. M-dwarf (Teff < 4000 K) planetary systems with at least one +detected planet in the conservative sample of potentially habitable exo- +planets (i.e., 0.5 R⊕ < Rp < 1.6 R⊕ or 0.1 M⊕ < M sin ip < 3 M⊕) defined +by the Habitable Exoplanet Catalog. The optimistic and conservative +HZ regions for a one Earth-mass planet following the definition as set +out by Kopparapu et al. (2013) are shaded with light and dark green, +respectively. Only the planets in either the conservative or optimistic +HZ of each planetary system are shown. White-filled and gray-filled +points indicate nontransiting and transiting detections, respectively. The +size of the circles is proportional to the planetary radius, estimated with +the mass–radius relationship of Zeng et al. (2016) for nontransiting RV +planets. The data used in this plot is further discussed in Appendix B. +Plot inspired by Zechmeister et al. (2019) and Dreizler et al. (2020). +RV-detected, single terrestrial planets (≲ 2 M⊕), which have all +been detected around M dwarfs less massive than 0.5 M⊙. These +objects are GJ 393 b (Amado et al. 2021) and Ross 128 b (Bon- +fils et al. 2018), where the former resides in the HZ region of its +host star (see also Sect. 5.1), whereas the latter receives too much +flux. Likewise, transiting planets following the same criteria in- +clude GJ 367 b (Lam et al. 2021) and GJ 1252 b (Shporer et al. +2020), where the latter has rather a tentative measured mass of +2.09 ± 0.56 M⊕ and both are not found in the HZ of their parent +stars. Nonetheless, the small sample size raises the question how +frequent the solitary occurrence of such a planet is. The overall +occurrence rate of small, rocky planets on close orbits has been +shown to be larger around host stars of later spectral type (e.g., +Mulders et al. 2015; Hardegree-Ullman et al. 2019; Hsu et al. +2020). However, there is some indication that this rule might not +apply for the latest M dwarfs (Gibbs et al. 2020; Sebastian et al. +2021; Sestovic & Demory 2020; Brady & Bean 2021; Mulders +et al. 2021) and that systems, such as the one presented here, +could be in fact rare. +Article number, page 14 of 26 + +ModernEarth-like +180° +60°W 0° 60°E +180° +60°N +60°N +30°N +30°N +0° +0° +30°S +30°S +S.09 +60°S +180° +60°W 0° +60°E +180° +min=178 Imean=233 Imax=286 +200 +225 +250 +275 +300 +Surface Temperature [K]D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +Planet formation models following the core accretion +paradigm (Pollack et al. 1996) generally suggest a high +multiplicity of Earth-mass planets around mid-M-dwarf host +stars (Burn et al. 2021). However, these models produce many +planets beyond current detection limits, and bias-corrected syn- +thetic populations show sufficiently reduced rates to be compat- +ible with the observations (Schlecker et al. 2022). The discovery +of a single planet comparable to Wolf 1069 b is consistent with +this picture. +We tested this scenario by applying the computed detec- +tion sensitivity (Fig. 10) to the synthetic planetary population +NGM10 around an 0.1 M⊙ star presented in Burn et al. (2021). +Using 50 % detection probability limits, 48 out of a total of 1000 +systems would result in a single detection. Out of those, we show +in Fig. 11 the three simulations that result in planets closest to +Wolf 1069 b on the period-versus-mass plane. Simulations lead- +ing to a single planet detection went through a stage of giant im- +pacts reducing the number of planets in the inner system and in- +creasing the mass of the detectable planet with respect to the rest +of the system. This is exemplified by the three best-fitting sim- +ulations which show three to four mergers with embryos more +massive than the lunar mass. +While the scenario of the formation of a single planet can- +not be ruled out, those simulations show that it is also possible +in ∼5 % of the cases to form a seemingly lone planet if multiple +embryos formed at the same time. However, if future observa- +tions extend the detection limits to larger orbits and lower plan- +etary masses, this formation theory will be more severely chal- +lenged. While a single, late stage giant impact with a similarly +massive body is currently in agreement with observations (e.g., +Sim 650), this could be ruled out with better sensitivity. Then, a +more dynamic history of the system is required (as in Sim 967 +where the complete inner system was ejected or accreted by the +detectable planet). A handicap of particular importance for thor- +ough analyses of planet multiplicity is the omission of early core +formation phases in current formation models (see e.g., Ormel +2017; Schlecker et al. 2021). Future planet population synthe- +sis studies have to take into account dust evolution, planetesimal +formation, and planetary embryo formation in a self-consistent +manner (Voelkel et al. 2020, 2021). +As for the observational prospects, dedicated measurements +with a high-precision spectrograph focused on searching for sub- +Earth-mass planets in the Wolf 1069 system could shed light on +a potential inner planet candidate (as was the case with Proxima +Centauri [d] first identified by Suárez Mascareño et al. 2020 and +later announced as a convincing planet candidate by Faria et al. +2022, with a periodicity of ∼5.12 d and K∼40 cm s−1), or even +further rule out this possibility. +5.3. Radio emission from star-planet interaction +Auroral radio emission from stars and planets is due to the elec- +tron cyclotron maser (ECM) instability (Melrose & Dulk 1982), +whereby plasma processes within the star (or planet) magneto- +sphere generate a population of unstable electrons that amplifies +the emission. The characteristic frequency of the ECM emis- +sion is given by the electron gyrofrequency, νG = 2.8 B MHz/G, +where B is the local magnetic field in the source region in Gauss. +ECM emission is a coherent mechanism that yields broadband +(∆ ν ∼ νG/2), highly polarized (sometimes reaching 100%), am- +plified nonthermal radiation. +For Jupiter-like planets, which have magnetic fields of Bpl ≃ +10 G, the direct detection of radio emission from them is plau- +sible, as the associated gyrosynchrotron frequency falls above +100 +101 +102 +103 +Period (d) +100 +101 +Mpl sini in M⊕ +0 +20 +40 +60 +80 +100 +Detection probability in % +Fig. 10. RV detection map of Wolf 1069 from an injection-and-retrieval +experiment after subtracting the 15.6 d, 90.3 d and 165 d signals. The red +circle indicates the planet Wolf 1069 b. +the ≃10 MHz ionosphere cutoff. However, the detection of radio +emission from Earth-sized exoplanets, which are also the type +of planets comprising a large majority of the CARMENES sam- +ple, is doomed to fail, as the associated frequency falls below the +ionosphere cutoff. +Fortunately, if the velocity, vrel, of the plasma relative to the +planetary body is less than the Alfvén speed, vA, in other words +MA = vrel/vA < 1, where MA is the Alfvén Mach number, then +energy and momentum can be transported upstream of the flow +along Alfvén wings. Jupiter’s interaction with its Galilean satel- +lites is a well-known example of sub-Alfvénic interaction, pro- +ducing auroral radio emission (Zarka 2007). In the case of star- +planet interaction, the radio emission arises from the magneto- +sphere of the host star, induced by the exoplanet crossing the +star magnetosphere, and the relevant magnetic field is that of +the star, B⋆, not the exoplanet magnetic field. Since M-dwarf +stars have magnetic fields ranging from about 100 G and up to +above 2-3 kG, their auroral emission falls in the range from a +few hundred MHz up to a few GHz. This interaction is expected +to yield detectable auroral radio emission via the cyclotron emis- +sion mechanism (e.g., Turnpenney et al. 2018; Pérez-Torres et al. +2021). +We followed the prescriptions in Appendix B of Pérez-Torres +et al. (2021) to estimate the flux density expected to arise from +the interaction between the planet Wolf 1069 b and its host star +at a frequency of 860 MHz, which corresponds to the cyclotron +frequency of the star magnetic field of 307 G, from Reiners et al. +(2022). We computed the radio emission arising from star-planet +interaction for two different magnetic field geometries: a closed +dipolar geometry, and an open Parker spiral geometry. For the +dipolar case, the motion of the plasma relative to Wolf 1069 b +happens in the supra-Alfvénic regime. Therefore no energy or +momentum can be transferred to the star through Alfvén waves. +In the open Parker spiral case, however, the plasma motion pro- +ceeds in the sub-Alfvénic regime. We show in Fig. 12 the pre- +dicted flux density as a function of orbital distance arising from +the interaction of a magnetized exoplanet (1 G) with its host star. +The yellow shaded areas encompass the range of values from +0.01 to 0.1 for the efficiency factor, ϵ, in converting Poynting +flux into ECM radio emission. The expected flux density is less +than 2 µ Jy. This is an extremely low value, which is not within +the reach of even the most sensitive radio interferometers. The +reasons behind this extremely faint signal are mainly two: First, +the relatively large distance to the system (9.6 pc away); and sec- +Article number, page 15 of 26 + +A&A proofs: manuscript no. MAIN +100 +101 +102 +103 +10 +2 +10 +1 +100 +101 +Mpl (M ) +Sim 625 +Detectable +Undetectable +Accreted +Wolf 1069 b +100 +101 +102 +103 +10 +2 +10 +1 +100 +101 +Mpl (M ) +Sim 967 +100 +101 +102 +103 +Period (d) +10 +2 +10 +1 +100 +101 +Mpl (M ) +Sim 650 +0 +20 +40 +60 +80 +100 +Detection probability in % +Fig. 11. Formation paths and final planets in planet formation simula- +tions taken from the population synthesis work of Burn et al. (2021). +We show the three simulations with a single detectable planet closest +to Wolf 1069 b in relative mass and orbital period. A planet is labeled +“undetectable” if the detection probability is below 50 %. Formation +tracks are shown as gray lines that can end in either a filled circles +(detectable planet), a triangle (accreted by detectable), a ring (unde- +tectable), or without a marker (accreted by other planets or ejected). +ond, the large separation of the planet from its host star (about 80 +stellar radii). Therefore, the chances of detecting radio emission +from star-planet interaction in Wolf 1069 are essentially null. +6. Conclusion +Using CARMENES spectroscopic measurements, we presented +the discovery of a nontransiting exoplanet, Wolf 1069 b, with a +period of 15.564±0.015 d, minimum mass of 1.26±0.21 M⊕, and +insolation of 0.652+0.029 +−0.027 S ⊕, putting it safely in the conservative +HZ around a low-mass M-dwarf star. This makes Wolf 1069 b +the sixth closest (d ∼ 9.6 pc), Earth-mass planet in the conser- +vative HZ from us, following Proxima Centauri b, GJ 1061 d, +Teegarden’s Star c, and GJ 1002 b and c. Preliminary investi- +gations of the potential habitability of the planet using GCM +climate simulations suggest the planet to be a promising addi- +3 +2 +1 +0 +log(MA) +Wolf 1069 b - Open field +Wolf 1069 b - Open field +20 +40 +60 +80 +100 +Distance / Stellar radius +3 +2 +1 +0 +1 +2 +3 +log (Flux density [mJy]) +Bcycl = 307.0 G +Bplanet = 1 G +ncorona = 1.0x107 cm−3 +Saur/Turnpenney model +1 +3 +5 +10 +15 +20 +Orbital period [days] +Fig. 12. Expected flux density for auroral radio emission arising from +star-planet interaction in the system Wolf 1069, as a function of orbital +distance. The interaction is expected to be in the sub-Alfvénic regime +(i.e., MA = vrel/vAlfv ≤ 1; top panel) at the location of each planet +(vertical dashed line). +tion to the group of current targets to search for biosignature +markers, such as Proxima Centauri b, TRAPPIST-1 e, and TOI- +700 d. Wolf 1069 b unfortunately does not transit its stellar host, +though future observations with thermal emission and reflected +light phase curves could shed light on the properties of its at- +mosphere. We additionally investigated whether star-planet in- +teractions in Wolf 1069 would be feasible to observe with radio +emissions, but found these potential observations unfruitful. +The Wolf 1069 system becomes more intriguing as there is +no significant evidence of closer-in planets (P < 10 d) greater +than one Earth mass, based on our detectability limits. This con- +figuration is a plausible outcome based on a select few synthetic +planetary population simulations, and even suggestive of a planet +formation history including a late giant impact phase. The detec- +tion of potential inner sub-Earth-mass planets with further sub- +m s−1 RV observations could then confirm or reject this forma- +tion theory. +The stellar host itself is a relatively inactive, low-mass M5.0 +dwarf, though exhibits periods of higher activity levels, for +which we determine its photometric rotation period to be 150– +170 d. This rotation period was also present in the CARMENES +RVs, and thus, modeled with a dSHO-GP in the final fit. The +RVs showed one more additional significant periodicity at 90.3 d +with a low amplitude (i.e., <1 m s−1), however, we demonstrate +Article number, page 16 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +that there is currently not enough supporting evidence in favor +of a planetary origin and it appears to be an effect of telluric +contamination. Further RV investigation could be beneficial. To +conclude, Wolf 1069 b is a noteworthy discovery that will al- +low further exploration into the habitability of Earth-mass plan- +ets around M dwarfs, as well as case study in testing planetary +formation theories. +Acknowledgements. Part of this work was supported by the German Deutsche +Forschungsgemeinschaft, DFG project number Ts 17/2–1. CARMENES is an +instrument at the Centro Astronómico Hispano-Alemán (CAHA) at Calar Alto +(Almería, Spain), operated jointly by the Junta de Andalucía and the Instituto +de Astrofísica de Andalucía (CSIC). CARMENES was funded by the Max- +Planck-Gesellschaft (MPG), the Consejo Superior de Investigaciones Cientí- +ficas (CSIC), the Ministerio de Economía y Competitividad (MINECO) and +the European Regional Development Fund (ERDF) through projects FICTS- +2011-02, ICTS-2017-07-CAHA-4, and CAHA16-CE-3978, and the members +of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Insti- +tuto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Cièn- +cies de l’Espai, Institut für Astrophysik Göttingen, Universidad Complutense +de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica +de Canarias, Hamburger Sternwarte, Centro de Astrobiología and Centro As- +tronómico Hispano-Alemán), with additional contributions by the MINECO, the +Deutsche Forschungsgemeinschaft through the Major Research Instrumentation +Programme and Research Unit FOR2544 “Blue Planets around Red Stars”, the +Klaus Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and +by the Junta de Andalucía. This work was based on data from the CARMENES +data archive at CAB (CSIC-INTA). Data were partly collected with the 90 cm +and 150 cm telescopes at Observatorio de Sierra Nevada (OSN), operated by the +Instituto de Astrofísica de Andalucí a (IAA, CSIC); we deeply acknowledge the +OSN telescope operators for their very appreciable support. The Telescopi Joan +Oró (TJO) of the Observatori Astronómic del Montsec is owned by the General- +itat de Catalunya and operated by the Institut d’Estudis Espacials de Catalunya +(IEEC). We acknowledge financial support from the Agencia Estatal de Investi- +gación of the Ministerio de Ciencia e Innovación (AEI/10.13039/501100011033) +and the ERDF “A way of making Europe” through projects PID2019-109522GB- +C5[1:4], PID2019-107061GB-C64, and PID2019-110689RB-100, and the Cen- +tre of Excellence “Severo Ochoa” and “María de Maeztu” awards to the Insti- +tuto de Astrofísica de Canarias (SEV-2015-0548), Instituto de Astrofísica de +Andalucía (SEV-2017-0709), and Centro de Astrobiología (MDM-2017-0737); +the European Research Council under the Horizon 2020 Framework Program +(ERC Advanced Grant Origins 832428 and under Marie Skłodowska-Curie grant +895525); the Generalitat de Catalunya/CERCA programme; the DFG through +the priority program SPP 1992 “Exploring the Diversity of Extrasolar Plan- +ets (JE 701/5-1)” and the Research Unit FOR 2544 “Blue Planets around Red +Stars” (KU 3625/2-1); the Bulgarian National Science Fund through program +“VIHREN-2021” (KP-06-DV/5); the SNSF under grant P2BEP2_195285; the +National Science Foundation under award No. 1753373, and by a Clare Boothe +Luce Professorship. 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Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +1 Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidel- +berg, Germany e-mail: kossakowski at mpia.de, kuerster +at mpia.de +2 Department of Astronomy, Sofia University “St Kliment Ohridski”, +5 James Bourchier Blvd, BG-1164 Sofia, Bulgaria +3 Landessternwarte, Zentrum für Astronomie der Universität Heidel- +berg, Königstuhl 12, 69117 Heidelberg, Germany +4 Centro de Astrobiología (CSIC-INTA), ESAC, Camino bajo del +castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain +5 NASA Goddard Space Flight Center, 8800 Greenbelt Road, Green- +belt, MD 20771, USA +6 NASA GSFC Sellers Exoplanet Environments Collaboration +7 Department of Astronomy, University of Maryland, College Park, +MD 20742, United States of America +8 Center for Research and Exploration in Space Science and Technol- +ogy, NASA/GSFC, Greenbelt, MD 20771, United States of America +9 American University, College of Arts and Sciences, Washington +DC, United States of America +10 Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Ger- +many +11 Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Taut- +enburg, Germany +12 Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, C/ de +Can Magrans s/n, 08193 Cerdanyola del Vallès, Spain +13 Institut d’Estudis Espacials de Catalunya (IEEC), C/ Gran Capità +2-4, 08034 Barcelona, Spain +14 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la As- +tronomía s/n, 18008 Granada, Spain +15 Centro de Astrobiología (CSIC-INTA), Carretera de Ajalvir km 4, +28850 Torrejón de Ardoz, Madrid, Spain +16 Institut für Astrophysik und Geophysik, Georg-August-Universität, +Friedrich-Hund-Platz 1, 37077 Göttingen, Germany +17 Centro Astronónomico Hispano en Andalucía, Observatorio de +Calar Alto, Sierra de los Filabres, E-04550 Gérgal, Spain +18 Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tener- +ife, Spain +19 Departamento de Astrofísica, Universidad de La Laguna, 38206 La +Laguna, Tenerife, Spain +20 Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig +Weg 3, 37077 Göttingen, Germany +21 Department of Physics, University of Warwick, Gibbet Hill Road, +Coventry CV4 7AL, United Kingdom +22 Departamento de Física de la Tierra y Astrofísica & IPARCOS- +UCM (Instituto de Física de Partículas y del Cosmos de la UCM), +Facultad de Ciencias Físicas, Universidad Complutense de Madrid, +E-28040 Madrid, Spain +23 Department of Physics, Ariel University, Ariel 40700, Israel +24 School of Sciences, European University Cyprus, Diogenes street, +Engomi, 1516 Nicosia, Cyprus +25 Department of Astronomy/Steward Observatory, The University of +Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, United +States of America +26 Centre for Earth Evolution and Dynamics, Department of Geo- +sciences, Universitetet i Oslo, Sem Sælands vei 2b, 0315 Oslo, Nor- +way +27 Department of Physics & Astronomy, University of California, +Irvine, CA 92697, Irvine +28 University of Colorado, Boulder Laboratory for Atmospheric and +Space Physics, Department of Atmospheric and Oceanic Sciences, +Boulder, CO 80303, United States of America +29 NASA NExSS Virtual Planetary Laboratory, Seattle, WA +Article number, page 19 of 26 + +A&A proofs: manuscript no. MAIN +Appendix A: AliasFinder figures +The RV data show a variety of aliases related to the 15.6 d sig- +nal. In order to establish that 15.6 d is indeed the true periodicity, +we tested the aliasing using AliasFinder (Stock & Kemmer +2020), which follows the methodology from Dawson & Fab- +rycky (2010). The essence behind the algorithm is to examine +the GLS periodograms of simulated data sets, in which either +of the two aliasing signals are injected, to the GLS periodogram +provided by the original data. The injected signal of whichever +periodogram matches best to the original one is defined to be the +true periodicity apparent in the data. The results of this method +by simulating 1000 time series for both the daily and yearly sam- +pling frequencies is shown in Fig. A.1, confirming the true signal +to be 15.6 d. +Appendix B: Known planets in the habitable zone +around M dwarfs +We started from the list collected by PHL at UPR, which ob- +tains its parameters from the NASA exoplanet archive. The list, +as last updated on 06 December 2021, comprises 21 planets that +are most probable to have a rocky composition and maintain sur- +face liquid water. We individually vetted each system using the +most up-to-date literature and updated the planetary and stellar +parameters. Most of the planets stayed consistent since the up- +date, though there are some modifications: +– GJ 667 C: We omit the planets d, e, f, and g proposed by +Anglada-Escudé et al. (2013) in which the second two would +reside in the HZ. They emerged as controversial when stel- +lar activity was also modeled within the RVs as a red-noise +component (Robertson & Mahadevan 2014; Feroz & Hob- +son 2014). Unfortunately, that leaves only planet c in the op- +timistic HZ in this planetary system. Nonetheless, there is +still an inner planet to that of the HZ one, planet b, with a +minimum mass of 5.6 M⊕. +– LP 890-9: We add the planet b recently unveiled by Del- +rez et al. (2022) around LP 890-9, which is next coolest star +found to host a HZ planet, after TRAPPIST-1. +– GJ 1002: We add planets b and c recently discovered around +the M5.5 V star, both of which reside in the conservative HZ +(Suárez Mascareño et al. 2022). +Appendix C: Priors and posteriors +Appendix D: Short tables and data tables +Article number, page 20 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +0.1 +0.2 +0.1 +0.2 +0.0630 +0.0645 +0.0660 +0.1 +0.2 +0.9340 +0.9355 +0.9370 +1.0630 +1.0645 +1.0660 +15.8730 +15.5039 +15.1515 +1.0707 +1.0689 +1.0672 +0.9407 +0.9394 +0.9381 +Frequency f [1/d] +Power (ZK) +Period [d] +0.15 +0.30 +0.15 +0.30 +0.0630 +0.0645 +0.0660 +0.15 +0.30 +0.0600 +0.0615 +0.0630 +0.0660 +0.0675 +0.0690 +15.8730 +15.5039 +15.1515 +16.6667 +16.2602 +15.8730 +15.1515 +14.8148 +14.4928 +Frequency f [1/d] +Power (ZK) +Period [d] +Fig. A.1. Plots generated by AliasFinder for the daily (top) and yearly (bottom) aliases for the 15.6 d signal. Each row illustrates the results +for one simulated frequency, as indicated by the dashed blue vertical line. Each column is centered on a frequency window corresponding to the +simulated frequencies. The red line represents the periodogram of the original data set, whereas the black line is the median of the simulations, and +the gray shaded regions depict the interquartile range and the confidence range of 90% and 99% of the simulations. The clock diagrams indicate +the phase. +Article number, page 21 of 26 + +A&A proofs: manuscript no. MAIN +Table C.1. Prior parameters for the photometric rotation period determination in Sect. 3.2. +Parameter name +Prior +Unit +Description +Photometric instrumental parameters +µOSN-V-T150 +U(0.9, 1.1) +ppm +Photometric normalization for OSN-V-T150 +σOSN-V-T150 +J(10−8, 10−1) +ppm +Extra jitter term for OSN-V-T150 +µOSN-R-T150 +U(0.9, 1.1) +ppm +Photometric normalization for OSN-R-T150 +σOSN-R-T150 +J(10−8, 10−1) +ppm +Extra jitter term for OSN-R-T150 +µOSN-I-T150 +U(0.9, 1.1) +ppm +Photometric normalization for OSN-I-T150 +σOSN-I-T150 +J(10−8, 10−1) +ppm +Extra jitter term for OSN-I-T150 +µOSN-V-T90 +U(0.9, 1.1) +ppm +Photometric normalization for OSN-V-T90 +σOSN-V-T90 +J(10−8, 10−1) +ppm +Extra jitter term for OSN-V-T90 +µOSN-R-T90 +U(0.9, 1.1) +ppm +Photometric normalization for OSN-R-T90 +σOSN-R-T90 +J(10−8, 10−1) +ppm +Extra jitter term for OSN-R-T90 +µTJO-R +U(0.9, 1.1) +ppm +Photometric normalization for TJO-R +σTJO-R +J(10−8, 10−1) +ppm +Extra jitter term for TJO-R +µSuperWASP +U(0.9, 1.1) +ppm +Photometric normalization for SuperWASP +σSuperWASP +J(10−8, 10−1) +ppm +Extra jitter term for SuperWASP +µMEarth-tel01-s1 +U(0.9, 1.1) +ppm +Photometric normalization for MEarth-tel01-s1 +σMEarth-tel01-s1 +J(10−8, 10−1) +ppm +Extra jitter term for MEarth-tel01-s1 +µMEarth-tel01-s2 +U(0.9, 1.1) +ppm +Photometric normalization for MEarth-tel01-s2 +σMEarth-tel01-s2 +J(10−8, 10−1) +ppm +Extra jitter term for MEarth-tel01-s2 +µMEarth-tel05-s2 +U(0.9, 1.1) +ppm +Photometric normalization for MEarth-tel05-s2 +σMEarth-tel05-s2 +J(10−8, 10−1) +ppm +Extra jitter term for MEarth-tel05-s2 +dSHO-GP parameters +Prot, GP, all(a) +J(10, 200) +d +Primary period of the dSHO-GP +δQGP, all(a) +J(102, 105) +. . . +Quality factor difference between the first +and second oscillations of the dSHO-GP +Q0 GP, all(a) +J(10−8, 103) +. . . +Quality factor for the secondary oscillation of the dSHO-GP +σGP, OSN-V-T150,OSN-V-T90 +U(0.0, 0.2) +. . . +��������������������������� +σGP, OSN-R-T150,OSN-R-T90,TJO-R +U(0.0, 0.2) +. . . +σGP, OSN-I-T150 +U(0.0, 0.2) +. . . +Amplitude of the dSHO-GP +σGP, MEarth-tel01-s1 +U(0.0, 0.2) +. . . +σGP, MEarth-tel01-s2,MEarth-tel05-s2 +U(0.0, 0.2) +. . . +σGP, SuperWASP +U(0.0, 0.2) +. . . +fGP, OSN-V-T150,OSN-V-T90 +U(0.1, 1.0) +. . . +��������������������� +fGP, OSN-R-T150,OSN-R-T90,TJO-R +U(0.1, 1.0) +. . . +Fractional amplitude of the +fGP, OSN-I-T150 +U(0.1, 1.0) +. . . +secondary oscillation of the dSHO-GP +fGP, MEarth-tel01-s1,MEarth-tel01-s2,MEarth-tel05-s2 +U(0.1, 1.0) +. . . +fGP, SuperWASP +U(0.1, 1.0) +. . . +Notes. (a) “all” comprises the following instruments: OSN-V-T150, OSN-R-T150,OSN-I-T150, OSN-V-T90, OSN-R-T90, TJO-R, SuperWASP, +MEarth-tel01-s1, MEarth-tel01-s2, MEarth-tel05-s2 +Article number, page 22 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +Table C.2. Priors for the RV fits for Wolf 1069 with juliet in Sect. 4.2. The 90.3 d signal is not speculated to have planetary origins (Sect. 4.3), +so we denote it as a signal “2”. +Parameter name +Prior +Units +Description +Parameters for planet b +Pb +U(15.4, 15.7) +d +Period. +t0,b +U(2458502.0, 2458515.0) +d +Time-of-transit center. +Kb +U(0.0, 5.0) +m s−1 +Radial velocity semi-amplitude. +S 1,b = √eb sin ωb +F (0.0) (circular) +. . . +Parametrization for e and ω +U(−1, 1) (eccentric) +. . . +Parametrization for e and ω +S 2,b = √eb cos ωb +F (0.0) (circular) +. . . +Parametrization for e and ω +U(−1, 1) (eccentric) +. . . +Parametrization for e and ω +Parameters for the 90.3 d signal +P2 +U(85.0, 95.0) +d +Period. +t0,2 +U(2458500.0, 2458590.0) +d +Time-of-transit center. +K2 +U(0.0, 5.0) +m s−1 +Radial velocity semi-amplitude. +S 1,2 = √eb sin ωb +F (0.0) (circular) +. . . +Parametrization for e and ω +S 2,2 = √eb cos ωb +F (0.0) (circular) +. . . +Parametrization for e and ω +RV instrumental parameters +γCARMENES-VIS +U(−20.0, 20.0) +m s−1 +Systemic velocity for CARMENES +σCARMENES-VIS +J(0.01, 50.0) +m s−1 +Extra jitter term for CARMENES +dSHO-GP parameters +σGP, CARMENES-VIS +U(0.0, 15.0) +m s−1 +Amplitude of the dSHO-GP +Q0 GP, CARMENES-VIS +J(10−8, 105) +. . . +Quality factor for the secondary oscillation of the dSHO-GP +fGP, CARMENES-VIS +U(0.1, 1.0) +. . . +Fractional amplitude of the secondary oscillation of the dSHO-GP +δQGP, CARMENES-VIS +J(102, 108) +. . . +Quality factor difference between the first +and second oscillations of the dSHO-GP +Prot, GP, CARMENES-VIS +U(155.0, 175.0) +d +Primary period of the dSHO-GP +Article number, page 23 of 26 + +A&A proofs: manuscript no. MAIN +Pb (d) = 15.56+0.02 +0.01 +1 +0 +1 +2 +3 +t0, b - 2458511 (d) +t0, b - 2458511 (d) = 0.63+0.45 +0.46 +0.4 +0.8 +1.2 +1.6 +Kb (m s +1) +Kb (m s +1) = 1.07+0.17 +0.17 +15.52 +15.55 +15.58 +15.61 +15.64 +Pb (d) +0.4 +0.8 +1.2 +1.6 +2.0 +Mbsin i (M +) +1 +0 +1 +2 +3 +t0, b - 2458511 (d) +0.4 +0.8 +1.2 +1.6 +Kb (m s +1) +0.4 +0.8 +1.2 +1.6 +2.0 +Mbsin i (M +) +Mbsin i (M +) = 1.27+0.21 +0.21 +Fig. C.1. Posterior distributions for the inner-most planet Wolf 1069 b from the final RV fit described in Sect. 4.2. +Article number, page 24 of 26 + +D. Kossakowski et al.: Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf +GP, CARMENES = 2.80+0.80 +0.53 +156 +160 +164 +168 +172 +Prot; GP, CARMENES +Prot; GP, CARMENES = 165.58+3.28 +3.44 +0.2 +0.4 +0.6 +0.8 +fGP, CARMENES +fGP, CARMENES = 0.61+0.25 +0.28 +7.5 +5.0 +2.5 +0.0 +log(Q0 GP, CARMENES) +log(Q0 GP, CARMENES) = +3.82+2.53 +2.60 +2.5 +5.0 +7.5 +10.0 +GP, CARMENES +3.0 +4.5 +6.0 +7.5 +log( QGP, CARMENES) +156 +160 +164 +168 +172 +Prot; GP, CARMENES +0.2 +0.4 +0.6 +0.8 +fGP, CARMENES +7.5 +5.0 +2.5 +0.0 +log(Q0 GP, CARMENES) +3.0 +4.5 +6.0 +7.5 +log( QGP, CARMENES) +log( QGP, CARMENES) = 4.90+1.97 +1.88 +Fig. C.2. Posterior distributions for the stellar rotation period using the dSHO-GP from the final RV fit described in Sect. 4.2. +Article number, page 25 of 26 + +A&A proofs: manuscript no. MAIN +Table C.3. Full set of posterior parameters used in the final model +choice for Wolf 1069 and described in Sect. 4.2. +Parameter +Posterior +Posterior parameters for planet b +Pb +15.564+0.015 +−0.015 +t0,b +2458511.63+0.45 +−0.46 +Kb +1.07+0.17 +−0.17 +RV instrumental parameters +γCARMENES (m s−1) +−10.64+0.3 +−0.39 +σCARMENES (m s−1) +0.47+0.36 +−0.41 +dSHO-GP parameters +σGP, CARMENES-VIS +2.80+0.80 +−0.53 +Q0 GP, CARMENES-VIS +0.00015+0.05076 +−0.00015 +fGP, CARMENES-VIS +0.61+0.25 +−0.28 +δQGP, CARMENES-VIS +80000+7300000 +−78000 +Prot, GP, CARMENES-VIS +165.6+3.3 +−3.4 +Table D.1. Multiband photometry of Wolf 1069a . +Band +Magnitude +Reference +(mag) +B +15.82 ± 0.10 +UCAC4 +g′ +14.78 ± 0.13 +UCAC4 +GBP +14.368 ± 0.004 +Gaia DR3 +V +13.99 ± 0.05 +UCAC4 +r′ +13.41 ± 0.05 +UCAC4 +i′ +11.58 ± 0.09 +UCAC4 +GRP +11.027 ± 0.004 +Gaia DR3 +J +9.029 ± 0.039 +2MASS +H +8.483 ± 0.073 +2MASS +KS +8.095 ± 0.021 +2MASS +W1 +7.877 ± 0.023 +AllWISE +W2 +7.717 ± 0.020 +AllWISE +W3 +7.545 ± 0.016 +AllWISE +W4 +7.445 ± 0.084 +AllWISE +Notes. (a) Gaia EDR3 G magnitude in Table 2. +References. 2MASS: Skrutskie et al. (2006); Gaia DR3: Gaia Collab- +oration et al. (2022); UCAC4: Zacharias et al. (2012); WISE/AllWISE: +Cutri & et al. (2012, 2014). +Table D.2. Telluric-corrected RV data used in this work for Wolf 1069. +Data will be available online in machine-readible format. +BJD (TDB* ) +RV (m s−1) +σRV (m s−1) +Instrument +2457563.66099 +−11.183 +1.547 +CARMENES +2457569.58800 +−10.331 +3.441 +CARMENES +2457575.61747 +−12.551 +1.584 +CARMENES +2457584.59445 +−15.557 +2.250 +CARMENES +2457591.51688 +−11.028 +1.915 +CARMENES +2457594.58057 +−14.606 +2.152 +CARMENES +2457596.52948 +−15.083 +1.468 +CARMENES +2457597.44527 +−16.279 +1.138 +CARMENES +2457610.51532 +−13.451 +1.577 +CARMENES +2457612.45812 +−12.912 +1.810 +CARMENES +2457613.43402 +−12.103 +1.540 +CARMENES +... +... +... +... +2458978.65464 +−12.167 +2.433 +CARMENES +2458988.61439 +−9.675 +1.770 +CARMENES +2458994.61685 +−9.450 +1.428 +CARMENES +2458999.63468 +−6.863 +1.591 +CARMENES +2459000.64327 +−7.105 +1.278 +CARMENES +2459001.64411 +−7.717 +1.318 +CARMENES +2459006.64529 +−5.812 +1.641 +CARMENES +2459010.59488 +−5.312 +1.664 +CARMENES +2459015.61493 +−7.113 +1.513 +CARMENES +2459017.64665 +−9.772 +3.050 +CARMENES +Notes. (*) Barycentric dynamical time. +Article number, page 26 of 26 + diff --git a/SNE0T4oBgHgl3EQfkwGA/content/tmp_files/load_file.txt b/SNE0T4oBgHgl3EQfkwGA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..80cc810ba1023c14d34c6fbda007cf55a2c93398 --- /dev/null +++ b/SNE0T4oBgHgl3EQfkwGA/content/tmp_files/load_file.txt @@ -0,0 +1,2959 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf,len=2958 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN ©ESO 2023 January 9, 2023 The CARMENES search for exoplanets around M dwarfs Wolf 1069 b: Earth-mass planet in the habitable zone of a nearby, very low-mass star⋆ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kürster1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Trifonov1,2, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Henning1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kemmer3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Caballero4, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Burn1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Sabotta3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Crouse5,6,7,8, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fauchez5,9,6, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Nagel10,11, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kaminski3, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Herrero12,13, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Rodríguez14, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' González-Álvarez15, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Quirrenbach3, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Amado14, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Ribas12,13, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Reiners16, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Aceituno17,14, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Béjar18,19, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Baroch12,13, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Bastelberger7,5,8,6, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Chaturvedi11, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Cifuentes4, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Dreizler16, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Jeffers20, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kopparapu5,6, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Lafarga21,12,13, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' López-González14, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Martín-Ruiz14, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Montes22, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Morales12,13, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Pallé18,19, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Pavlov1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Pedraz17, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Perdelwitz23,10, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Pérez-Torres14,24, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Perger12,13, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Reffert3, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Rodríguez López14, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Schlecker25,1, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Schöfer14,16, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Schweitzer10, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Shan26,16, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Shields27, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stock3, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf28,5,29, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Zapatero Osorio15, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Zechmeister16 (Affiliations can be found after the references) Received 28 October 2022 / accepted 21 December 2022 ABSTRACT We present the discovery of an Earth-mass planet (Mb sin i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 M⊕) on a 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d orbit of a relatively nearby (d ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 pc) and low-mass (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='167 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='011 M⊙) M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 V star, Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Sitting at a separation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0672 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0014 au away from the host star puts Wolf 1069 b in the habitable zone (HZ), receiving an incident flux of S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='652 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='029 S ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The planetary signal was detected using telluric-corrected radial-velocity (RV) data from the CARMENES spectrograph, amounting to a total of 262 spectroscopic observations covering almost four years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There are additional long-period signals in the RVs, one of which we attribute to the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This is possible thanks to our photometric analysis including new, well-sampled monitoring campaigns undergone with the OSN and TJO facilities that supplement archival photometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', from MEarth and SuperWASP), and this yielded an updated rotational period range of Prot = 150 − 170 d, with a likely value at 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The stellar activity indicators provided by the CARMENES spectra likewise demonstrate evidence for the slow rotation period, though not as accurately due to possible factors such as signal aliasing or spot evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Our detectability limits indicate that additional planets more massive than one Earth mass with orbital periods of less than 10 days can be ruled out, suggesting that perhaps Wolf 1069 b had a violent formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This planet is also the sixth closest Earth-mass planet situated in the conservative HZ, after Proxima Centauri b, GJ 1061 d, Teegarden’s Star c, and GJ 1002 b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Despite not transiting, Wolf 1069 b is nonetheless a very promising target for future three-dimensional climate models to investigate various habitability cases as well as for sub-m s−1 RV campaigns to search for potential inner sub-Earth-mass planets in order to test planet formation theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' methods: data analysis – planetary systems – stars: individual: Wolf 1069 – stars: low-mass – techniques: radial velocities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Introduction An impressive 5000 exoplanets and counting have been detected thus far1, largely thanks to the past and ongoing radial-velocity (RV) and transit surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' On the hunt for an Earth analog, out of these thousands of planets, only ∼50 have been found to sit in the so-called habitable zone (HZ) of their stellar host2, which is defined to be the circumstellar region in which liquid water could potentially exist on the surface of the planet (Kasting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Only 20 of these are considered to be Earth-sized, defined by radii of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 R⊕ < Rp < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 R⊕ or by masses of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 M⊕ < M sin ip < 3 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Moreover, a major- ity of them have been discovered around M-dwarf stellar hosts, ⋆ RVs and additional data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', stellar activity indicators as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5 are available in electronic form at the CDS via anonymous ftp to cdsarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='fr (TBD)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1 https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='edu/, accessed on 16 September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2 The Habitable Exoplanet Catalog: http://phl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='upr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='edu/ projects/habitable-exoplanets-catalog, last updated on 6 December 2021 and further discussed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' most likely due to the ease in detectability considering the higher planet-to-star mass and radius ratios (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Zechmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Seager 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Bonfils et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Shields et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Per- ryman 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The definition of the HZ is not a definite implication for a life-hosting world, but rather acts as a good indicator for a planet to show further promising potential for markers of sur- face habitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There are in fact a variety of factors that affect its habitability potential, such as the X-ray/UV emission or age in regards to the stellar host, or processes due to the planet it- self including its atmospheric composition or its ability to retain certain elements in its atmosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kop- parapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For this reason, it is not only crucial to first gather planets that are situated within the HZ, but also, it is nec- essary to build a better understanding of the stellar effects on the planet’s habitability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Segura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2003, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Hilton 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Chadney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016), and to also characterize its atmosphere observability, with instruments such as the James Webb Space Telescope (JWST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 1 of 26 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='02477v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='EP] 6 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Even though most HZ planets around low-mass M dwarfs are RV-only detections that do not transit, we can nonetheless generate useful indicators to investigate their habitability further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Three-dimensional (3D) general circulation model (GCM) cli- mate simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Way et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022) can produce predictions to investigate various atmosphere composi- tions to test how durably habitable the planet is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' These analyses, along with a push for improvements in thermal emission and re- flected light phase curve observations, are crucial given the rise of nontransiting planets found in the HZ of stellar hosts within the solar neighborhood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Anglada-Escudé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Drei- zler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In this paper, we turn our attention to the discovery of Wolf 1069 b, an Earth-mass planetary companion orbiting a mid- type M dwarf within the conservative HZ limits, as defined by Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This planet could very well possess the key factors in making it indeed a habitable world according to preliminary 3D GCM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Also, in contrast to other habit- able worlds in the conservative HZ with similar host stars (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Kepler-1649, Proxima Centauri, GJ 1061, Teegarden’s Star, and GJ 1002), Wolf 1069 b is the only one within the conservative HZ without an inner planet, based on our current detection lim- its.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This notion is supported by the works of Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021), Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021), and Schlecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021), where we ex- pect a lower planet occurrence rate for stars with M⋆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 M⊙ than for stars with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 M⊙ < M⋆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 M⊙ for both the peb- ble and core accretion scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Granted, these are theoretical predictions as more observation-based evidence is required to confirm this, and Wolf 1069 b could still be accompanied by closer-in and outer planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Nevertheless, the concept that only one planet survives is predicted by formation models if there were at least one giant impact at the late stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This would en- hance the chance of having a massive moon similar to the Earth and might also stir up the interior of the planet to prevent strati- fication and sustain a magnetic field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Jacobson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' As remote as this appears, the search for exo-moons is no longer so far-fetched in recent times (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Martínez-Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Dobos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The paper is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Section 2 first presents the comprehensive spectroscopic and photometric data collected for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Then, the host star and its properties are introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3, in which we determine and update its rotational period using newly taken photometry from our facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The various signals in the RVs for this system are investigated and modeled in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4, where the results and prospects for Wolf 1069 b are then discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We finally display our conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Observational data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' CARMENES high-resolution spectroscopy The CARMENES3 instrument is located at the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 m telescope at the Calar Alto Observatory in Spain and consists of two sep- arate spectrographs residing in two channels: the visual (VIS), which covers the spectral range 520–960 nm (spectral resolution of R = 94 600), and the near-infrared (near-IR), which covers the 960–1710 nm range (R = 80 400) (Quirrenbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2014, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 (Karmn J20260+585) was one of the ∼300 stars initially chosen as part of the CARMENES Guaranteed 3 Calar Alto high-Resolution search for M dwarfs with Exo-earths with Near-infrared and optical Échelle Spectrographs, http://carmenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' caha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='es Time Observation (GTO) program (Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018), and 276 observations were since accumulated spanning 1450 days (June 2016 – June 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There were six measurements that were missing a drift correction as well as an additional eight that had low signal-to-noise ratio, and were, for this reason, discarded, resulting in 262 usable RV measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Further- more, the spectra were notably affected by telluric absorption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We corrected them by employing the template division telluric modeling methodology, a technique to remove telluric absorption lines from stellar spectra with numer- ous intrinsic lines (Nagel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This technique is suit- able for separating telluric and spectral components based on the Earth’s barycentric motion throughout the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The telluric- free spectra of Wolf 1069 are then produced by fitting a syn- thetic transmission model of the Earth’s atmosphere to each in- dividually extracted telluric spectrum with molecfit (Smette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The weighted root mean square (wrms) and the me- dian uncertainty of the remaining 262 data points are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='66 m s−1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='67 m s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The simultaneously taken near-IR measurements were not considered as part of the analysis given the notably higher wrms of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 m s−1 along with the significantly higher mean uncertainty of each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Therefore, we con- tinue the analysis with the 262 telluric-absorption-corrected VIS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The CARMENES RV data with their uncertainties are displayed in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The raw data are first pipelined through the standard guar- anteed time observations via caracal (Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Then, the RVs are determined using serval4 (Zechmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018), where the spectra are corrected for barycentric motion, secular acceleration, and instrumental drift, and then nightly zero-points were calculated and applied (Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In addition, serval produces various stellar activity indica- tors such as the chromatic index (CRX), the differential line width (dLW), the Hα index, the Ca ii IR triplet (IRT) lines, and the Na i D doublet lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We also obtained the photospheric absorption band indices TiO λ 7050 Å, TiO λ 8430 Å, and TiO λ 8860 Å from the nontelluric-corrected spectra using spec- tral ranges without notable telluric contamination, as defined by Schöfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Lastly, we computed the cross-correlation function (CCF) and its full-width half-maximum (FWHM), con- trast (CTR), and bisector velocity span (BVS) values computed with the raccoon5 code, adopting the approach of using bi- nary masks as explained in Lafarga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' These indi- cators were investigated for the rotation period of the stellar host (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Our photometric campaigns We carried out simultaneous, continuous photometric follow-up of Wolf 1069 from 2017 to 2020 with the photometric facilities as listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A summary of the various photometric data sets is found in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' They are also displayed as a time series in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Observatorio de Sierra Nevada (OSN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The Observatorio de Sierra Nevada (OSN)6, currently maintained by the Instituto de Astrofísica de Andalucía (IAA) and situated at Loma de Dilar in Granda, Spain, hosts two Nasmyth optical telescopes with apertures of 90 cm (T90) and 150 cm (T150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Both telescopes 4 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='com/mzechmeister/serval 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='com/mlafarga/raccoon 6 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='osn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='iaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='es/en Article number, page 2 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf 10 5 0 5 10 RV (m/s) GP component CARMENES 600 800 1000 1200 1400 1600 1800 2000 Time (BJD - 2457000) 10 0 10 RV resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (m/s) 10 5 0 5 10 RV (m/s) P = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='56 d t0 = 2458511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 Phase 10 0 10 RV resid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (m/s) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' RV time series and phase-folded plots for Wolf 1069 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Top panel: CARMENES VIS RV measurements for Wolf 1069 along with the best-fit model (dark gray line) and the stellar rotation period modeled by a dSHO-GP (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The light gray band indicates the 68% confidence interval of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Bottom panel: RVs phase-folded to the period of Wolf 1069 b at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d (Kb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 m s−1) and with the GP component subtracted out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The black circles represent the data points binned to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 in phase space for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The bottom panel for each plot represents the residuals after subtracting out the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There are two data points that did not fit within the boundary for visual reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' are equipped with similar VersArray 2k × 2k CCD cameras, which deliver images with fields of view (FOV) of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 arcmin × 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 arcmin (T90;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Amado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021) or 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='92 arcmin × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='92 arcmin (T150;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Rodríguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each camera is based on a high quantum efficiency back-illuminated CCD chip, type Marconi-EEV CCD42-4, with optimized response in the ultravi- olet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We monitored Wolf 1069 using both telescopes and various observing runs between the years 2017 and 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The observa- tions with the T90 telescope were collected using both Johnson V and R filters during three observing runs as tabulated in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each epoch typically consisted of 20 exposures of 80 s in each filter per night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The resulting light curves were obtained by the method of synthetic aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each CCD frame was corrected in a standard way for bias and flat-fielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Dif- ferent aperture sizes were tested in order to choose the best one for our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A number of nearby and relatively bright stars within the frames were selected as reference stars to pro- duce differential photometry of Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Outliers due to poor observing conditions or very high airmass were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The observations with the T150 telescope were collected during a single observing run and were reduced in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In this case, each epoch typically consisted of 30 exposures of 80 s, 35 s, and 10 s in each V, R, and I filter, respectively, per night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Telescopi Joan Oró (TJO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Simultaneous to the OSN pho- tometry, observations for Wolf 1069 were carried out with the 80 cm Telescopi Joan Oró (TJO)7 at the Observatori del Montsec in Lleida, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The images were obtained with an exposure time of 60 seconds using the Johnson R filter of the LAIA im- ager, a 4k × 4k CCD with a FOV of 30 arcmin and a plate scale of 4 arcsec/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The images were calibrated with darks, bias, and flat fields with the ICAT pipeline (Colome & Ribas 2006) of the TJO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The differential photometry was extracted with AstroImageJ (Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2017) using the aperture size that minimized the rms of the resulting relative fluxes, and a selection 7 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='ieec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='cat/content/206/what-s-the-oadm/ Article number, page 3 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN of the 20 brightest comparison stars in the field, which did not show variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Then, data points with a low S/N were filtered from the light curve, and any points with relative fluxes greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 or less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='96 were removed before nightly binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Photometric monitoring surveys Additionally, we compiled a collection of archival long-term photometric data taken by monitoring surveys as described be- low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' They are also listed in Table 1 and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' SuperWASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The SuperWASP8 project is led by a chiefly UK- based consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Using two arrays of robotic telescopes oper- ating in the northern and southern hemispheres, the survey ob- tains light curves for millions of objects at high cadence to look for transiting planets and study other astrophysical phenomena across the entire sky (Pollacco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Butters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' SuperWASP-North is located at the Roque de los Muchachos Observatory in La Palma, whereas SuperWASP-South is at the South African Astronomical Observatory near Sutherland, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each observatory consists of eight wide-angle cameras with Canon 200 mm, f/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 lenses that feed into 2048 × 2048 CCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The pixel scale is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 was monitored for four seasons with SuperWASP-North from 2007 to 2010, though only sparsely in the last two seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The usable data spans from June 2007 to August 2008 with a ∼6 month gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We received the complete light curve, corrected for instrumental and atmospheric system- atics, from the SuperWASP team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The detrending procedure is nonaggressive and expected to preserve true astrophysical signals (including rotational modulation), as documented by Tamuz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' After clipping the final two seasons and iteratively rejecting outliers commensurate with the size of the data set in each season, we binned the data nightly such that the weighted mean and error of all the data points that go into each bin constitutes the flux and error of that bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MEarth-North.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 was observed by the MEarth9 project (Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2015), specifically with the MEarth-North array composed of eight 40 cm telescopes, each equipped with a 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 arcmin × 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 arcmin FOV Apogee U42 camera, and located at the Fred Lawrence Whipple Observatory on Mount Hopkins nearby Tucson, Arizona, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Using the light curves from the latest data release10, the target was observed for more than six years with telescope one (“MEarth-tel01”) and tele- scope five (“MEarth-tel05”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The MEarth project generally uses a RG715 long-pass filter, except for the 2010–2011 season when an I715−895 interference filter was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In the case of Wolf 1069, with observations collected from both telescopes later than October 2011, the RG715 filter was always used, and thus, we consider each photometric light curve as its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The data were nightly binned following the same procedure as for the SuperWASP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Particularly for first season with MEarth- tel05, we excluded certain nightly measurements where only one observation was taken to ensure accurate data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This con- stituted ∼15 nights out of the final 228 (Table 1), which were in the end not considered for the final rotational period determina- tion due to large noisiness (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 8 Super-Wide Angle Search for Planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 9 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='edu/MEarth/Welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='html 10 DR10: https://lweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='edu/MEarth/DR10/ north2011-2020/ TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 was thus far observed in three of the north- ern sectors (15 – camera #2 CCD #4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 16 – camera #2 CCD #3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 17 – camera #3 CCD #4, 15 August – 2 November 2019) during the nominal mission of the Transiting Exoplanet Survey Satellite (TESS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2015) with the short 2-minute cadence pho- tometry, as well as in three sectors (41 – camera #2 CCD #1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 23 July 2021 – 20 August 2021, 55 – camera #3 CCD #3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 05 August 2022 – 01 September 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 56 – camera#3 CCD #3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 01 Septem- ber 2022 – 30 September 2022) during the extended mission with 2-minute and 20-second cadence11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The target is being currently observed in one sector (57 – camera #3 30 September 2022 – 29 October 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The publicly available data from all sectors were downloaded from the Mikulski Archive for Space Tele- scopes12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Following the typical procedure, these data are cor- rected for artifacts and systematic trends (Presearch Data Con- ditioning, PDC_SAP flux – Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stumpe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2012, 2014), provided by the Science Processing Operations Center (SPOC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We use these data for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stellar properties 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Basic astrophysical properties Wolf 1069 (GJ 1253, Karmn J20260+585) is a slowly rotating, high-proper motion M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 V star at less than 10 pc discovered by Wolf (1920).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For decades, it was the subject only of astro- photometric analysis of stars in the solar neighborhood (Luyten 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Gliese & Jahreiß 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Probst 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Weis 1984), with the first spectral analysis by Bidelman (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Since the end of the 20th century, Wolf 1069 was more investigated in X-rays (Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stelzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013), with high-resolution imaging (Dieterich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Jódar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Janson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Lam- man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020), and especially for determining its astrophysical stellar parameters (Rojas-Ayala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Rajpurohit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Marfil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We summarize the stellar properties of Wolf 1069 in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Following Cifuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020), we integrated the spectral en- ergy distribution and computed the bolometric luminosity using broadband photometry and the latest parallactic distance from Gaia Data Release 3 (DR3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' From this value and the effective temperature of the star, deter- mined spectroscopically by Passegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019), we derived the radius from the Stefan-Boltzmann law, and the mass from the mass-radius relation by Schweitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' All tabu- lated parameters agree within uncertainties with published val- ues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Terrien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Lafarga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020), except for the rotation period, which we concluded to be 150–170 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Our Prot determination is explained in detail and compared with the literature in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Although Wolf 1069 kinematically belongs to the Galactic thin disk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', younger ages of ∼1-5 Gyr) according to the galactocentric velocities cal- culated as in Montes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2001) with a custom made python code (Cortés-Contreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ), the very long Prot de- termined by us points toward older ages (∼7-11 Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Newton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The low activity indicators (Hα and X-ray emis- sion) and slow rotational velocity, v sin i, are also in line with a long Prot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' As a result, Wolf 1069 is a very weakly active star, which facilitates the identification of RV signals with very low semi-amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 11 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='gov/cgi-bin/tess/webtess/ wtv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='py 12 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='edu Article number, page 4 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf −3000 −2800 −2600 −2400 −2200 −2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='025 −800 −600 −400 −200 0 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='025 1000 1200 1400 1600 1800 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='975 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Time - 2457000 (days) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Normalized flux SuperWASP MEarth-tel01-s1 MEarth-tel05-s1 OSN-V-T150 OSN-R-T150 OSN-I-T150 OSN-V-T90 OSN-R-T90 TJO-R MEarth-tel01-s2 MEarth-tel05-s2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Time series of the long-term photometry for Wolf 1069 color coded by instrument and filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The time range for each panel is consistent among all panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The MEarth-tel05-s1 data were not included in the final rotational period determination but are faintly shown for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Given that the GP model is unique to each instrument with its own amplitude hyperparamter (see for example Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 8 in Kemmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020), the extrapolated GP models of two instruments (MEarth-tel-05 and OSN-R-T90) are overplotted with the same color as their respective data sets for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Observing log of ground-based long-term photometric observations acquired for Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Instrument Date Filter ∆ta Nobs Nnightsb rmsc Begin End (d) (ppt) SuperWASP 13 June 2007 12 August 2008 400–700 nm 426 10 436 172 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 MEarth-tel01 � 29 September 2012 08 July 2015 ������������� RG715 1 012 2 595 183 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='02 13 September 2017 19 November 2018 431 6 237 177 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='85 MEarth-tel05 � 28 May 2012† 10 November 2015† 1 260 2 716 228 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='44 13 September 2017 19 November 2018 431 5 665 175 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='52 OSN-T150 ��������� 31 August 2017 06 December 2017 V 97 1 322 41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='65 31 August 2017 06 December 2017 R 97 1 276 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='90 14 September 2017 06 December 2017 I 84 1 078 37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='38 OSN-T90 ��������������������� 29 June 2017 04 September 2017 ��������� V 67 439 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='07 21 June 2019 29 October 2019 130 719 45 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='42 26 June 2020 02 November 2020 129 1187 64 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='73 29 June 2017 04 September 2017 ��������� R 67 442 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='68 21 June 2019 29 October 2019 130 716 45 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 26 June 2020 02 November 2020 129 1207 64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='40 TJO-T80 19 December 2018 12 January 2020 R 388 1 345 79 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='02 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (a) Time span of the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (b) Number of nightly binned observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (c) Root mean square in parts-per-thousand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Data sets that were not used for the photometric rotational period determination are indicated by a dagger †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 5 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 SuperWASP SuperWASP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 MEarth-tel01-s1 MEarth-tel01-s1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 MEarth-tel01-s2, MEarth-tel05-s2 MEarth-tel01-s2, MEarth-tel05-s2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 OSN-T150-I OSN-T150-I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 OSN-T150-V, OSN-T90-V OSN-T150-V, OSN-T90-V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 OSN-T150-R, OSN-T90-R, TJO-T80-R OSN-T150-R, OSN-T90-R, TJO-T80-R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='15 All instruments All instruments 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 100 100 150 200 300 400 f [1/d] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 GLS Power (ZK) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Period [d] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GLS periodograms for the long-term photometric data of Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The right panels are zoomed in to the longer-period regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each horizontal panel represents an effective instrument that was considered for the determination of the stellar rotation period (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The normal- ized GLS power of the sampling of the data for each row is shown in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The range for the photometric rotation period of 150–170 d is shaded in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Some significant alias signals due to the sampling of each respective data set are illustrated with a vertical dashed line, whereas the true signal is marked with a solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stellar rotation period Using MEarth light curves from 2011 to 2014, Díez Alonso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019) determined a photometric rotation period of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 d for Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, the same data were analyzed earlier by Newton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2016), who reported only a tentative signal at 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 d and noted that it did not meet their criteria for a con- fident detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To further establish the stellar rotation period, we examined all available photometric measurements, including the newer-taken observations from the OSN and TJO telescopes, along with various stellar activity indicators available from the CARMENES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 6 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stellar parameters of Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parameter Value Reference Basic identifiers and data Name Wolf 1069 Wolf1920 GJ 1253 Gli57 Karmn J20260+585 Cab16 Gaia DR3 2188318517720321664 Gaia DR3 G (mag) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='352 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='003 Gaia DR3a Coordinates and spectral classification α (ICRS) 20:26:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='80 Gaia DR3 δ (ICRS) +58:34:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 Gaia DR3 Sp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' type M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 V Reid95 Parallax and kinematics π (mas) 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='441 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='026 Gaia DR3 d (pc) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5747 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0024 Gaia DR3 µα cos δ (mas yr−1) 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='038 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='032 Gaia DR3 µδ (mas yr−1) 542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='906 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='031 Gaia DR3 γ (km s−1) –60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='020 Lafa20 U (km s−1) –23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1335 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0057 This work V (km s−1) –61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2071 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0200 This work W (km s−1) –8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0060 This work Galactic population Thin disk This work Photospheric parameters Teff (K) 3158 ± 54 Pass19 log g⋆ (cgs) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='06 Pass19 [Fe/H] (dex) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='19 Pass19 v sin i⋆ (km s−1) <2 Rein18 pEW(Hα) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='045 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='082 Fuhr22 ⟨B⟩ (G) <620 Rein22 log FX (mW m−2) −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='27 Stel13 Prot (d) 150–170 This workb Physical parameters L⋆ (10−4 L⊙) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='28 This work R⋆ (R⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1813 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0063 This work M⋆ (M⊙) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='167 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='011 This work Conservative HZ (au)c [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='056,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='111] This work Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (a) See Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 for multiband photometry different from Gaia DR3 G band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (b) See Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 for the Prot determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (c) For planets with Mp = 1M⊕ References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Cab16: Caballero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2016a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Cif20: Cifuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Gaia DR3: Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fuhr20: Fuhrmeis- ter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fuhr22: Fuhrmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Gli57: Gliese (1957);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Lafa20: Lafarga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Mar21: Marfil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Pas19: Passegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Reid95: Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Rein22: Rein- ers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Schw19: Schweitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Stel13: Stelzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf1920: Wolf (1920).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Long-term photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Considering each photometric data set alone, namely, OSN, TJO, SuperWASP, and MEarth, all indi- cate hints toward a prominent peak of ∼150–165 d when taking a look at the generalized Lomb-Scargle (GLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Zechmeister & Kürster 2009) periodograms, along with strong peaks present at the respective alias frequencies (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Interestingly, while this periodicity does present itself significantly in most photometric data sets, the most prominent peak in some is rather at a longer period (∼200–300 d), which can also be connected to an alias signal of the presumed rotational period due to the sampling for each of the respective data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Focusing on the first season of the photometry from MEarth, solely from tel-01 as the data from tel-05 are rather noisy, the highest peak is around 140 d, similar to Newton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' On the other hand, the second observ- ing block of the MEarth data, considering now data from both the tel-01 and tel-05 instruments, peaks clearly at ∼158 d, with present alias signals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', ∼110 d and ∼300 d) due to the sam- pling of 320 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To determine the rotational period of Wolf 1069, we per- formed a fit with juliet13 (Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019) using the sum of two stochastically driven, damped harmonic oscillator (SHO) terms, or a double SHO Gaussian process (dSHO-GP), as done so in previous works such as, David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019), Gillen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020), and Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021) first considering all the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A summary of the priors used for the analy- sis is found in Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To set this up, we treated the OSN data as effectively different instruments arranged together only if the telescope size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', T90 and T150) and filter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', V, R, and I) were the same across multiple observational seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each telescope of the MEarth data were separated into two tempo- ral subsets, namely MEarth-tel05-s1 and MEarth-tel05-s2, and the same for MEarth-tel01 though without tel05, to account for possible changes in stellar activity behavior over long periods of time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', ∼800 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The SuperWASP data were kept as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We im- posed a log-uniform prior for the rotational period, Prot, shared among all instruments ranging from 10 d to 200 d to avoid sam- ples from populating near the longer-period alias signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Like- wise, the quality factor for the secondary oscillation, Q0, and the difference between the quality factors of the first and second os- cillations, δQ, were also shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' As for the fractional amplitude of the secondary oscillation with respect to the primary one, f, this parameter was shared among instruments with the same fil- ter, that is, wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The amplitude of the dSHO-GP, σGP, followed suit as the dSHO-GP is trying to model the underlying physical behavior of the star that is wavelength dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To then account for any individual instrumental systematics, each respective instrument had its own offset value and jitter term14 (Baluev 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' From the posterior results of this fit, we obtain a rotational period of 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, compatible with the peaks in the peri- odograms and their widths (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To further experiment, we also tested out the same model setup, but this time solely on the photometry contemporaneously taken with the RVs, mean- ing those data comprising the last panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The reason for considering just these data is that, it could be possible that the star underwent some changes in activity on the long scale due to spot migration or differential rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The results of this run give a rotational period of 170+15 −11 d, which is 1-σ well in agreement when considering all photometry, ensuring that the behavior of the star is still applicable to today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Therefore, we determine the photometric stellar rotation period to be 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We furthermore explored the various stellar ac- tivity indicators provided by the CARMENES spectra as well as the RVs themselves to look for agreement with the pho- tometric rotational period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 is an Hα-inactive star (pEW’(Hα) Å > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Schöfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' While there are no strong or moderate correlations found between the RVs and the stellar activity indicators using the Pearson’s p-probability, the GLS periodograms of the indica- tors nonetheless consist of a variety of prominent peaks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' While some peaks found in the periodograms do coin- cide with the rotation period derived from the photometry, that 13 https://juliet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='io/en/latest/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='html 14 The jitter term is an additional noise count that is added in quadrature to the nominal uncertainty values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 7 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN is, ∼169 d, there are nonetheless other existing signals with even higher significance ranging in periods from ∼200–260 d, or even in the longer-period regime of up to a few hundred days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Par- ticularly, these signals are present in the CRX, dLW, Hα index, TiO λ 7050 Å, TiO λ 8430 Å, TiO λ 8860 Å, CTR, and FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There are still instances where the significant peaks coincide with the expected alias frequencies, however, the power at the rotational period is sometimes consistent with zero, as for in- stance in the Ca ii IRT3 indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This could be pure coinci- dental, we cannot exclude the possibility of it being a chance alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There are also peaks at even longer-period regimes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', >1000 d), which could be due to a stellar magnetic cycle not related to the stellar rotation itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This behavior appears similar to EV Lac (see Jeffers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022, for a detailed analy- sis of the various periodicites), where it was shown that different indicators can respond with a phase lag, or nonuniformly with the rotational variation of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In this work, we focus only on signals pertaining to the stellar rotation period, as well as those connected to the other RV signals (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 is considered to be a low-activity, low-mass star following the categorizations in Lafarga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Hence, ap- plying the findings made by Lafarga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021), the most effec- tive indicators for identifying the stellar rotation period comprise the dLW, CTR, and FWHM, which are tracing the varying width of the absorption lines in the spectra, as well as the indices of the chromospheric lines, Hα and Ca ii IRT, and sometimes the RVs themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The CRX and BVS in this instance are not as beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Taking a closer look at these, we see that in fact, the dLW does peak at the expected rotation period and fits quite well to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Likewise, when fitting for 169 d, the CTR and FWHM also match quite well in the phase-folded plots (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The Hα index and Ca ii IRT, however, do not agree with this periodicity, but rather show long-term trends, that may be related to a longer magnetic cycle as these indicators are also associated with mag- netic cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Another significant signal to mention in the dLW and the CTR appears at ∼29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 d, which is close to twice the or- bital period of the planetary signal of interest (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, this periodicity is most likely due to instru- mental effects, namely, the contamination from the light of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Within the RVs, there is a peak at 165 d, though not sig- nificant, after subtracting out the other more prominent peaks (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' When performing the final fit on the RVs (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2), we apply a dSHO-GP on this signal arriving at a value of Prot, RV = 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To bring all this evidence together, the photometric data point to a rotational period of ∼169 d, the RVs to ∼165 d, and a por- tion of stellar activity indicators similarly hint toward this value, though sometimes exhibit higher peaks at longer periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Based on the photometry, it is evident that the variability of the star is changing from season to season (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2), so the period- icities detected in the indices skewing away from the predicted rotational period may be exhibiting the evolution of the spot con- figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For example, some activity indices are evidently still consistent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', same period or aliasing due to the sampling), although others are not as consistent, which could be a result of other issues, stellar cycles, spot evolution, data precision and sampling, that make the interpretation nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We therefore adopt the rotational period of Wolf 1069 to be within the range of 150−170 d, with indications toward Prot = 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, though values outside this constrained range cannot be easily excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This is consistent with the predictions for a low-mass, inactive M dwarf (Newton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2017, their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Analysis and results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Signal detection We first computed a GLS periodogram on the RVs using the Exo-Striker15 tool (Trifonov 2019) to initially identify poten- tially interesting signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A series of GLS periodograms after se- quentially subtracting out the most prominent sinusoidal signals, along with the discrete Fourier transform of the window func- tion, can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The sampling of the data showed three notable peaks at 899 d, 365 d (the yearly sampling), and 270 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To quantify the significance of a signal’s peak, we com- puted the false alarm probability (FAP) levels of 10, 1, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 % using 10 000 randomizations of the input data where the fre- quency ranged from 1/tbaseline to 1, with a frequency resolution of 1/tbaseline and oversampling factor of ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Already, in the original RV data set there are three prominent peaks at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, ∼90–100 d and ∼400 d, all hovering the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 % FAP level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To also mention, there are equally promi- nent peaks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='94 d and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='066 d, both aliases of the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal due to the daily sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Furthermore, there are strong yearly aliases for the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal, particularly at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9 d and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Per- forming tests with the AliasFinder (Stock & Kemmer 2020), we find that the true periodicity is the one at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1), which we adopt when considering this signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Continuing on, there are an additional three peaks around the 10 % FAP level, namely at 165 d, ∼190 d, and ∼145 d, in order of significance, where the first one corresponding to the rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The other two can be explained as aliases due to the 365-d and 270- d sampling of the ∼400 d and ∼90–100 d signal, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A full outline of all the aliases is tabulated in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' As it does not matter which of the peaks of equal signif- icance is chosen first to subtract out for the prewhitening, we opt for the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d considering its aliases are also those that are most prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' After subtracting the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4 c), all corresponding aliases disappear, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Both the ∼90– 100 d and ∼400 d signal reach above the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 % FAP level, where the 165 d, ∼190 d, and ∼145 d signals also become more promi- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In the next succession of simultaneously modeling a two- sinusoidal model with the next highest peak (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d), there are no longer any signals that have an FAP level less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The highest peak at ∼400 d is above the 10 % FAP level, and its alias due to the yearly sampling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', ∼190 d) is appar- ent but no longer significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The 165 d is still above 10 % FAP level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The next signal to subtract for would be the one at ∼400 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, we have evidence that this signal is related to tel- luric contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We specifically compared the nontelluric- corrected RVs with those corrected for tellurics, finding that there is a prominent peak at 388 d in the telluric component (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6, top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We conclude that even though we use the telluric-corrected spectra, the contamination still likely man- ifests as this process of telluric removal is nontrivial and, nonetheless, introduces residuals in the corrected spectra (Nagel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Moreover, the peak in the RVs is more so at 404 d, whereas the peak in the periodogram of the telluric component is actually at 388 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We presume that this could be due to the fact that the alias due to the 270-d sampling of the 165 d signal is 420 d, thus, adding some power there, in turn broadening the peak and veering the periodicity away from 388 d to a value in between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' With this in mind, and considering the photo- metrically determined rotation period, we proceed with simul- taneously subtracting the 165 d signal next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Finally, there are no 15 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='com/3fon3fonov/exostriker Article number, page 8 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf Sampling periods (d) Alias order Present periods (d) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 165 388 ( f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='06410 d−1) (f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01107 d−1) ( f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00602 d−1) (f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00258 d−1) m = 1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 † 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 † 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2∗ 270 m = −1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0∗ 887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 ( fs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00370 d−1) m = 2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 † 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 † 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 m = −2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 m = 1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 † 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 † 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 365 m = −1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 † 6157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 ( fs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00274 d−1) m = 2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 †∗ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 ∗ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 m = −2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 1836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 m = 1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 † 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 † 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 899 m = −1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9 † 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 ( fs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00111 d−1) m = 2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 †∗ 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 m = −2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 2835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Aliases of the significant peaks found in the RVs given the sampling period tabulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The periods of the aliases are computed using Palias = 1/ falias, where falias = f + m · fs following f as the frequency of the true signal, m as the alias order, and fs as the sampling frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Aliases that are significant (FAP > 10%) are bolded, whereas those that are present but not significant are indicated with a dagger †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Ones marked with an asterisk signify peaks that are close to, but not exactly, the center of the expected alias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' more signals above 10 % FAP after removing three simultaneous sinusoidal signals (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d, 165 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' RV model comparison Out of the three prominent signals, only the one at 165 d does not reach our significance criterium (FAP < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 %, though greater than 10 %), but is however related to stellar activity and thus might be important to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We therefore continue the analysis by testing out “three-signal models” while properly considering the activity, with comparison to “two-signal models” by ignor- ing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For reasons explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3, the origin of the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d is ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Thus, we model it solely with a sinusoid when including this signal in a fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal, there is no evidence against it to not be planetary, therefore, we apply cir- cular or eccentric Keplerian orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For eccentric fits, the eccen- tricity was parametrized with S1 = √e sin ω and S2 = √e cos ω with uniform priors, U(−1, 1) as recommended by Eastman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The modeling and comparison of models is all performed with juliet, a versatile python package for simultaneous transit and RV fitting, as already described in depth in other works such as Kemmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020), Stock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020), Bluhm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020), and Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Following the same recipe as in the mentioned references, we use the computation of the Bayesian log evidence (ln Z) to compare models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Models with a ∆ ln Z ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 in comparison to another one show moderate ev- idence in favor of the winning model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Trotta 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Feroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Any value greater than 5 signifies strong evidence toward the winning model and anything below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 indicates that neither of the two models are favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To begin, we first ran a flat model with a white-noise term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', jitter term, σCARMENES-VIS) to act as a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Likewise, we ran a red-noise model using a dSHO-GP centered around the stellar rotation period (see below for setup details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal, we imposed an appropriately sized uniform prior for the period, U(15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 d, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7 d) in order to cover the width of the peak as in the periodogram and to ensure that this true signal would be picked up rather than its nearby aliases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The time-of-transit center, which is used in juliet to parametrize the phase of the orbit, was chosen to be uniform and set to cover one cycle during the most-sampled time epoch of the RV data set, U(2458502 d, 2458515 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Likewise, for the ∼90 d signal, the prior on the pe- riod was kept wide enough to capture the tails of the posterior sample distribution, U(85 d, 95 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To account for the stellar rotation period, we experimented modeling it with a sinusoid and with a dSHO-GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The photometrically determined rotational period of 150–170 d (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) was taken into consideration for the prior setup on the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Within the RV periodograms (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1), the periodicity shows up as closer to 165 d and en- counters various neighboring signals due to aliasing of the other signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For this reason, the prior on the period was kept rela- tively narrow and uniform from 155 d to 175 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 show- cases a full overview of the employed priors, including those for the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A table showcasing the Bayesian log evidence for the assortment of the tested models tested can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The winning model comprises one Keplerian for the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d sig- nal alongside a dSHO-GP centered on 165 d to describe the be- havior of the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We note that including an extra sinusoid term for the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal to this model is equiv- alent in terms of the Bayesian log evidence, even though this signal is evidently prominent in the GLS periodogram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Given the ambiguity of the nature of this signal (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3), we find it appropriate to omit it from the final model and allow the dSHO-GP to moderately absorb it for the time being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The crucial aspect is that the interpretation of this signal and, thus, how we consider it in our models, which we acknowledge may adjust in the future, does not drastically alter the planetary parameters of the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal, an eccentric Keplerian orbit was con- sistent with a circular one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The distribution of the eccentricity was consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Focusing on the stellar rotation period, the dSHO-GP was preferred over a sinusoid (∆ ln Z > 10) as it is most likely better suited in describing the quasi-periodic be- Article number, page 9 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 (a) window function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 899d 365d 270d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 (b) original 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 (c) 1 sinusoid (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 (d) 2 sinusoids (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 (e) 3 sinusoids (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d, 165 d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='06 (f) 1 Kep (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d) + dSHO-GP165d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 388 165 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 388 165 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 f [1/d] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 GLS Power (ZK) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Period [d] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GLS periodograms for the RVs of Wolf 1069 after sequentially subtracting out the most prominent signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The horizontal dashed, dot- dashed, and dotted lines represent the 10 %, 1 %, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 % FAP levels (from bottom to top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' There were no significant signals with periods shorter than 10 d other than the aliases due to the daily sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The right panel is a closer zoom-in of the left panel to highlight the longer-period signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d planetary signal is illustrated with a vertical blue solid line, and the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal and its alias due to the 270-d sampling with a green solid and dashed line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The range for the photometric rotation period is shaded in orange, where the stellar rotation period within the RVs is marked with an orange solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The component of residual telluric contamination at 388 d and its alias due to the 365-d sampling period are also represented with a vertical magenta solid and dashed line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Panel (a): the window function of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Panel (b): no signal fitted, solely the original RVs with an offset and jitter term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Panel (c): residuals after subtracting the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Panel (d): residuals after subtracting a simultaneous model fit of two sinusoids at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Panel (e): residuals after subtracting a simultaneous model fit of three sinusoids at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d, and 165 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Panel (f): residuals after subtracting the final model choice including 1 Keplerian at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d (further described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' havior of the stellar activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Furthermore, this corresponds well to the fact that there seems to be a high level of spot evolution as we have encountered in the stellar activity indicators (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) such that this is also demonstrated as an effect in the RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To conclude, the final model consists of ten free parameters applied on the 262 RV data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal is best de- scribed by a circular Keplerian model and the 165 d signal by a dSHO-GP to account for the stellar rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The best RV model from the posteriors is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1, where values for the derived planetary parameters can be found in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The full posterior overview for all model parameters is located in Ta- ble C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A visual inspection of the posterior probability densities for key parameters are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Investigating the low-amplitude 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal The 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal is significant with an FAP level near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 % in the GLS periodograms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4), yet it does not statistically im- prove the fit when including it in the RV models including also the rotation period (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This could be an artifact that the semi-amplitude hovers around the 1 m s−1 limit (K90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d = 91 ± 38 cm s−1), thus, making it possibly difficult to justify including such a low-amplitude signal with a relatively large uncertainty in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Likewise, the dSHO-GP kernel used in the model is equipped to pick up the rotational period, Prot, and half of the rotational period, Prot/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' It could be that 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d is decently close to 83 d (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Prot/2) and, thus, the dSHO-GP is performing its job of absorbing this signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In fact, this is demonstrated in the GP component of the RV model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' It is nonetheless evident that this periodicity is present in the RV data as can be seen in the residuals of the RV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Its nature, however, ap- pears to be quite dubious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Below, we explore the signal further to better understand its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Its periodicity is near a quarter of one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A suspicion could be that this signal should be present in the telluric- contamination-only component of the RVs, though, it is not Article number, page 10 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 CRX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='15 dLW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 Caii-IRT1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 Caii-IRT2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 Caii-IRT3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='15 Hα 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 NaD1 5890˚A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 NaD2 5896˚A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 TiO 7050˚A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='25 TiO 8430˚A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 TiO 8860˚A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 BVS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 CTR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='15 FWHM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 388 165 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 388 165 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 f [1/d] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 GLS Power (ZK) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Period [d] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GLS periodograms for various known stellar activity indicators from the CARMENES spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The window function of the data sampling can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For consistency, the colored vertical lines and the frequency width of the panels on the left correspond exactly to those in the RV GLS periodograms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The horizontal dashed, dot-dashed, and dotted blue lines represent the 10 %, 1 %, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 % FAP levels (from bottom to top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 11 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Model comparison using the Bayesian log evidences on the RVs for Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Model ln Z ∆ ln Z Base models Flat −640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00 −33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='14 dSHO-GP165 d −618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='51 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='65 One-signal model 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d −631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='49 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='63 Two-signal models 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + 1 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d −620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='62 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='76 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + dSHO-GP165 d −606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Three-signal models 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + 2 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d, 165 d −616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='83 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='96 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + 1 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d + dSHO-GP165 d −606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The chosen model was the 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + dSHO-GP165 d, as marked as the bold-faced row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A better model would have a larger, more positive ∆ ln Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Regarding the model names, “Kep” refers to a circular Keplerian orbit and “Sin” to a sinusoidal signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The period values are quoted as the median of the posterior distribution and can vary slightly depending on the model choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Derived posterior parameters for Wolf 1069 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parameter name Posterior(a) Unit b Pp 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='564+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='015 d t0,p (BJD) 2458511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='46 d Kp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 m s−1 S 1,p = √ep sin ωp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 (fixed) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' S 2,p = √ep cos ωp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 (fixed) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' M sin ip 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 M⊕ ap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0672+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0014 au Teq(b) 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 K S p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='652+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='029 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='027 S ⊕ Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (a) Error bars denote the 68% posterior credibility intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (b) The equilibrium temperature of the planet assuming zero Bond Albedo and one emissivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6, top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We additionally tested breaking up both the telluric-corrected (TC) and nontelluric-corrected (nonTC) RVs into “blue” and “red” subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To do this, we can recompute the RVs by selecting certain orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The CARMENES VIS channel consists of 55 RV orders, 42 of which are used to compute the RV measurement via a weighted mean through serval (Zech- meister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For the blue and red subset, we consid- ered the first 21 and last 21 orders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Thus, the blue and red subsets span roughly 570 nm to 700 nm and 700 nm to 910 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A GLS periodogram for both subsets and for both data sets is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Taking a look at the nonTC spectra, the 388 d (telluric-attributed) signal, its yearly alias at 188 d, and the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d, as well as a neighboring signal at 97 d, are substantially stronger in the red than in the blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This is in agreement that the telluric contamination is stronger in the red than in the blue, given that there are sharper, deeper telluric- absorption features in the redder part of the VIS channel (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='065 388 165 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 f [1/d] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 GLS Power (ZK) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Period [d] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GLS periodograms of the telluric-only (nonTC subtracted from TC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' top), nonTC (middle) and TC (bottom) RVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The red and blue colors represent the “blue” and “red” subsets within the VIS channel of the CARMENES instrument (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The vertical lines match those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4 for consistency and the horizontal lines are identical in the bottom panels but differ slightly from the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' in Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Meanwhile, the power of the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d and 165 d signals is consistent within one another in both subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' After the telluric correction, some residual telluric contamina- tion remains, though dampened, indicating that the correction for tellurics was indeed effective but left some residual effect as can be seen in the RV periodograms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The power of the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d and 165 d signals is still compatible, which is most im- portant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Therefore, even though the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d has no appearance in the telluric-contamination-only component, it does exhibit chro- maticity and behavior similar to other telluric signals, pointing less in favor for a planetary signal and potentially more in favor of telluric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' It is, however, puzzling as to why the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal does not peak in the telluric-only RVs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To summarize, we are not able to distinctly decipher the ori- gin of the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Even if it were planetary, we currently do not have enough evidence to support this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' As our main concern is the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal and we presented that its planetary parameters are independent of whether the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal is con- sidered or not (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 7, particularly between 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + 1 dSHO- GP165 d and 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d + 1 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d + dSHO-GP165 d), we choose not to include this signal in the final model as a precaution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fur- ther investigation or RV monitoring may be beneficial, however, this lies beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Transit search within TESS With a minimum mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 M⊕ and assuming an Earth- like core-mass fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='26, we obtained a radius estimate of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='08 R⊕ for Wolf 1069 b using the mass-radius relation as given by Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This then translates to an expected transit depth of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 ppt, which should be easily detectable with TESS, though not with the other available photometric facilities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', SuperWASP and MEarth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The transit probability is however rather low at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2% (p ≈ R⋆/ap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Nonetheless, propagating the orbital period and t0 from the RV fit with their 1-σ uncertainties (taken from Table 5), we unfortunately did not find any hint of a possible transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We additionally checked to confirm that the transits could not have happened during the data gaps, where Article number, page 12 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 K (m s 1) 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6d + dSHO-GP165d 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6d + 1 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3d 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6d + 1 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3d + dSHO-GP165d 1 Kep15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6d + 2 Sin90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3d, 165d TC non-TC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Box plot of the posteriors for the distributions of the minimum mass for the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal based on the model choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The gray and blue boxes represent the 25 % and 75 % quartiles of the posterior from the telluric-corrected (TC) and nontelluric-corrected (nonTC) RVs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The red vertical line represents the median value of the minimum mass of the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal when applying the most favored model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The extending gray lines depict the rest of the distribution and the dots are deemed as “outliers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The models named here match those in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' only one would fall within an observational gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Likewise, we checked with the transit-least-squares16 method (Hippke & Heller 2019), though no interesting signals popped up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Given this information, we were able to obtain a maximum inclination for Wolf 1069 b of imax = arccos � R⋆/ap � = 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='35 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' On the promising habitability of Wolf 1069 b Plugging in the stellar luminosity and effective temperature (Ta- ble 2), Wolf 1069 b, with a distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0672 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0014 au to the star, sits comfortably within the conservative HZ lim- its, namely, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='056 au to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='111 au given the runaway-greenhouse and maximum-greenhouse limits, respectively (Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Even more so, it is very likely that Wolf 1069 b is indeed an Earth-like planet with Earth-like composition (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5% iron mass fraction and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5% silicates) and radius around one Earth radii (following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1 in Luque & Pallé 2022), as we also estimated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Figure 9 puts Wolf 1069 b in context with other planets around M-dwarf stars that are most likely to have a rocky composition and maintain surface liquid water as listed in the Habitable Exoplanet Catalog17 with some modifications (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To this effect, Wolf 1069 b resembles best Prox- ima Centauri b, GJ 1061 d, Teegarden’s Star c, Kepler-1649 c, and GJ 1002 b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' With the exception of Kepler-1649 c, all are RV-only detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Furthermore, all 14 planetary sys- tems illustrated contain more than one planetary companion, ex- cluding Wolf 1069, Ross 128, and Kepler-1229, as discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' When considering the occurrence rate for planets with 1–10 Earth masses on periods shorter than 10 d around later- type M dwarfs, this value lies between ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='56–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='75 planets per star (Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Hardegree-Ullman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Regard- ing the proximity of these systems, Dressing & Charbonneau (2015) estimated that the nearest nontransiting HZ planet could be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 pc away, and is within 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 pc with 95% confidence for potentially habitable 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 R⊕ planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Soon after, Proxima Cen- tauri b was discovered at a distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='30 pc (Anglada-Escudé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016), GJ 1061 d at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='67 pc (Dreizler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020), Teegar- 16 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='com/hippke/tls 17 https://personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='ems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='edu/~ruk15/planets/ den’s Star c at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='83 pc (Zechmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019), and GJ 1002 b and c at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='85 pc (Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 b is located at a distance of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='57 pc, making it the sixth closest, con- servative HZ Earth-mass planet to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Other closer contenders included Ross 128 b (d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='38 pc) and GJ 273 b (d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='83 pc), though these planets lie in the optimistic HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 b is in the slow rotator regime, and possibly in tidal equilibirum rotation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Heller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2011), that can lead to unique atmospheric circulation pathways (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Dole 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The impacts of this slow rotation on both the potential habitability and impact on observations have been discussed in detail by several 3D GCMs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Edson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Leconte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Preliminary results from GCMs climate simulations using both the ExoCAM model (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022) and the ROCKE-3D model (Way et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2017) suggest that Wolf 1069 b could maintain moderate temper- atures and surface liquid water for a large range of atmospheric compositions and surface types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Simulations explore a variety of surface pressures, N2, CO2, CH4, and H2O abundances, along with desert, solid rock, slab ocean, and dynamic ocean surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The comprehensive analysis of these 3D climate results and the observational signals that could be used to differentiate between climate states of Wolf 1069 b show the planet to be durably hab- itable (Crouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Figure 8 shows the surface temper- ature map produced with the ExoCAM GCM assuming a Mod- ern Earth-like atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The red line delimiting the open ocean shows that a significant fraction of the day side surface could maintain liquid water, therefore day-side habitable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' While the presence and nature of any atmosphere on Wolf 1069 b (and existing M-dwarf planets in general) remains theoretical, the support of habitable conditions over such a wide range of possible atmospheric states puts Wolf 1069 b in the same elevated class as Proxima Centauri b (Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Del Genio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019a), TRAPPIST-1 e (Wolf 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fauchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019), and TOI-700 d (Suissa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020) as a primary target to search for habitability and biosignature markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Similar to Proxima Centauri b, Wolf 1069 b does not transit its host star, meaning that observation and analysis of thermal emission and reflected light phase curves will need to be em- ployed to probe its atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Given the brightness of the host Article number, page 13 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Surface temperature map of Wolf 1069 b produced by the Ex- oCAM GCM, assuming a Modern Earth-like atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The map is centered at the substellar point and the red line delimits the area where water is at the liquid phase on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' star and the distance to Earth, and assuming atmosphere mod- els and albedo similar to the ones predicted for the TRAPPIST- 1 planets and Proxima Centauri b (Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022), the at- mospheric characterization of Wolf 1069 b might be within the reach of the ELT18 instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ANDES19 (formerly known as HIRES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Maiolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2013) will be the first instru- ment theoretically capable of detecting the reflected light from HZ rocky planet atmosphere in the early 2030’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, for Wolf 1069 b, the small angular separation in the sky between the planet and the host stars, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 milliarcsec, makes these ob- servations very challenging, even with the use of extreme adap- tive optics systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Further instrumental advances, such as the proposed PCS20 instrument for the ELT (Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021) or space-based coronographic/interFerometric missions, might be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' While such observations are very challenging, many of the nearest planets found in the conservative HZ around M dwarfs are nontransiting, RV detections, indicating that perhaps more time and investment into the development of such obser- vations should be considered if we want to establish ground statistics using all of the thus-far detected, potentially habitable worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The case of Wolf 1069 b as a lone, short-period planet Our comprehensive analysis of the RV and photometric data sug- gests that Wolf 1069 b is the only bonafide planet in the sensitive domain of the planetary parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We characterized this domain with injection-and-retrieval tests by taking the residu- als of the winning model without a GP (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) and creating simulated RV time series using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2 in Sabotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This was repeated 50 times on 30 log-uniformly distributed grid points in mass and 60 in period, allowing us to rule out additional planets with at least one Earth mass and periods of less than 10 days (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 b joins a sample of currently two 18 Extremely Large Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 19 ArmazoNes high Dispersion Echelle Spectrograph, https://elt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='org/instrument/ANDES/ 20 Planetary Camera and Spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Insolation (S⊕) 2400 2600 2800 3000 3200 3400 3600 3800 4000 Teff (K) Teegarden’s Star b Teegarden’s Star c Proxima Centauri b TOI-700 d GJ 1061 c GJ 1061 d Kepler-1649 c TRAPPIST-1 d TRAPPIST-1 e TRAPPIST-1 f TRAPPIST-1 g Ross 128 b Kepler-296A e Kepler-296A f Kepler-1229 b K2-72 e GJ 273 b LP 890-9 c GJ 1002 b GJ 1002 c Wolf 1069b Radius (R⊕) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' M-dwarf (Teff < 4000 K) planetary systems with at least one detected planet in the conservative sample of potentially habitable exo- planets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 R⊕ < Rp < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 R⊕ or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 M⊕ < M sin ip < 3 M⊕) defined by the Habitable Exoplanet Catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The optimistic and conservative HZ regions for a one Earth-mass planet following the definition as set out by Kopparapu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2013) are shaded with light and dark green, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Only the planets in either the conservative or optimistic HZ of each planetary system are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' White-filled and gray-filled points indicate nontransiting and transiting detections, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The size of the circles is proportional to the planetary radius, estimated with the mass–radius relationship of Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2016) for nontransiting RV planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The data used in this plot is further discussed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Plot inspired by Zechmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2019) and Dreizler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' RV-detected, single terrestrial planets (≲ 2 M⊕), which have all been detected around M dwarfs less massive than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' These objects are GJ 393 b (Amado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021) and Ross 128 b (Bon- fils et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018), where the former resides in the HZ region of its host star (see also Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1), whereas the latter receives too much flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Likewise, transiting planets following the same criteria in- clude GJ 367 b (Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021) and GJ 1252 b (Shporer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020), where the latter has rather a tentative measured mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='56 M⊕ and both are not found in the HZ of their parent stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Nonetheless, the small sample size raises the question how frequent the solitary occurrence of such a planet is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The overall occurrence rate of small, rocky planets on close orbits has been shown to be larger around host stars of later spectral type (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Hardegree-Ullman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, there is some indication that this rule might not apply for the latest M dwarfs (Gibbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Sebastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Sestovic & Demory 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Brady & Bean 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Mulders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021) and that systems, such as the one presented here, could be in fact rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 14 of 26 ModernEarth-like 180° 60°W 0° 60°E 180° 60°N 60°N 30°N 30°N 0° 0° 30°S 30°S S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='09 60°S 180° 60°W 0° 60°E 180° min=178 Imean=233 Imax=286 200 225 250 275 300 Surface Temperature [K]D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf Planet formation models following the core accretion paradigm (Pollack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1996) generally suggest a high multiplicity of Earth-mass planets around mid-M-dwarf host stars (Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, these models produce many planets beyond current detection limits, and bias-corrected syn- thetic populations show sufficiently reduced rates to be compat- ible with the observations (Schlecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The discovery of a single planet comparable to Wolf 1069 b is consistent with this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We tested this scenario by applying the computed detec- tion sensitivity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 10) to the synthetic planetary population NGM10 around an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 M⊙ star presented in Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Using 50 % detection probability limits, 48 out of a total of 1000 systems would result in a single detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Out of those, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 11 the three simulations that result in planets closest to Wolf 1069 b on the period-versus-mass plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Simulations lead- ing to a single planet detection went through a stage of giant im- pacts reducing the number of planets in the inner system and in- creasing the mass of the detectable planet with respect to the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This is exemplified by the three best-fitting sim- ulations which show three to four mergers with embryos more massive than the lunar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' While the scenario of the formation of a single planet can- not be ruled out, those simulations show that it is also possible in ∼5 % of the cases to form a seemingly lone planet if multiple embryos formed at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, if future observa- tions extend the detection limits to larger orbits and lower plan- etary masses, this formation theory will be more severely chal- lenged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' While a single, late stage giant impact with a similarly massive body is currently in agreement with observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Sim 650), this could be ruled out with better sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Then, a more dynamic history of the system is required (as in Sim 967 where the complete inner system was ejected or accreted by the detectable planet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A handicap of particular importance for thor- ough analyses of planet multiplicity is the omission of early core formation phases in current formation models (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Ormel 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Schlecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Future planet population synthe- sis studies have to take into account dust evolution, planetesimal formation, and planetary embryo formation in a self-consistent manner (Voelkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' As for the observational prospects, dedicated measurements with a high-precision spectrograph focused on searching for sub- Earth-mass planets in the Wolf 1069 system could shed light on a potential inner planet candidate (as was the case with Proxima Centauri [d] first identified by Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2020 and later announced as a convincing planet candidate by Faria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022, with a periodicity of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='12 d and K∼40 cm s−1), or even further rule out this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Radio emission from star-planet interaction Auroral radio emission from stars and planets is due to the elec- tron cyclotron maser (ECM) instability (Melrose & Dulk 1982), whereby plasma processes within the star (or planet) magneto- sphere generate a population of unstable electrons that amplifies the emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The characteristic frequency of the ECM emis- sion is given by the electron gyrofrequency, νG = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 B MHz/G, where B is the local magnetic field in the source region in Gauss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ECM emission is a coherent mechanism that yields broadband (∆ ν ∼ νG/2), highly polarized (sometimes reaching 100%), am- plified nonthermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For Jupiter-like planets, which have magnetic fields of Bpl ≃ 10 G, the direct detection of radio emission from them is plau- sible, as the associated gyrosynchrotron frequency falls above 100 101 102 103 Period (d) 100 101 Mpl sini in M⊕ 0 20 40 60 80 100 Detection probability in % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' RV detection map of Wolf 1069 from an injection-and-retrieval experiment after subtracting the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d and 165 d signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The red circle indicates the planet Wolf 1069 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the ≃10 MHz ionosphere cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' However, the detection of radio emission from Earth-sized exoplanets, which are also the type of planets comprising a large majority of the CARMENES sam- ple, is doomed to fail, as the associated frequency falls below the ionosphere cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fortunately, if the velocity, vrel, of the plasma relative to the planetary body is less than the Alfvén speed, vA, in other words MA = vrel/vA < 1, where MA is the Alfvén Mach number, then energy and momentum can be transported upstream of the flow along Alfvén wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Jupiter’s interaction with its Galilean satel- lites is a well-known example of sub-Alfvénic interaction, pro- ducing auroral radio emission (Zarka 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In the case of star- planet interaction, the radio emission arises from the magneto- sphere of the host star, induced by the exoplanet crossing the star magnetosphere, and the relevant magnetic field is that of the star, B⋆, not the exoplanet magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Since M-dwarf stars have magnetic fields ranging from about 100 G and up to above 2-3 kG, their auroral emission falls in the range from a few hundred MHz up to a few GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This interaction is expected to yield detectable auroral radio emission via the cyclotron emis- sion mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', Turnpenney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Pérez-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We followed the prescriptions in Appendix B of Pérez-Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021) to estimate the flux density expected to arise from the interaction between the planet Wolf 1069 b and its host star at a frequency of 860 MHz, which corresponds to the cyclotron frequency of the star magnetic field of 307 G, from Reiners et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We computed the radio emission arising from star-planet interaction for two different magnetic field geometries: a closed dipolar geometry, and an open Parker spiral geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' For the dipolar case, the motion of the plasma relative to Wolf 1069 b happens in the supra-Alfvénic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Therefore no energy or momentum can be transferred to the star through Alfvén waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In the open Parker spiral case, however, the plasma motion pro- ceeds in the sub-Alfvénic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 12 the pre- dicted flux density as a function of orbital distance arising from the interaction of a magnetized exoplanet (1 G) with its host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The yellow shaded areas encompass the range of values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 for the efficiency factor, ϵ, in converting Poynting flux into ECM radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The expected flux density is less than 2 µ Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This is an extremely low value, which is not within the reach of even the most sensitive radio interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The reasons behind this extremely faint signal are mainly two: First, the relatively large distance to the system (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 pc away);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' and sec- Article number, page 15 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN 100 101 102 103 10 2 10 1 100 101 Mpl (M ) Sim 625 Detectable Undetectable Accreted Wolf 1069 b 100 101 102 103 10 2 10 1 100 101 Mpl (M ) Sim 967 100 101 102 103 Period (d) 10 2 10 1 100 101 Mpl (M ) Sim 650 0 20 40 60 80 100 Detection probability in % Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Formation paths and final planets in planet formation simula- tions taken from the population synthesis work of Burn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We show the three simulations with a single detectable planet closest to Wolf 1069 b in relative mass and orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A planet is labeled “undetectable” if the detection probability is below 50 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Formation tracks are shown as gray lines that can end in either a filled circles (detectable planet), a triangle (accreted by detectable), a ring (unde- tectable), or without a marker (accreted by other planets or ejected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ond, the large separation of the planet from its host star (about 80 stellar radii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Therefore, the chances of detecting radio emission from star-planet interaction in Wolf 1069 are essentially null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Conclusion Using CARMENES spectroscopic measurements, we presented the discovery of a nontransiting exoplanet, Wolf 1069 b, with a period of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='564±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='015 d, minimum mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 M⊕, and insolation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='652+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='029 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='027 S ⊕, putting it safely in the conservative HZ around a low-mass M-dwarf star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This makes Wolf 1069 b the sixth closest (d ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 pc), Earth-mass planet in the conser- vative HZ from us, following Proxima Centauri b, GJ 1061 d, Teegarden’s Star c, and GJ 1002 b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Preliminary investi- gations of the potential habitability of the planet using GCM climate simulations suggest the planet to be a promising addi- 3 2 1 0 log(MA) Wolf 1069 b - Open field Wolf 1069 b - Open field 20 40 60 80 100 Distance / Stellar radius 3 2 1 0 1 2 3 log (Flux density [mJy]) Bcycl = 307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 G Bplanet = 1 G ncorona = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0x107 cm−3 Saur/Turnpenney model 1 3 5 10 15 20 Orbital period [days] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Expected flux density for auroral radio emission arising from star-planet interaction in the system Wolf 1069, as a function of orbital distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The interaction is expected to be in the sub-Alfvénic regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', MA = vrel/vAlfv ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' top panel) at the location of each planet (vertical dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' tion to the group of current targets to search for biosignature markers, such as Proxima Centauri b, TRAPPIST-1 e, and TOI- 700 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Wolf 1069 b unfortunately does not transit its stellar host, though future observations with thermal emission and reflected light phase curves could shed light on the properties of its at- mosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We additionally investigated whether star-planet in- teractions in Wolf 1069 would be feasible to observe with radio emissions, but found these potential observations unfruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The Wolf 1069 system becomes more intriguing as there is no significant evidence of closer-in planets (P < 10 d) greater than one Earth mass, based on our detectability limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This con- figuration is a plausible outcome based on a select few synthetic planetary population simulations, and even suggestive of a planet formation history including a late giant impact phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The detec- tion of potential inner sub-Earth-mass planets with further sub- m s−1 RV observations could then confirm or reject this forma- tion theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The stellar host itself is a relatively inactive, low-mass M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 dwarf, though exhibits periods of higher activity levels, for which we determine its photometric rotation period to be 150– 170 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This rotation period was also present in the CARMENES RVs, and thus, modeled with a dSHO-GP in the final fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The RVs showed one more additional significant periodicity at 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d with a low amplitude (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=', <1 m s−1), however, we demonstrate Article number, page 16 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf that there is currently not enough supporting evidence in favor of a planetary origin and it appears to be an effect of telluric contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Further RV investigation could be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' To conclude, Wolf 1069 b is a noteworthy discovery that will al- low further exploration into the habitability of Earth-mass plan- ets around M dwarfs, as well as case study in testing planetary formation theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG project number Ts 17/2–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' CARMENES is an instrument at the Centro Astronómico Hispano-Alemán (CAHA) at Calar Alto (Almería, Spain), operated jointly by the Junta de Andalucía and the Instituto de Astrofísica de Andalucía (CSIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' CARMENES was funded by the Max- Planck-Gesellschaft (MPG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the Consejo Superior de Investigaciones Cientí- ficas (CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the Ministerio de Economía y Competitividad (MINECO) and the European Regional Development Fund (ERDF) through projects FICTS- 2011-02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ICTS-2017-07-CAHA-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' and CAHA16-CE-3978,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' and the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Insti- tuto de Astrofísica de Andalucía,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Landessternwarte Königstuhl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Institut de Cièn- cies de l’Espai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Institut für Astrophysik Göttingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Universidad Complutense de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Thüringer Landessternwarte Tautenburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Instituto de Astrofísica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Hamburger Sternwarte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Centro de Astrobiología and Centro As- tronómico Hispano-Alemán),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' with additional contributions by the MINECO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the Deutsche Forschungsgemeinschaft through the Major Research Instrumentation Programme and Research Unit FOR2544 “Blue Planets around Red Stars”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the Klaus Tschira Stiftung,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the states of Baden-Württemberg and Niedersachsen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' and by the Junta de Andalucía.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' This work was based on data from the CARMENES data archive at CAB (CSIC-INTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Data were partly collected with the 90 cm and 150 cm telescopes at Observatorio de Sierra Nevada (OSN), operated by the Instituto de Astrofísica de Andalucí a (IAA, CSIC);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' we deeply acknowledge the OSN telescope operators for their very appreciable support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The Telescopi Joan Oró (TJO) of the Observatori Astronómic del Montsec is owned by the General- itat de Catalunya and operated by the Institut d’Estudis Espacials de Catalunya (IEEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We acknowledge financial support from the Agencia Estatal de Investi- gación of the Ministerio de Ciencia e Innovación (AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='13039/501100011033) and the ERDF “A way of making Europe” through projects PID2019-109522GB- C5[1:4], PID2019-107061GB-C64, and PID2019-110689RB-100, and the Cen- tre of Excellence “Severo Ochoa” and “María de Maeztu” awards to the Insti- tuto de Astrofísica de Canarias (SEV-2015-0548), Instituto de Astrofísica de Andalucía (SEV-2017-0709), and Centro de Astrobiología (MDM-2017-0737);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the European Research Council under the Horizon 2020 Framework Program (ERC Advanced Grant Origins 832428 and under Marie Skłodowska-Curie grant 895525);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the Generalitat de Catalunya/CERCA programme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the DFG through the priority program SPP 1992 “Exploring the Diversity of Extrasolar Plan- ets (JE 701/5-1)” and the Research Unit FOR 2544 “Blue Planets around Red Stars” (KU 3625/2-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the Bulgarian National Science Fund through program “VIHREN-2021” (KP-06-DV/5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the SNSF under grant P2BEP2_195285;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' the National Science Foundation under award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1753373, and by a Clare Boothe Luce Professorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We thank the anonymous referee for the insightful com- ments that helped improve the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Software: astropy Astropy Collaboration et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='de 2 Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Sofia University “St Kliment Ohridski”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 5 James Bourchier Blvd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' BG-1164 Sofia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Bulgaria 3 Landessternwarte,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Zentrum für Astronomie der Universität Heidel- berg,' metadata={'source': 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Partículas y del Cosmos de la UCM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Facultad de Ciencias Físicas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Universidad Complutense de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' E-28040 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Spain 23 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Ariel University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Ariel 40700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Israel 24 School of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' European University Cyprus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Diogenes street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Engomi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 1516 Nicosia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Cyprus 25 Department of Astronomy/Steward Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 933 North Cherry Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Tucson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' AZ 85721,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' United States of America 26 Centre for Earth Evolution and Dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Department of Geo- sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Universitetet i Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Sem Sælands vei 2b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 0315 Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Nor- way 27 Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Irvine,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' United States of America 29 NASA NExSS Virtual Planetary Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' WA Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' page 19 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Appendix A: AliasFinder figures The RV data show a variety of aliases related to the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d sig- nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' In order to establish that 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d is indeed the true periodicity, we tested the aliasing using AliasFinder (Stock & Kemmer 2020), which follows the methodology from Dawson & Fab- rycky (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The essence behind the algorithm is to examine the GLS periodograms of simulated data sets, in which either of the two aliasing signals are injected, to the GLS periodogram provided by the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The injected signal of whichever periodogram matches best to the original one is defined to be the true periodicity apparent in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The results of this method by simulating 1000 time series for both the daily and yearly sam- pling frequencies is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, confirming the true signal to be 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Appendix B: Known planets in the habitable zone around M dwarfs We started from the list collected by PHL at UPR, which ob- tains its parameters from the NASA exoplanet archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The list, as last updated on 06 December 2021, comprises 21 planets that are most probable to have a rocky composition and maintain sur- face liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' We individually vetted each system using the most up-to-date literature and updated the planetary and stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Most of the planets stayed consistent since the up- date, though there are some modifications: – GJ 667 C: We omit the planets d, e, f, and g proposed by Anglada-Escudé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2013) in which the second two would reside in the HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' They emerged as controversial when stel- lar activity was also modeled within the RVs as a red-noise component (Robertson & Mahadevan 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Feroz & Hob- son 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Unfortunately, that leaves only planet c in the op- timistic HZ in this planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Nonetheless, there is still an inner planet to that of the HZ one, planet b, with a minimum mass of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' – LP 890-9: We add the planet b recently unveiled by Del- rez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2022) around LP 890-9, which is next coolest star found to host a HZ planet, after TRAPPIST-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' – GJ 1002: We add planets b and c recently discovered around the M5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 V star, both of which reside in the conservative HZ (Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Appendix C: Priors and posteriors Appendix D: Short tables and data tables Article number, page 20 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2602 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8730 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1515 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8148 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4928 Frequency f [1/d] Power (ZK) Period [d] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Plots generated by AliasFinder for the daily (top) and yearly (bottom) aliases for the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 d signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each row illustrates the results for one simulated frequency, as indicated by the dashed blue vertical line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Each column is centered on a frequency window corresponding to the simulated frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The red line represents the periodogram of the original data set, whereas the black line is the median of the simulations, and the gray shaded regions depict the interquartile range and the confidence range of 90% and 99% of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The clock diagrams indicate the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 21 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Prior parameters for the photometric rotation period determination in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parameter name Prior Unit Description Photometric instrumental parameters µOSN-V-T150 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for OSN-V-T150 σOSN-V-T150 J(10−8, 10−1) ppm Extra jitter term for OSN-V-T150 µOSN-R-T150 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for OSN-R-T150 σOSN-R-T150 J(10−8, 10−1) ppm Extra jitter term for OSN-R-T150 µOSN-I-T150 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for OSN-I-T150 σOSN-I-T150 J(10−8, 10−1) ppm Extra jitter term for OSN-I-T150 µOSN-V-T90 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for OSN-V-T90 σOSN-V-T90 J(10−8, 10−1) ppm Extra jitter term for OSN-V-T90 µOSN-R-T90 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for OSN-R-T90 σOSN-R-T90 J(10−8, 10−1) ppm Extra jitter term for OSN-R-T90 µTJO-R U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for TJO-R σTJO-R J(10−8, 10−1) ppm Extra jitter term for TJO-R µSuperWASP U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for SuperWASP σSuperWASP J(10−8, 10−1) ppm Extra jitter term for SuperWASP µMEarth-tel01-s1 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for MEarth-tel01-s1 σMEarth-tel01-s1 J(10−8, 10−1) ppm Extra jitter term for MEarth-tel01-s1 µMEarth-tel01-s2 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for MEarth-tel01-s2 σMEarth-tel01-s2 J(10−8, 10−1) ppm Extra jitter term for MEarth-tel01-s2 µMEarth-tel05-s2 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1) ppm Photometric normalization for MEarth-tel05-s2 σMEarth-tel05-s2 J(10−8, 10−1) ppm Extra jitter term for MEarth-tel05-s2 dSHO-GP parameters Prot, GP, all(a) J(10, 200) d Primary period of the dSHO-GP δQGP, all(a) J(102, 105) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Quality factor difference between the first and second oscillations of the dSHO-GP Q0 GP, all(a) J(10−8, 103) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Quality factor for the secondary oscillation of the dSHO-GP σGP, OSN-V-T150,OSN-V-T90 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ��������������������������� σGP, OSN-R-T150,OSN-R-T90,TJO-R U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' σGP, OSN-I-T150 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Amplitude of the dSHO-GP σGP, MEarth-tel01-s1 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' σGP, MEarth-tel01-s2,MEarth-tel05-s2 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' σGP, SuperWASP U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' fGP, OSN-V-T150,OSN-V-T90 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' ��������������������� fGP, OSN-R-T150,OSN-R-T90,TJO-R U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fractional amplitude of the fGP, OSN-I-T150 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' secondary oscillation of the dSHO-GP fGP, MEarth-tel01-s1,MEarth-tel01-s2,MEarth-tel05-s2 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' fGP, SuperWASP U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (a) “all” comprises the following instruments: OSN-V-T150, OSN-R-T150,OSN-I-T150, OSN-V-T90, OSN-R-T90, TJO-R, SuperWASP, MEarth-tel01-s1, MEarth-tel01-s2, MEarth-tel05-s2 Article number, page 22 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Priors for the RV fits for Wolf 1069 with juliet in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' The 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal is not speculated to have planetary origins (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3), so we denote it as a signal “2”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parameter name Prior Units Description Parameters for planet b Pb U(15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='7) d Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' t0,b U(2458502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 2458515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) d Time-of-transit center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kb U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) m s−1 Radial velocity semi-amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' S 1,b = √eb sin ωb F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) (circular) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parametrization for e and ω U(−1, 1) (eccentric) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parametrization for e and ω S 2,b = √eb cos ωb F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) (circular) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parametrization for e and ω U(−1, 1) (eccentric) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parametrization for e and ω Parameters for the 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 d signal P2 U(85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) d Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' t0,2 U(2458500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 2458590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) d Time-of-transit center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' K2 U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) m s−1 Radial velocity semi-amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' S 1,2 = √eb sin ωb F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) (circular) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parametrization for e and ω S 2,2 = √eb cos ωb F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) (circular) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parametrization for e and ω RV instrumental parameters γCARMENES-VIS U(−20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) m s−1 Systemic velocity for CARMENES σCARMENES-VIS J(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01, 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) m s−1 Extra jitter term for CARMENES dSHO-GP parameters σGP, CARMENES-VIS U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) m s−1 Amplitude of the dSHO-GP Q0 GP, CARMENES-VIS J(10−8, 105) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Quality factor for the secondary oscillation of the dSHO-GP fGP, CARMENES-VIS U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Fractional amplitude of the secondary oscillation of the dSHO-GP δQGP, CARMENES-VIS J(102, 108) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Quality factor difference between the first and second oscillations of the dSHO-GP Prot, GP, CARMENES-VIS U(155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0, 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0) d Primary period of the dSHO-GP Article number, page 23 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Pb (d) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='56+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='01 1 0 1 2 3 t0, b - 2458511 (d) t0, b - 2458511 (d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 Kb (m s 1) Kb (m s 1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='52 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='55 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='61 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='64 Pb (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Mbsin i (M ) 1 0 1 2 3 t0, b - 2458511 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 Kb (m s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 Mbsin i (M ) Mbsin i (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Posterior distributions for the inner-most planet Wolf 1069 b from the final RV fit described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 24 of 26 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Kossakowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' : Wolf 1069 b: Earth-mass planet in the habitable zone of an M-dwarf GP, CARMENES = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='53 156 160 164 168 172 Prot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GP, CARMENES Prot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GP, CARMENES = 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='58+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 fGP, CARMENES fGP, CARMENES = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 log(Q0 GP, CARMENES) log(Q0 GP, CARMENES) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='82+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 GP, CARMENES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 log( QGP, CARMENES) 156 160 164 168 172 Prot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' GP, CARMENES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='8 fGP, CARMENES 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 log(Q0 GP, CARMENES) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='5 log( QGP, CARMENES) log( QGP, CARMENES) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='90+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='88 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Posterior distributions for the stellar rotation period using the dSHO-GP from the final RV fit described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 25 of 26 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' MAIN Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Full set of posterior parameters used in the final model choice for Wolf 1069 and described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Parameter Posterior Posterior parameters for planet b Pb 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='564+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='015 t0,b 2458511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='46 Kb 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='07+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='17 RV instrumental parameters γCARMENES (m s−1) −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='64+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='39 σCARMENES (m s−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='41 dSHO-GP parameters σGP, CARMENES-VIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='53 Q0 GP, CARMENES-VIS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00015+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05076 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='00015 fGP, CARMENES-VIS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='28 δQGP, CARMENES-VIS 80000+7300000 −78000 Prot, GP, CARMENES-VIS 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='6+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='3 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='4 Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Multiband photometry of Wolf 1069a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Band Magnitude Reference (mag) B 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='10 UCAC4 g′ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='13 UCAC4 GBP 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='368 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='004 Gaia DR3 V 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 UCAC4 r′ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='05 UCAC4 i′ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='09 UCAC4 GRP 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='027 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='004 Gaia DR3 J 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='039 2MASS H 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='483 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='073 2MASS KS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='095 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='021 2MASS W1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='877 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='023 AllWISE W2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='717 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='020 AllWISE W3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='545 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='016 AllWISE W4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='445 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='084 AllWISE Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (a) Gaia EDR3 G magnitude in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' 2MASS: Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Gaia DR3: Gaia Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' UCAC4: Zacharias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' WISE/AllWISE: Cutri & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (2012, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Telluric-corrected RV data used in this work for Wolf 1069.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Data will be available online in machine-readible format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' BJD (TDB* ) RV (m s−1) σRV (m s−1) Instrument 2457563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='66099 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='183 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='547 CARMENES 2457569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='58800 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='331 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='441 CARMENES 2457575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='61747 −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='551 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='584 CARMENES 2457584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='59445 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='557 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='250 CARMENES 2457591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='51688 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='028 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='915 CARMENES 2457594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='58057 −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='606 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='152 CARMENES 2457596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='52948 −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='083 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='468 CARMENES 2457597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='44527 −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='279 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='138 CARMENES 2457610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='51532 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+page_content='103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='540 CARMENES .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' .' 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+page_content='591 CARMENES 2459000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='64327 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='278 CARMENES 2459001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='64411 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='717 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='318 CARMENES 2459006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='64529 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='812 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='641 CARMENES 2459010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='59488 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='312 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='664 CARMENES 2459015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='61493 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='113 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='513 CARMENES 2459017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='64665 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='772 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content='050 CARMENES Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' (*) Barycentric dynamical time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} +page_content=' Article number, page 26 of 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNE0T4oBgHgl3EQfkwGA/content/2301.02477v1.pdf'} diff --git a/StAyT4oBgHgl3EQfVfcN/content/tmp_files/2301.00143v1.pdf.txt b/StAyT4oBgHgl3EQfVfcN/content/tmp_files/2301.00143v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7053f1872ac7edc0387d6584b8177f6eeffbe7d2 --- /dev/null +++ b/StAyT4oBgHgl3EQfVfcN/content/tmp_files/2301.00143v1.pdf.txt @@ -0,0 +1,1466 @@ +An implementation of the density functional perturbation theory +in the PAW framework +Xiaoqiang Liu,1 Yihao Lin,1 and Ji Feng1, 2, ∗ +1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China +2Hefei National Laboratory, Hefei 230088, China +(Dated: January 3, 2023) +Quantifying materials’ dynamical responses to external electromagnetic fields is central to un- +derstanding their physical properties. Here we present an implementation of the density functional +perturbation theory for the computation of linear susceptibilities using the projector augmented- +wave method. The Sternheimer equations are solved self-consistently through a nested iterative +procedure to compute the first-order wavefunctions, from which the linear susceptibilities are ob- +tained. As a demonstration, we compute the spin wave spectral functions of two magnetic metals. +The computed magnon spectra for half-metallic CrO2 and a Heusler intermetallic Cu2MnAl show +gapless Goldstone modes when spin rotation symmetry is preserved and display reasonable agree- +ment with available experimental data. The Landau damping is computed to be small in CrO2, +but significant in Cu2MnAl producing an asymmetric Lorentzian spectral lineshape. The access +to linear susceptibilities as well as first-order wavefunctions offers a range of novel possibilities in +quantitative understanding of materials’ electronic properties from ab initio methods. +I. +INTRODUCTION +A microscopic understanding of electrical and mag- +netic characteristics of materials plays a key role in con- +densed matter physics, furnishing a unified insight into a +wide range of phenomena. Indeed, the generalized den- +sity response functions (a.k.a. susceptibilities) of a many- +electron system to external electromagnetic fields, [1] in +a broad sense encompasses the information of the usual +longitudinal dielectric function and magnetic permeabil- +ity, as well as the cross terms for electromagnetic cou- +pling. The properties of collective excitations, e.g. plas- +mon and magnon, can also be procured from the sus- +ceptibilities. +Therefore, computing the susceptibilities, +which involve both charge and spin degrees of freedom +of electrons, is essential to a full characterization of elec- +tronic properties. Though relatively straightforward for +a non-interacting system, computing the susceptibilities +for an interacting many-electron system is a nontrivial +task due to interaction effects. +The Kohn-Sham density-functional theory (DFT) [2] is +by far the most widely employed ground state electronic +structure method for materials and molecules. By map- +ping the ground state energy to a non-interacting sys- +tem described by Kohn-Sham equations, the Kohn-Sham +ansatz enables the variational formulation of the static re- +sponses of a many-electron system. The time-dependent +(td) DFT [3] is subsequently developed, in which the elec- +trodynamics is described by td Kohn-Sham equations. +In these theories, the Hartree and exchange-correlation +potentials are functionals of density, and treated as self- +consistent fields. The self-consistent first-order perturba- +tion in td DFT leads to the density functional perturba- +tion theory (DFPT), [4] which is then the machinery for +∗ jfeng11@pku.edu.cn +linear response calculations leading directly to the full +response functions. +DFPT-based methods have been successfully applied +to calculate the dielectric function [5, 6] and phonon dis- +persion, [7, 8] with the results in quantitative agreement +with experimental observations. In the computations of +dielectric function, the first-order wavefunctions with re- +spect to k are solved, while deformation potential from +frozen atomic displacements is used as the external field +to compute phonon dispersions. +Dynamical responses +to external electromagnetic fields from DFPT, which ac- +count for the screening of both charge and spin, have at- +tracted considerable interest recently. [9–16] In addition +to quantifying electrical and magnetic properties of ma- +terials, the linear susceptibilities computed from DFPT +also find applications in the GW approximations. [17–19] +Approaches to the DFPT can be broadly grouped into +two categories: a Dyson-like equation is solved in the +first, [10, 11, 16] whereas the Sternheimer equations are +solved in the second. [9, 12–15] The Dyson-like equation +approach starts with response functions computed for the +Kohn-Sham ground state. Though formally transparent +and amenable to various iterative techniques, the Dyson- +like equation approach suffers from two shortcomings. It +requires a large number of unoccupied states and huge +planewave bases for adequate convergence. [16] More se- +rious is the subtle basis set incompatibility between the +Kohn-Sham states and the DFPT process, which gives +rise to an artifactual spin excitation gap in systems with +spin rotation symmetry. The latter problem can only be +partially mended with delicate engineering of the inter- +action kernel. [11, 20] In the second category, the first- +order wavefunctions are procured (often iteratively) by +solving the Sternheimer equations, from which the den- +sity is updated with charge mixing iterations and the re- +sponse functions are computed upon convergence. In this +case, one is forced to deal with wavefunctions and var- +ious pseudopotentials with all the technicalities, [21–24] +arXiv:2301.00143v1 [cond-mat.mtrl-sci] 31 Dec 2022 + +2 +in addition to the nested iterative procedure. Since the +full Kohn-Sham response functions are never required, +this method is free of the burden of summation over +a huge number of empty states. In addition, the first- +order wavefunctions and densities computed are actually +bonus, which can be useful for computing a variety of +properties. +A particularly popular planewave-based approach to +DFT is based on the projector augmented-wave (PAW) +method. [23, 24] Combining the formal simplicity of +pseudopotentials and the versatility of the linearized +augmented-planewave method, the PAW method offers +both efficiency and accuracy to Kohn-Sham DFT calcula- +tions on extended solids, and a wide range of capabilities +in various implementations. [25–27] Despite its popular- +ity, DFPT in the PAW framework has remained to be +developed, which is accomplished in this work by solving +the td Sternheimer equations to compute the linear sus- +ceptibilities of crystalline materials accounting for both +charge and spin degrees of freedom. The paper is orga- +nized as follows. In Sec. II A the general theory of DFPT +is introduced, from the viewpoint of the dressed spin in +td external electromagnetic fields. The screening built on +the notion of dressed spin leads to the Sternheimer equa- +tions in the frequency and momentum domain and ex- +plicit formula for the linear susceptibilities. In Sec. II B +the PAW method is reviewed, based on which the formu- +lation of DFPT in the PAW framework is described, along +with a few implementation details. As a first calibration +our implementation, the spin wave spectral functions are +extracted from the computed linear susceptibilities. Two +examples are presented: half-metallic CrO2 (Sec. III A) +shows a clean spin wave spectrum and minimal Landau +damping, and a full Heusler intermetallic Cu2MnAl (Sec. +III B) shows significant Landau damping in the spin ex- +citations that can be quantified with a simple asymmet- +ric Lorentzian lineshape. Lastly, a summary is provided +with an eye on room for development from algorithm and +physics points of view. +II. +THEORY AND IMPLEMENTATION +A. +Density functional perturbation theory +The td DFT offers an efficient description of the dy- +namics of an interacting many-electron system in the +presence of external fields. [3] As a self-consistent per- +turbation theory of td DFT, DFPT is introduced in this +section, wherein the Sternheimer equations are special- +ized to crystalline systems under a monochromatic, peri- +odic external electromagnetic field. +In td DFT, to account for both charge and spin degrees +of freedom, the generalized density is given by +ρ(r, t) = +� +n +θn tr{ψ† +n(r, t)σψn(r, t)} += (ρ0, m) = (ρ0, ρ1, ρ2, ρ3) +(1) +where θn is the occupancy of the spinor single-particle +state ψn, and the four-vector spin σ = (σ0, σ1, σ2, σ3) +(σ0 is the identity matrix and σα with α = 1, 2, 3 are +the Pauli matrices). The atomic units [28] are adopted +in this paper, so ρ0 is the total charge density, and m is +magnetization density. The dynamics of ψn is prescribed +by the td Kohn-Sham equations [3] +i∂t|ψn(t)⟩ = [H + δH(t)]|ψn(t)⟩. +(2) +The ground state Kohn-Sham Hamiltonian [2] in Eq. (2) +is H = − 1 +2∇2 + v[ρ(0)](r) where the self-consistent po- +tential is a functional of the ground state density ρ(0), +composed of ionic, Hartree and exchange-correlation (xc) +potentials, namely, vi, vH and vxc. Though the xc poten- +tial vxc as a functional of density is in principle nonlocal +in space and time, [29–33] the commonly adopted local +and adiabatic approximation (ALDA) is assumed in this +work. [34–36] +The first-order Hamiltonian δH comprises two contri- +butions. The first arises from the coupling of the four- +vector spin σ with external fields +vext(r, t) = −Bα(r, t)σα, +(3) +where the four-vector electromagnetic field B(r, t) = +(−φ, 1 +2B). [37] The indices α, β = 0, 1, 2, 3 are implicitly +summed over when repeated, but we will keep other sum- +mations explicit. The self-consistent inclusion of the den- +sity dependence in the Hartree and xc potentials means +that δH also includes a second contribution from induced +density δρ(r, t) that screens vext. In the adiabatic linear +response theory, this can be formulated in terms of a +dressed spin τα +τα(r, r′, t − t′) = − δH(r, t) +δBα(r′, t′) = σαδ(r − r′)δ(t − t′) +− σγ +� +fγβ(r, r′′)χβα(r′′, r′, t − t′)dr′′. +(4) +Here χαβ(r, r′, t) is the linear susceptibility that we pur- +sue in this work, defined via +δρα(r, t) = +� +χαβ(r, r′, t − t′)Bβ(r′, t′)dr′dt′. +(5) +The interaction kernel has two components, fαβ = f H +αβ + +f xc +αβ, namely, the Hartree and xc kernels in the ALDA +fαβ(r, r′) = δα0δβ0 +|r−r′| + 1 +2δ(r − r′) tr +� +σα +∂vxc +∂ρβ +� +. +(6) +In terms of the dressed spin, the first-order Hamiltonian +in unscreened external fields is written +δH(r, t) = − +� +τα(r, r′, t − t′)Bα(r′, t′)dr′dt′. +(7) +Now with the first-order Hamiltonian, the first-order +wavefunctions can be obtained by solving the Stern- +heimer equations [8, 9, 12, 38–40] +(i∂t − H)|ψ(1) +n (t)⟩ = δH(t)|ψ(0) +n (t)⟩, +(8) + +3 +FIG. 1. Flow chart for a nested-loop DFPT calculation in the +Sternheimer equation approach. The outer loop depicted here +is the charge mixing. The inner loop is incurred in solving the +Sternheimer equations in each step of the outer iteration. +in which ψ(ℓ) +n (t) is the ℓth-order wavefunction, with +|ψ(0) +n (t)⟩ = e−iεnt|ψn⟩. +With the first-order wavefunc- +tions, we will be able to compute the induced density +via the variation of Eq. (1), from which the first-order +Hamiltonian will be updated. +With the above introduction, a DFPT calculation can +then be performed in a nested iterative process depicted +in Fig. 1. In the initializing step, δH is constructed from +the external electromagnetic fields, whence the Stern- +heimer equations are solved in the inner iteration. This +produces a set of first-order wavefunctions, from which +the induced density is calculated. A charge mixing strat- +egy [41, 42] is employed to revise the induced density, +with the linear susceptibility and dressed spin computed +subsequently. Then a new δH is constructed from the up- +dated spin to enter a second round of the outer iteration. +The above process is repeated till convergence. +Now we will explain how the Sternheimer equations are +solved in a crystalline solid. For electrons in a crystal, +the initial Kohn-Sham states are Bloch functions such +that |ψn⟩ �→ |ψnk⟩ = eik·r|unk⟩, where n is the band +index and k is the crystal momentum, and unk is the +cell-periodic part of the Bloch function. The system is +subject to spatially periodic and monochromatic external +electromagnetic fields +Bα(r, t) = Bαei(p·r−ωt) + c.c., +(9) +where p = q + g with g being a reciprocal lattice vector +for q to in the first Brillouin zone. +In this case, the +following expansion is useful +ζ(r, t) = +� +l +ei(l·r−νt)ζ(r, l) +(10) +for ζ = δH, δρα, in which ζ(r, l) is a cell-periodic function +with +l ≡ (ν, l) = ±(ω, q). +Then the first-order Hamiltonian in the l channel is +δH(r, l) = −e−il·rτα(r, sign(νω)p, ν)Bα(l), +(11) +where Bα(ω, q) = Bα, Bα(−ω, −q) = (Bα)∗, and +τα(r, p′, ν) = +� +τα(r, r′, t)ei(p′·r′+νt)dr′dt. +(12) +The first-order wavefunction is expanded as +|ψ(1) +nk(t)⟩ = +� dν +2π e−i(ν+εnk)t|ψ(1) +nk(ν)⟩. +(13) +For the fields in Eq. (9), |ψ(1) +nk(ν)⟩ is nonzero only when +ν = ±ω and can be written as +|ψ(1) +nk(ν)⟩ = ei(k+l)·r|u(1) +nk(l)⟩. +(14) +Then the Sternheimer equations become +(ν + iη + εnk − Hk+l) |u(1) +nk(l)⟩ = δH(r, l)|unk⟩, +(15) +in which Hk = e−ik·rHeik·r is the Bloch Hamiltonian. A +positive infinitesimal η is introduced on the left-hand side +as a convergence factor to embody the causality struc- +ture of the linear susceptibility. In actual calculations, +η takes finite values to ensure convergence with finite k- +mesh especially for metals and bestows finite broadening +on spectral peaks. +From the first-order wavefunctions, we compute the +first-order induced density as +δρα(r, l) = +� +nk +θnk tr +� +u† +nk(r)σαu(1) +nk(r, l) + u(1) +nk +†(r, −l)σαunk(r) +� +. +(16) +The linear susceptibility can be extracted from the Fourier components of δρα(r, l) +δρα(g′, l) = χαβ(g′, sign(νω)g, l)Bβ(l), + +B +op +8H(r, +charge +Sternheimer equations +(io,-H)(1) =8H(0) +mixing +No +convergence? +Yes4 +whence +χαβ(g′, g′′, l) = +� +e−i(g′+l)·r′χαβ(r′, r′′, ν)ei(g′′+l)·r′′dr′dr′′. +(17) +It is worth mentioning that the Sternheimer equations in (15) admit the following formal solutions in the l channel +|u(1) +nk(l)⟩ = −Bα(l) +� +n′ +|un′k+l⟩τα(k, l)n′n +ν + εnk−εn′k+l + iη , +(18) +in which the dressed spin matrix element is +τα(k, l)n′n = +� +u† +n′k+l(r)e−il·rτα(r, sign(νω)p, ν)unk(r)dr. +(19) +Because it requires the summation over a large number of empty states, this formal solution is not used in practice. [43] +The screened susceptibility then has the same expression as the (bare) Kohn-Sham susceptibility, except that the +(unscreened) external fields are now coupled to the dressed spin; that is, +χαβ(g′, sign(νω)g, l) = − +� +nn′k +(θnk − θn′k+l)⟨unk|e−ig′·rσα|un′k+q⟩τβ(k, l)n′n +ν + εnk − εn′k+l + iη +. +(20) +In arriving at the last expression, we have used the fact that τα(k, l)n′n = τα(k + l, −l)∗ +nn′. +B. +DFPT with PAW method +The previous Subsection presents a sketch of the DFPT +for periodic systems without recourse to computational +details. In practical calculations, however, various tech- +nologies have been developed such that one can per- +form DFT (and therefore DFPT) calculations on valence +electrons only for crystalline materials using planewave +bases with the aid of pseudopotentials, to yield satisfac- +tory accuracy with high efficiencies. +It is well known +that the norm-conserving pseudopotential [21] requires +large planewave bases particularly for localized orbitals +in transition elements, while the application of ultrasoft +pseudopotential [22] is partly limited by the rather labo- +rious construction. In contrast, the PAW method, com- +bining the pseudopotential and linearized augmented- +plane-wave methods, is free from the above difficulties +and has been used widely. +Thus, performing DFPT +within the PAW framework [23, 24] for inhomogeneous +and td electromagnetic fields is evidently useful though +notably nontrivial. +DFPT with the PAW method has +been implemented within the Vienna ab initio simula- +tion package (VASP) [25] for atomic displacements in +the static and long-wavelength limit to calculate zone- +center phonon energies. The extension to inhomogeneous +and td electromagnetic fields is accomplished in this work +based on VASP 5.4.4, and a few implementation details +warrant further clarification. +Here, we will briefly re- +view the PAW method and describe how it is used in +our DFPT calculations. Although the formalism for the +ground state quantities is identical to those in the lit- +erature and notationally notorious, we feel compelled to +provide some of these details, particularly in view of the +time- and position-dependent external fields involved in +our implementation. +The PAW method is based on a linear transformation +between the all-electron (AE) Hilbert space orthogonal +to core states and pseudo (PS) Hilbert space. The AE +and PS wavefunctions are related by +|ψnk⟩ = T| ˜ψnk⟩, +(21) +with the linear operator defined as +T = 1 + +� +i +(|φi⟩ − |˜φi⟩)⟨˜pi|. +(22) +The index i is a shorthand encapsulating the atomic site +located at Ri as well as the quantum numbers (nlm) of +the local orbitals and spin. φ, ˜φ and ˜p are AE partial +waves, PS partial waves and projector functions, respec- +tively, and should all be understood as spinors. In order +to perform the DFPT calculations on a crystalline solid +in the electromagnetic fields in Eq. (9), the key is to find +the cell-periodic part of each PAW-pseudized quantity, +especially the nonlocal ones. +Upon application of the time-independent transforma- +tion T to the first-order wavefunctions the pseudized +Sternheimer equations read +(i∂tS − ˜H)| ˜ψ(1) +nk(t)⟩ = δ ˜H(t)| ˜ψ(0) +nk(t)⟩, +(23) +with S = T †T, ˜H = T †HT and δ ˜H(t) = T †δH(t)T. +According to the implementation of PAW method in + +5 +VASP, [24] we have +S = 1 + +� +ij +|˜pi⟩ qij ⟨˜pj| , +˜H = −1 +2∆ + ˜veff + +� +ij +|˜pi⟩ Dij ⟨˜pj| , +(24) +in which the nonlocal potential Dij = ˆDij + ˜Dij with +˜Dij = D1 +ij − ˜D1 +ij. The quantities qij, D1 +ij and ˜D1 +ij are +defined in reference. [24] Apparently, the local potential +˜veff(r) is a functional of pseudo density ˜n(r) and com- +pensation density ˆn(r), while ˜Dij is a function of density +matrix ϱ, i.e. +˜veff =˜veff[˜n + ˆn], +ˆDij = +� 1 +2 tr{σα˜veff(r)}Qα +ij(r)dr, +˜Dij = ˜Dij(ϱ). +(25) +We have hidden the functional dependence on the +pseudized core densities in ˜veff(r), which are kept frozen +during the DFPT calculations. +˜n(r), ˆn(r) and ϱij are +given by, respectively, +˜n(r) = +� +nk +θnk tr{ ˜ψ† +nk(r)σ ˜ψnk(r)}, +ˆn(r) = +� +i,j +ϱijQij(r), +ϱij = +� +nk +θnk⟨ ˜ψnk|˜pi⟩⟨˜pj| ˜ψnk⟩. +(26) +Here Qij(r) = � +L tr{σ ˆQL +ij(r)} with ˆQL +ij(r) defined to +construct the compensation density ˆn(r) in the refer- +ence. [24] It should be noticed that for ϱij, only elements +with Ri = Rj are useful in our calculations, while Dij +are nonzero only when Ri = Rj. +Now we derive the expression for the first-order Hamil- +tonian δ ˜H(t). For the fields in Eq. (9), the first-order +local densities and potentials follow the same expansion +as in Eq. (10), while the first-order density matrix δϱij +and nonlocal potential δDij can be expanded as +δζij(t) = +� +l +ei(l·Ri−νt)δζij(l). +(27) +Here, the factor eil·Ri in the l channel is introduced such +that δζij(l) is cell-periodic, i.e., δζij(l) = δζi′j′(l) if the +positions of atomic site for i, j and i′, j′ differ by a lattice +vector. +The contribution of external electromagnetic fields in δ ˜H(t) can be calculated directly +δ ˜Hext(r, t) = +� +l +e−iνt � +eil·rvext(r, l) + +� +ij +eil·Ri |˜pi⟩ Dext +ij (l) ⟨˜pj| +� +, +(28) +with +vext(r, l) = − Bα(l)σαesign(νω)ig·r, +Dext +ij (l) =⟨φi|eil·(r−Ri)vext(r, l)|φj⟩ − ⟨˜φi|eil·(r−Ri)vext(r, l)|˜φj⟩. +(29) +The contribution to δ ˜H(t) from the induced densities has a similar expression as in Eq. (28) and can be calculated +via an explicit finite difference, as ˜H is a functional of ˜n, ˆn and ρij. The first-order densities are found to be +δ˜n(r, l) = +� +nk +θnk tr{˜u† +nk(r)σ˜u(1) +nk(r, l) + ˜u(1),† +nk (r, −l)σ˜unk(r)}, +δˆn(r, l) = +� +i,j +eil(Ri−r)δϱij(l)Qij(r), +δϱij(l) = +� +nk +θnk[⟨˜unk|˜pik⟩⟨˜pjk+l|˜u(1) +nk(l)⟩ + ⟨˜u(1) +nk(−l)|˜pik−l⟩⟨˜pjk|˜unk⟩], +(30) +where we define |˜pik⟩ = e−ik·(r−Ri)|˜pi⟩. To linear order in external fields, the first order effective local potential is +given by +δ˜veff(l) ≈ ˜veff[˜n + ˆn + δ˜n(l) + δˆn(l)] − ˜veff[˜n + ˆn]. +(31) +Similarly, the first-order nonlocal potentials can be approximated as +δ ˆDij(l) ≈ +� +eil·(r−Ri) 1 +2 tr{σαδ˜veff(r, l)}Qα +ij(r)dr, +δ ˜Dij(l) ≈ ˜Dij(ϱ + δϱ(l)) − ˜Dij(ϱ). +(32) + +6 +Though introduced as forward differences in Eq.(31) and (32), these quantities are evaluated using 4th-order centered +finite differences, with a step length of a thousandth the density variables. +With the above results, the final Sternheimer equations become +� +νSk+l + εnkSk+l − ˜Hk+l +� +|˜u(1) +nk(l)⟩ = δ ˜Hk(l)|˜unk⟩, +(33) +with +Sk+l = 1 + +� +ij +|˜pik+l⟩ qij ⟨˜pjk+l| , +˜Hk+l = −1 +2∆k+l + ˜veff + +� +ij +|˜pik+l⟩ Dij ⟨˜pjk+l| , +δ ˜Hk(l) = vext(l) + δ˜veff(l) + +� +ij +|˜pik+l⟩ [Dext +ij (l) + δ ˆDij(l) + δ ˜Dij(l)] ⟨˜pjk| . +(34) +Here ˜unk and ˜u(1) +nk(l) are the cell-periodic parts of corresponding pseudo wavefunctions, respectively. +The pseudized Sternheimer equations in ±(ω, q) channels are solved separately in each iteration using a variant +of residual minimization method with a direct inversion in the iterative subspace (RMM-DIIS), [44, 45] which is +already implemented in VASP 5.4.4. The L¨owdin perturbation theory is also performed to correct the first-order +wavefunctions in the subspace of occupied states and low-lying excitations to speed up convergence +|˜u(1) +nk(l)⟩ → |˜u(1) +nk(l)⟩ − +� +n′ +|˜un′k+l⟩⟨˜un′k+l|Sk+l|˜u(1) +nk(l)⟩ + +� +n′ +|˜un′k+l⟩⟨˜un′k+l|δ ˜H|˜unk⟩ +ν + εnk − εn′k+l +. +(35) +In the last equation above, the summations on n′ run +over the occupied bands plus a few empty bands. +Apparently, solving Eq. (33) requires a k-grid supple- +mented by two additional grids shifted by ±q when q +itself is not on the k-grid. Doing so, however, not only +increases the computational burden but also, more seri- +ously, obliterates the exact cancellation of the contribu- +tion of the occupied manifold to the density change due +to a k-grid discretization error. The latter can be eas- +ily avoided by employing a pair of grids with a q shift, +which also reduces the calculation partly. Then Eq. (33) +is solved on the k-grid in +q channel, and on the k + q +grid in −q channel. It is observed that in this dual grid +setup, the above cancellation is well preserved. +The xc potentials in ALDA are functionals of real- +valued densities. +Thus, calculating δ˜veff(l) like in Eq. +(31) requires caution as δ˜n(l) and δˆn(l) are usually com- +plex. In fact, the real and imaginary part of δ˜veff(l) are +calculated separately, +Fδ˜veff(l) ≈ ˜veff[˜n+ˆn+Fδ˜n(l)+Fδˆn(l)]−˜veff[˜n+ˆn] (36) +where F = Re, Im takes the real or imaginary part, re- +spectively. In the case of nonlocal potential, δ ˜Dij(l) and +δϱij(l) are first decomposed into two independent Her- +mitian matrices (i.e. Hermitian part and anti-Hermitian +part multiplied by −i), and then finite differenced sepa- +rately in an analogous fashion. +Symmetry reduction is also performed in our imple- +mentation, where the summation over k points in Eq. +(30) is restricted to the symmetry-irreducible part of the +Brillouin zone. The symmetry group here is the subgroup +of the magnetic group of the studied crystal in which the +external electromagnetic fields in Eq. (9) is invariant. +III. +APPLICATION TO SPIN-WAVE +SPECTRUM CALCULATION +Our implementation enables computing the linear sus- +ceptibilities χαβ with the self-consistent inclusion of the +interaction kernel. Directly inverting χαβ yields the di- +electric tensor, which is composed of the usual charge +sector ϵ00, spin sector ϵαβ, and the spin-charge sector +ϵ0β, each embodying unique physics. +Computing χαβ +then can have diverse applications in evaluating materi- +als properties pertaining to both charge and spin fluctu- +ations, or in subsequent many-body calculations beyond +the Kohn-Sham mean fields. +One immediate applica- +tion that has received considerable attention is the cal- +culation of spin-wave excitation. [9–16] According to the +fluctuation-dissipation theorem, the spin-spin correlation +function, directly accessible by various spin-sensitive in- +elastic scattering probes, [46–48] is related to the imagi- +nary part of the linear susceptibilities, +S+−(p, ω) = Im χ+−(g, g, ω, q) +1 − e−ℏω/kBT +. +(37) + +7 +TABLE I. Reported implementations of DFPT for spin wave spectra calculations by solving +the Sternheimer equations. +Authors (Year) +Potential +Basis set +Software +Savrasov (1998) [9] +Full potential +LMTO1 +LMTO Magnons +Cao et al. (2018) [12] +NCPP2, USPP3 +Planewave +QE4 +Gorni et al. (2018) [13] +NCPP +Planewave +QE +Tancogne-Dejean et al. (2020) [14] +NCPP +Real space +Octopus +Singh et al. (2020) [15] +Full potential +LAPW5 +Elk +1Linear muffin-tin orbital. 2Norm-conserving pseudopotential. 3Ultrasoft pseudopotential. +4Quantum ESPRESSO. 5Linearised augmented-plane wave. +Although for magnetic systems dominated by local mo- +ments the magnon can be described effectively by lo- +calized spin models, this method is subject to debate +when delocalization sets in, and ultimately of question- +able validity for itinerant magnetism. +In these latter +cases, which include a wide range of magnetic mate- +rials, the DFPT route becomes invaluable for comput- +ing the spin wave spectra ab initio. To our knowledge, +there have been just a handful of works devoted to im- +plementing DFPT scheme for this purpose by solving +the Sternheimer equations, as summarized in Table I. +In these efforts, implementations are limited to the full- +potential, [9, 15] or norm-conserving and ultrasoft pseu- +dopotentials. [12–14] +In this section, we present an initial application of our +implementation of DFPT in the PAW framework to the +calculations of spin-wave spectra for a couple of mag- +netic materials. For ferromagnets with the spin polar- +ized along z direction, transverse magnetic field can be +applied by choosing B = (0, 1, −i, 0) in Eq. +(9), from +which χ+− is calculated directly. +In these cases, the +only remaining symmetry operation keeping the crys- +tal and the transverse magnetic field unchanged is iden- +tity transformation. +Thus, there is no room for sym- +metry reduction. For notationaly convenience, we define +χ+−(p, ω) ≡ χ+−(g, g, ω, q) as p = q + g. +A. +Half-metallic chromium dioxide +As shown in the inset in Fig. 2, chromium dioxide, +CrO2, is a ferromagnetic half-metallic oxide with a ru- +tile crystal structure, where each Cr atom is situated +at the center of an octahedral cage formed by oxygen +atoms. [49, 50] Widely used as a magnetic recording ma- +terial, CrO2 also has various potential applications in +spintronics and magnetoelectronics [51, 52] due to its +half-metallic properties. +The experimental lattice parameters a += +b += +4.4218 ˚A and c = 2.9182 ˚A [50] are used in our calcu- +lations. +The planewave energy cutoff is set to be 500 +eV and a 13 × 13 × 20 Γ-centered mesh of k-points is +FIG. 2. +Spin-resolved densities of states of CrO2. The spin- +flip gap is found to be about 310 meV. Inset: the crystal +structure of CrO2, highlighting the CrO6 octahedra. +used. The spin-resolved density of states of CrO2 is com- +puted from the collinear spin-polarized calculation and +shown in Fig. 2, where the half-metallic band structure +is clearly seen. The magnetic moment of Cr is found to +be 2 µB. We then turn to the noncollinear calculations, +and compute the transverse spin susceptibility χ+−(p, ω) +along [100] and [001] directions for ω ≤ 400 meV on 10 +meV intervals. The broadening parameter η introduced +in Eq. (15) is set to be 50 meV in the calculations in this +subsection. +Fig. 3(a) shows the computed Imχ+−(p, ω), without +spin-orbit interaction, along two p paths. +In general, +χ+−(p, ω) is not periodic in p. +Then the branch in +the first Brillouin Zone is composed of acoustic magnon +modes, while the branch in the second Brillouin zone op- +tical modes. The profile of the magnon peak at a given p +is nearly perfect Lorentzian over the entire energy range. +The extracted half width at half maximum ηp are almost +a constant and equal to the artificial broadening param- +eter η, indicating that the Landau damping in CrO2 is +negligible. This is expected given that half-metallic CrO2 +has a large spin-flip gap around 310 meV, as shown in Fig. +2. + + up +- down +N(e) (States/eV/u.c.) +-5 +-3 +-2 +0 +28 +FIG. 3. (a) Imχ+−(p, ω) of CrO2 along [100] and [001] di- +rections. The black vertical line indicates the center of the +first Brillouin zone. (b) The folded magnon energy dispersion +ωq of CrO2 in the first Brillouin zone, extracted from Im χ+− +shown in (a). +The inset shows the quadratic fits to ωq at +small q along [100] and [001] directions, respectively, for cal- +culations without Hubbard U correction and with U eff = 2.1 +eV. The squares are the calculated data. +The location of the maxima of magnon peaks, ωp, +are then recorded and folded to the first Brillouin zone +(ωq). +As shown in Fig. +3(b), we find one acoustic +magnon branch and one optical branch, consistent with +the fact that the unit cell in CrO2 contains two mag- +netic Cr atoms. There is no magnon gap at the Brillouin +zone boundaries, which is result of the n glide symme- +try. +The energy of long-wavelength acoustic magnons +is quadratic in q with a gapless Goldstone mode, ωq = +D∥(q2 +x + q2 +y) + Dzq2 +z, as expected for a ferromagnet with +spin rotation symmetry. The spin stiffness coefficients are +found to be D∥ = 82 meV·˚A2 along [100] and Dz = 92 +meV·˚A2 along [001] directions, respectively. From this, +we can estimate the average spin stiffness coefficient to be +85 meV·˚A2, which is close to the experimental measured +result (∼ 112.5 meV·˚A2 [53]). +For comparison, additional electron correlation is in- +cluded statically within the LSDA+U formalism [54] in +both the ground state calculation and subsequent DFPT +calculations, with U eff = 2.1 eV. [55] As shown in the +inset of Fig. +3(b), the gapless Goldstone mode is ob- +tained again, accompanied by an average spin stiffness +FIG. 4. (a) Imχ+−(p = 0, ω) as a function of ω for different λ +values. λ = 1 corresponds to the actual strength of spin-orbit +coupling in CrO2. The squares are the calculated data and +the solid lines are the fits with Lorentzian line shape. (b) The +Goldstone gap as a function of λ2 in CrO2. The solid line is +the linear fit. +coefficient D = 391 meV·˚A2, almost five times as large +as the one without Hubbard U correction. The energy +of magnon in CrO2 seems to be highly overestimated in +LSDA+U calculations. +As a further test, we examine the Goldstone gap as +a result of breaking the spin rotation symmetry by in- +troducing spin-orbit interaction. The atomic spin-orbit +interaction for Cr is fairly weak (on the order of tens of +meV). Since the gap in the Goldstone magnon is second +order in the spin-orbit coupling, it is small for CrO2. In +order to visualize the effect of spin-orbit coupling, we in- +troduce a parameter λ to artificially tune its strength (or +speed of light), as in H = H0 + λHsoc, where λ = 1 cor- +responds to the actual strength of spin-orbit coupling in +CrO2. The calculated Imχ+−(p = 0, ω) as a function of ω +for different λ values are shown in Fig. 4(a). Apparently +there is blue shift of the Goldstone mode with increas- +ing λ, indicating the emergence of a Goldstone gap. The +Goldstone gap indeed shows a quadratic dependence on +λ, as demonstrated by the gap-vs-λ2 plot in Fig. 4(b). +The extrapolated Goldstone gap in CrO2 is about 0.1 +meV. +B. +Heusler intermetallic Cu2MnAl +The ternary intermetallic Cu2MnAl is a Mn-based +full-Heusler alloy with the L21 structure type (see in- +set in Fig. +5). +The experimental lattice parameter +for the conventional cubic cell (space group Fm¯3m) is +a = 5.968 ˚A. [56] Cu2MnAl is ferromagnetic below the +relatively high Curie temperature (603 K). [56] Apart +from being regarded as a prototype for understanding +the electronic correlations in Heusler intermetallics, [57] +Cu2MnAl is also being used as a neutron polarizer and +monochromator material. [58, 59] +A planewave energy cutoff of 350 eV and a 15×15×15 +Γ-centered k-grid are used in our calculations. +Spin- +orbit coupling is not included. +The spin-resolved den- +sities of states of Cu2MnAl computed from the collinear + +(a) +X102 +400 +4 +300 +3 +Imx+-(p, w) (a.u.) +(meV) +200 +2 +3 +100 +1 +0 +(2,0,0) +(0,0,0) +(0,0,2) +(b) +400 +350 +300 +250 +80 +(meV) +U +0eV +口 +200 +60 +U +2.1 eV +40 +20中 +口 +100 +0 +(0.3, 0, 0) +(0, 0, 0) +(0, 0, 0.2) +50 +0 +(0.5, 0, 0) +(0, 0, 0) +(0, 0, 0.5)(a) +X102 +(b) +3.5 +40 +>=0 +Imx+-(p = O, w) (a.u.) +=1 +— >= 2.5 +>=5 +30 +日 >= 7.5 +neV) +=10 +中 >= 15 +2.5 +00 +10 +1.5 +0 +0 +20 +40 +0 52 +102 +152 +w (meV) +129 +FIG. 5. +Spin-resolved densities of states of Cu2MnAl. +In- +set: the crystal structure of full Heusler Cu2MnAl, showing a +conventional cubic unit cell for the L21 structure. +spin-polarized calculation confirms the ferromagnetism +of Cu2MnAl, as shown in Fig. 5. The magnetic moment +is carried primarily by Mn atoms and computed to be +3.4 µB/Mn. The transverse spin susceptibility χ+−(p, ω) +along [100], [110] and [111] crystallographic directions is +then computed in noncollinear calculations for ω ≤ 300 +meV on 5 meV intervals, with a broadening parameter η +of 50 meV. +Fig. 6(a) shows the computed magnon spectral func- +tion Imχ+−(p, ω) along the three principal directions. +The acoustic magnon branch is seen clearly only at low +energies near the Brillouin zone center. +The spectral +peaks of these low-energy modes can be adequately fit- +ted with the Lorentzian lineshape as in the CrO2 case. +The dispersion of the long-wavelength modes is quadratic +and isotropic, as demonstrated in the inset of Fig. 7 (a). +A spin stiffness coefficient D = 268 meV·˚A2 is procured +from the quadratic fit, which is about 1.5 times larger +than the experimental value of 175 meV·˚A2. [60] +Notably, at higher energies and near Brillouin zone +boundaries, +the magnon peaks become fuzzier and +broader, attesting to substantial Landau damping in this +material. In stark contrast to the CrO2 case with almost +no Landau damping, the coupling to the Stoner contin- +uum in Cu2MnAl bestows a magnon peak at a given p an +asymmetric profile that defies a Lorentzian fit, as shown +in Fig. 6(b). The stronger Landau damping in this sys- +tem is consistent with the absence of the spin-flip gap +as shown in Fig.5. Viewing the coupling to the Stoner +continuum as a Fano-type resonance, we superimpose a +linear function on the Lorentzian to describe the asym- +metric line shape, as +A(p, ω) = +apηp +(ω − ωp)2 + η2p ++ ξp(ω − ωp) +(38) +with ωp, ap, ηp and ξp as fitting parameters. +As it turns out, this simple modification leads to sat- +isfactory fitting for the entire spectrum, as evidenced in +Fig.6(c). The extracted magnon dispersion is then shown +FIG. 6. (a) Imχ+−(p, ω) of Cu2MnAl along [100], [110] and +[111] directions. +Reciprocal lattice vectors of conventional +cubic cell are adopted here. (b,c) Imχ+−(p, ω) as a function +of ω for p = (1, 0, 0), (1, 1, 0) and (1, 1, 1). The squares are the +calculated data. The solid lines are the fits with (b) symmetric +or (c) asymmetric Lorentzian function. +in Fig. 7(a), which coincides with the calculated results +of Buczek et al. [10, 61] and agrees well with the ex- +perimental observations along [100] direction. [60] Along +the [110] and [111] directions where the Landau damp- +ing seems more pronounced, our computed dispersion +shows significant discrepancy from the experimental one. +For low-energy modes, ηp is dominated by the artificial +broadening parameter η and the asymmetry is small, as +shown in Fig. 7(b). With increasing energies, however, +the broadening quickly exceeds η and the asymmetry be- +comes pronounced, especially near Brillouin zone bound- +aries, both providing quantitative characterization of the +Landau damping. +IV. +SUMMARY AND OUTLOOK +In conclusion, we report an implementation of the +DFPT method in the PAW framework, which is capable +of computing the full linear susceptibilities of real mate- +rials. A nested iterative procedure is employed to self- +consistently solve the Sternheimer equations, to procure +linear susceptibilities along with the first-order wavefunc- +tions and densities in monochromatic and periodic exter- +nal electromagnetic fields. The time cost of each DFPT +calculation (given an external field direction, momentum +and frequency) is comparable to that of a corresponding +Kohn-Sham DFT calculation. + +- up +- down +Mn +N(e) (States/eV/u.c.) +Al +-5 +-3 +-2 +-1 +0 +2(a) +×102 +300 +3 +250 +Imx+-(p, w) (a.u.) +200 +2 +(meV) +150 +3 +100 +50 +0 +(0,0,0) +(1,0,0)/(1,1,0) +(0,0,0) +(1,1,1) +(b) +(c) +×102 +1.5 +Imx+-(p, w) (a.u.) +Imx+-(p,w) (a.u.) +(1,0,0) +(1,0,0) +(1,1,0) +(1,1,0) +(1,1,1) +(1,1,1) +0.5 +0.5 +0 +0 +0 +100 +200 +300 +0 +100 +200 +300 +(meV) +w (meV) +310 +FIG. 7. (a) The magnon energy dispersion ωp of Cu2MnAl +along [100], [110] and [111] directions, extracted from the +asymmetric Lorentzian fits. +△ is the experimental data of +Tajima et al. [60] Inset is the quadratic fit to the long- +wavelength acoustic magnons. (b) Broadening ηp and asym- +metry ξp of the magnon peaks in Cu2MnAl along [100], [110] +and [111] directions. +The error bars show 95% confidence +bounds on the fitting parameters from the fits. +As a demonstration, we compute the spin wave spec- +tra for CrO2 and Cu2MnAl. +Gapless magnon disper- +sion is obtained for both materials from the calculations +without spin-orbit coupling. +The spin stiffness coeffi- +cient extracted from the quadratic fit is in agreement +with experimental value for CrO2 but 1.5 times larger +for Cu2MnAl. The Landau damping in CrO2 is insignifi- +cant due to its half-metallic nature, while in Cu2MnAl is +remarkable at high energies and can be quantified with +a simple asymmetric Lorentzian fit. LSDA+U method +as well as the effect of spin-orbit coupling are examined +in CrO2, from which the former highly overestimates the +magnon energy, while the latter gives rise to a Goldstone +gap quadratic in spin-orbit coupling strength λ. +There is clearly room for future developments, to make +the current implementation more efficient and versatile. +From an algorithm viewpoint, the occupied subspace is +not projected out in Sternheimer equations in the cur- +rent implementation. +As the contribution to the first- +order wavefunctions from the occupied states does not +contribute to the first-order densities, projecting out the +occupied subspace [8] can potentially improve the effi- +ciency and stability of the nested iteration. As an addi- +tional benefit of the projection, it also renders the prin- +ciple integrals explicit and amenable to analytic tech- +niques, which can further reduce the number of k-points +required and improve efficiencies. Alternative iterative +techniques should be tested in general, for both inner +and outer loops, especially in conjunction with the pro- +jection. +From a physics viewpoint, a few tasks are on immediate +agenda and new possibilities are clearly on the horizon, +beyond the initial demonstrations presented herein. For +the spin-wave spectral functions, it will be valuable to +compare the computed spectra with experimental results +for more materials. A particularly interesting comparison +can be made between the dispersion relations obtained ab +initio from our DFPT implementation and those from +Heisenberg models parametrized from constrained DFT +energies on the basis of the magnetic force theorem. [62– +64] Such comparisons should be examined in detail for +materials in the localized and the itinerant limits, as well +as for the continuum falling in between. Further system- +atic studies for the gradient correction (as in generalized +gradient approximations) and for the Hubbard correction +in LSDA+U method can reveal the effect of correlation +on the spin-wave spectra. As the first-order wavefunc- +tions are also produced in our code, it is also tempting to +evaluate other physical properties, related to density and +current responses, such as the magnetoelectric coupling +and related transport coefficients. A particular connec- +tion may be made by observing that +W = f + fχf +(39) +is the screened kernel, which now includes the charge, +spin and cross screening effects. +This will enable an- +alyzing the many-electron effects in magnetic materials +with strong spin-orbit coupling, and potentially evalu- +ating novel bound states from the screened charge/spin +interactions. +ACKNOWLEDGMENTS +We acknowledge the financial support from the Na- +tional Natural Science Foundation of China (Grant No. +11725415), the National Key R&D Program of China +(Grant Nos. +2018YFA0305601 and 2021YFA1400100), +and the Innovation Program for Quantum Science and +Technology (Grant No. 2021ZD0302600). +[1] R. Kubo, Statistical-mechanical theory of irreversible +processes. I. general theory and simple applications to +magnetic and conduction problems, J. Phys. Soc. 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Buczek, Spin dynamics of complex itinerant mag- +nets, Ph.D. thesis, Universit¨ats- und Landesbibliothek +Sachsen-Anhalt (2009). +[62] A. I. Liechtenstein, M. I. Katsnelson, and V. A. Gubanov, +Exchange interactions and spin-wave stiffness in ferro- +magnetic metals, J. Phys. F 14, L125 (1984). +[63] A. Liechtenstein, M. Katsnelson, V. Antropov, and +V. Gubanov, Local spin density functional approach to +the theory of exchange interactions in ferromagnetic met- +als and alloys, J. Magn. Magn. Mater. 67, 65 (1987). +[64] P. Bruno, Exchange interaction parameters and adiabatic +spin-wave spectra of ferromagnets: +A “renormalized +magnetic force theorem”, Phys. Rev. Lett. 90, 087205 +(2003). + diff --git a/StAyT4oBgHgl3EQfVfcN/content/tmp_files/load_file.txt b/StAyT4oBgHgl3EQfVfcN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..84ceabfafc023e96974a97aa7f86b7cafc6aa3a4 --- /dev/null +++ b/StAyT4oBgHgl3EQfVfcN/content/tmp_files/load_file.txt @@ -0,0 +1,892 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf,len=891 +page_content='An implementation of the density functional perturbation theory in the PAW framework Xiaoqiang Liu,1 Yihao Lin,1 and Ji Feng1, 2, ∗ 1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 2Hefei National Laboratory, Hefei 230088, China (Dated: January 3, 2023) Quantifying materials’ dynamical responses to external electromagnetic fields is central to un- derstanding their physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Here we present an implementation of the density functional perturbation theory for the computation of linear susceptibilities using the projector augmented- wave method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The Sternheimer equations are solved self-consistently through a nested iterative procedure to compute the first-order wavefunctions, from which the linear susceptibilities are ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As a demonstration, we compute the spin wave spectral functions of two magnetic metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The computed magnon spectra for half-metallic CrO2 and a Heusler intermetallic Cu2MnAl show gapless Goldstone modes when spin rotation symmetry is preserved and display reasonable agree- ment with available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The Landau damping is computed to be small in CrO2, but significant in Cu2MnAl producing an asymmetric Lorentzian spectral lineshape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The access to linear susceptibilities as well as first-order wavefunctions offers a range of novel possibilities in quantitative understanding of materials’ electronic properties from ab initio methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' INTRODUCTION A microscopic understanding of electrical and mag- netic characteristics of materials plays a key role in con- densed matter physics, furnishing a unified insight into a wide range of phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Indeed, the generalized den- sity response functions (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' susceptibilities) of a many- electron system to external electromagnetic fields, [1] in a broad sense encompasses the information of the usual longitudinal dielectric function and magnetic permeabil- ity, as well as the cross terms for electromagnetic cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The properties of collective excitations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' plas- mon and magnon, can also be procured from the sus- ceptibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Therefore, computing the susceptibilities, which involve both charge and spin degrees of freedom of electrons, is essential to a full characterization of elec- tronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Though relatively straightforward for a non-interacting system, computing the susceptibilities for an interacting many-electron system is a nontrivial task due to interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The Kohn-Sham density-functional theory (DFT) [2] is by far the most widely employed ground state electronic structure method for materials and molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' By map- ping the ground state energy to a non-interacting sys- tem described by Kohn-Sham equations, the Kohn-Sham ansatz enables the variational formulation of the static re- sponses of a many-electron system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The time-dependent (td) DFT [3] is subsequently developed, in which the elec- trodynamics is described by td Kohn-Sham equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In these theories, the Hartree and exchange-correlation potentials are functionals of density, and treated as self- consistent fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The self-consistent first-order perturba- tion in td DFT leads to the density functional perturba- tion theory (DFPT), [4] which is then the machinery for ∗ jfeng11@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='cn linear response calculations leading directly to the full response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' DFPT-based methods have been successfully applied to calculate the dielectric function [5, 6] and phonon dis- persion, [7, 8] with the results in quantitative agreement with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In the computations of dielectric function, the first-order wavefunctions with re- spect to k are solved, while deformation potential from frozen atomic displacements is used as the external field to compute phonon dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Dynamical responses to external electromagnetic fields from DFPT, which ac- count for the screening of both charge and spin, have at- tracted considerable interest recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [9–16] In addition to quantifying electrical and magnetic properties of ma- terials, the linear susceptibilities computed from DFPT also find applications in the GW approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [17–19] Approaches to the DFPT can be broadly grouped into two categories: a Dyson-like equation is solved in the first, [10, 11, 16] whereas the Sternheimer equations are solved in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [9, 12–15] The Dyson-like equation approach starts with response functions computed for the Kohn-Sham ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Though formally transparent and amenable to various iterative techniques, the Dyson- like equation approach suffers from two shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' It requires a large number of unoccupied states and huge planewave bases for adequate convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [16] More se- rious is the subtle basis set incompatibility between the Kohn-Sham states and the DFPT process, which gives rise to an artifactual spin excitation gap in systems with spin rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The latter problem can only be partially mended with delicate engineering of the inter- action kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [11, 20] In the second category, the first- order wavefunctions are procured (often iteratively) by solving the Sternheimer equations, from which the den- sity is updated with charge mixing iterations and the re- sponse functions are computed upon convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In this case, one is forced to deal with wavefunctions and var- ious pseudopotentials with all the technicalities, [21–24] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='00143v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='mtrl-sci] 31 Dec 2022 2 in addition to the nested iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Since the full Kohn-Sham response functions are never required, this method is free of the burden of summation over a huge number of empty states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In addition, the first- order wavefunctions and densities computed are actually bonus, which can be useful for computing a variety of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A particularly popular planewave-based approach to DFT is based on the projector augmented-wave (PAW) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [23, 24] Combining the formal simplicity of pseudopotentials and the versatility of the linearized augmented-planewave method, the PAW method offers both efficiency and accuracy to Kohn-Sham DFT calcula- tions on extended solids, and a wide range of capabilities in various implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [25–27] Despite its popular- ity, DFPT in the PAW framework has remained to be developed, which is accomplished in this work by solving the td Sternheimer equations to compute the linear sus- ceptibilities of crystalline materials accounting for both charge and spin degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The paper is orga- nized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' II A the general theory of DFPT is introduced, from the viewpoint of the dressed spin in td external electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The screening built on the notion of dressed spin leads to the Sternheimer equa- tions in the frequency and momentum domain and ex- plicit formula for the linear susceptibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' II B the PAW method is reviewed, based on which the formu- lation of DFPT in the PAW framework is described, along with a few implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As a first calibration our implementation, the spin wave spectral functions are extracted from the computed linear susceptibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Two examples are presented: half-metallic CrO2 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' III A) shows a clean spin wave spectrum and minimal Landau damping, and a full Heusler intermetallic Cu2MnAl (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' III B) shows significant Landau damping in the spin ex- citations that can be quantified with a simple asymmet- ric Lorentzian lineshape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Lastly, a summary is provided with an eye on room for development from algorithm and physics points of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' THEORY AND IMPLEMENTATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Density functional perturbation theory The td DFT offers an efficient description of the dy- namics of an interacting many-electron system in the presence of external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [3] As a self-consistent per- turbation theory of td DFT, DFPT is introduced in this section, wherein the Sternheimer equations are special- ized to crystalline systems under a monochromatic, peri- odic external electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In td DFT, to account for both charge and spin degrees of freedom, the generalized density is given by ρ(r, t) = � n θn tr{ψ† n(r, t)σψn(r, t)} = (ρ0, m) = (ρ0, ρ1, ρ2, ρ3) (1) where θn is the occupancy of the spinor single-particle state ψn, and the four-vector spin σ = (σ0, σ1, σ2, σ3) (σ0 is the identity matrix and σα with α = 1, 2, 3 are the Pauli matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The atomic units [28] are adopted in this paper, so ρ0 is the total charge density, and m is magnetization density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The dynamics of ψn is prescribed by the td Kohn-Sham equations [3] i∂t|ψn(t)⟩ = [H + δH(t)]|ψn(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (2) The ground state Kohn-Sham Hamiltonian [2] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (2) is H = − 1 2∇2 + v[ρ(0)](r) where the self-consistent po- tential is a functional of the ground state density ρ(0), composed of ionic, Hartree and exchange-correlation (xc) potentials, namely, vi, vH and vxc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Though the xc poten- tial vxc as a functional of density is in principle nonlocal in space and time, [29–33] the commonly adopted local and adiabatic approximation (ALDA) is assumed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [34–36] The first-order Hamiltonian δH comprises two contri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The first arises from the coupling of the four- vector spin σ with external fields vext(r, t) = −Bα(r, t)σα, (3) where the four-vector electromagnetic field B(r, t) = (−φ, 1 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [37] The indices α, β = 0, 1, 2, 3 are implicitly summed over when repeated, but we will keep other sum- mations explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The self-consistent inclusion of the den- sity dependence in the Hartree and xc potentials means that δH also includes a second contribution from induced density δρ(r, t) that screens vext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In the adiabatic linear response theory, this can be formulated in terms of a dressed spin τα τα(r, r′, t − t′) = − δH(r, t) δBα(r′, t′) = σαδ(r − r′)δ(t − t′) − σγ � fγβ(r, r′′)χβα(r′′, r′, t − t′)dr′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (4) Here χαβ(r, r′, t) is the linear susceptibility that we pur- sue in this work, defined via δρα(r, t) = � χαβ(r, r′, t − t′)Bβ(r′, t′)dr′dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (5) The interaction kernel has two components, fαβ = f H αβ + f xc αβ, namely, the Hartree and xc kernels in the ALDA fαβ(r, r′) = δα0δβ0 |r−r′| + 1 2δ(r − r′) tr � σα ∂vxc ∂ρβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (6) In terms of the dressed spin, the first-order Hamiltonian in unscreened external fields is written δH(r, t) = − � τα(r, r′, t − t′)Bα(r′, t′)dr′dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (7) Now with the first-order Hamiltonian, the first-order wavefunctions can be obtained by solving the Stern- heimer equations [8, 9, 12, 38–40] (i∂t − H)|ψ(1) n (t)⟩ = δH(t)|ψ(0) n (t)⟩, (8) 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Flow chart for a nested-loop DFPT calculation in the Sternheimer equation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The outer loop depicted here is the charge mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The inner loop is incurred in solving the Sternheimer equations in each step of the outer iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' in which ψ(ℓ) n (t) is the ℓth-order wavefunction, with |ψ(0) n (t)⟩ = e−iεnt|ψn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' With the first-order wavefunc- tions, we will be able to compute the induced density via the variation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (1), from which the first-order Hamiltonian will be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' With the above introduction, a DFPT calculation can then be performed in a nested iterative process depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In the initializing step, δH is constructed from the external electromagnetic fields, whence the Stern- heimer equations are solved in the inner iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' This produces a set of first-order wavefunctions, from which the induced density is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A charge mixing strat- egy [41, 42] is employed to revise the induced density, with the linear susceptibility and dressed spin computed subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Then a new δH is constructed from the up- dated spin to enter a second round of the outer iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The above process is repeated till convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Now we will explain how the Sternheimer equations are solved in a crystalline solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For electrons in a crystal, the initial Kohn-Sham states are Bloch functions such that |ψn⟩ �→ |ψnk⟩ = eik·r|unk⟩, where n is the band index and k is the crystal momentum, and unk is the cell-periodic part of the Bloch function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The system is subject to spatially periodic and monochromatic external electromagnetic fields Bα(r, t) = Bαei(p·r−ωt) + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=', (9) where p = q + g with g being a reciprocal lattice vector for q to in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In this case, the following expansion is useful ζ(r, t) = � l ei(l·r−νt)ζ(r, l) (10) for ζ = δH, δρα, in which ζ(r, l) is a cell-periodic function with l ≡ (ν, l) = ±(ω, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Then the first-order Hamiltonian in the l channel is δH(r, l) = −e−il·rτα(r, sign(νω)p, ν)Bα(l), (11) where Bα(ω, q) = Bα, Bα(−ω, −q) = (Bα)∗, and τα(r, p′, ν) = � τα(r, r′, t)ei(p′·r′+νt)dr′dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (12) The first-order wavefunction is expanded as |ψ(1) nk(t)⟩ = � dν 2π e−i(ν+εnk)t|ψ(1) nk(ν)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (13) For the fields in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (9), |ψ(1) nk(ν)⟩ is nonzero only when ν = ±ω and can be written as |ψ(1) nk(ν)⟩ = ei(k+l)·r|u(1) nk(l)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (14) Then the Sternheimer equations become (ν + iη + εnk − Hk+l) |u(1) nk(l)⟩ = δH(r, l)|unk⟩, (15) in which Hk = e−ik·rHeik·r is the Bloch Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A positive infinitesimal η is introduced on the left-hand side as a convergence factor to embody the causality struc- ture of the linear susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In actual calculations, η takes finite values to ensure convergence with finite k- mesh especially for metals and bestows finite broadening on spectral peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' From the first-order wavefunctions, we compute the first-order induced density as δρα(r, l) = � nk θnk tr � u† nk(r)σαu(1) nk(r, l) + u(1) nk †(r, −l)σαunk(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (16) The linear susceptibility can be extracted from the Fourier components of δρα(r, l) δρα(g′, l) = χαβ(g′, sign(νω)g, l)Bβ(l), B op 8H(r, charge Sternheimer equations (io,-H)(1) =8H(0) mixing No convergence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Yes4 whence χαβ(g′, g′′, l) = � e−i(g′+l)·r′χαβ(r′, r′′, ν)ei(g′′+l)·r′′dr′dr′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (17) It is worth mentioning that the Sternheimer equations in (15) admit the following formal solutions in the l channel |u(1) nk(l)⟩ = −Bα(l) � n′ |un′k+l⟩τα(k, l)n′n ν + εnk−εn′k+l + iη , (18) in which the dressed spin matrix element is τα(k, l)n′n = � u† n′k+l(r)e−il·rτα(r, sign(νω)p, ν)unk(r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (19) Because it requires the summation over a large number of empty states, this formal solution is not used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [43] The screened susceptibility then has the same expression as the (bare) Kohn-Sham susceptibility, except that the (unscreened) external fields are now coupled to the dressed spin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' that is, χαβ(g′, sign(νω)g, l) = − � nn′k (θnk − θn′k+l)⟨unk|e−ig′·rσα|un′k+q⟩τβ(k, l)n′n ν + εnk − εn′k+l + iη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (20) In arriving at the last expression, we have used the fact that τα(k, l)n′n = τα(k + l, −l)∗ nn′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' DFPT with PAW method The previous Subsection presents a sketch of the DFPT for periodic systems without recourse to computational details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In practical calculations, however, various tech- nologies have been developed such that one can per- form DFT (and therefore DFPT) calculations on valence electrons only for crystalline materials using planewave bases with the aid of pseudopotentials, to yield satisfac- tory accuracy with high efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' It is well known that the norm-conserving pseudopotential [21] requires large planewave bases particularly for localized orbitals in transition elements, while the application of ultrasoft pseudopotential [22] is partly limited by the rather labo- rious construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In contrast, the PAW method, com- bining the pseudopotential and linearized augmented- plane-wave methods, is free from the above difficulties and has been used widely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Thus, performing DFPT within the PAW framework [23, 24] for inhomogeneous and td electromagnetic fields is evidently useful though notably nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' DFPT with the PAW method has been implemented within the Vienna ab initio simula- tion package (VASP) [25] for atomic displacements in the static and long-wavelength limit to calculate zone- center phonon energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The extension to inhomogeneous and td electromagnetic fields is accomplished in this work based on VASP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='4, and a few implementation details warrant further clarification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Here, we will briefly re- view the PAW method and describe how it is used in our DFPT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Although the formalism for the ground state quantities is identical to those in the lit- erature and notationally notorious, we feel compelled to provide some of these details, particularly in view of the time- and position-dependent external fields involved in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The PAW method is based on a linear transformation between the all-electron (AE) Hilbert space orthogonal to core states and pseudo (PS) Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The AE and PS wavefunctions are related by |ψnk⟩ = T| ˜ψnk⟩, (21) with the linear operator defined as T = 1 + � i (|φi⟩ − |˜φi⟩)⟨˜pi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (22) The index i is a shorthand encapsulating the atomic site located at Ri as well as the quantum numbers (nlm) of the local orbitals and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' φ, ˜φ and ˜p are AE partial waves, PS partial waves and projector functions, respec- tively, and should all be understood as spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In order to perform the DFPT calculations on a crystalline solid in the electromagnetic fields in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (9), the key is to find the cell-periodic part of each PAW-pseudized quantity, especially the nonlocal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Upon application of the time-independent transforma- tion T to the first-order wavefunctions the pseudized Sternheimer equations read (i∂tS − ˜H)| ˜ψ(1) nk(t)⟩ = δ ˜H(t)| ˜ψ(0) nk(t)⟩, (23) with S = T †T, ˜H = T †HT and δ ˜H(t) = T †δH(t)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' According to the implementation of PAW method in 5 VASP, [24] we have S = 1 + � ij |˜pi⟩ qij ⟨˜pj| , ˜H = −1 2∆ + ˜veff + � ij |˜pi⟩ Dij ⟨˜pj| , (24) in which the nonlocal potential Dij = ˆDij + ˜Dij with ˜Dij = D1 ij − ˜D1 ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The quantities qij, D1 ij and ˜D1 ij are defined in reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [24] Apparently, the local potential ˜veff(r) is a functional of pseudo density ˜n(r) and com- pensation density ˆn(r), while ˜Dij is a function of density matrix ϱ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' ˜veff =˜veff[˜n + ˆn], ˆDij = � 1 2 tr{σα˜veff(r)}Qα ij(r)dr, ˜Dij = ˜Dij(ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (25) We have hidden the functional dependence on the pseudized core densities in ˜veff(r), which are kept frozen during the DFPT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' ˜n(r), ˆn(r) and ϱij are given by, respectively, ˜n(r) = � nk θnk tr{ ˜ψ† nk(r)σ ˜ψnk(r)}, ˆn(r) = � i,j ϱijQij(r), ϱij = � nk θnk⟨ ˜ψnk|˜pi⟩⟨˜pj| ˜ψnk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (26) Here Qij(r) = � L tr{σ ˆQL ij(r)} with ˆQL ij(r) defined to construct the compensation density ˆn(r) in the refer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [24] It should be noticed that for ϱij, only elements with Ri = Rj are useful in our calculations, while Dij are nonzero only when Ri = Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Now we derive the expression for the first-order Hamil- tonian δ ˜H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For the fields in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (9), the first-order local densities and potentials follow the same expansion as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (10), while the first-order density matrix δϱij and nonlocal potential δDij can be expanded as δζij(t) = � l ei(l·Ri−νt)δζij(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (27) Here, the factor eil·Ri in the l channel is introduced such that δζij(l) is cell-periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=', δζij(l) = δζi′j′(l) if the positions of atomic site for i, j and i′, j′ differ by a lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The contribution of external electromagnetic fields in δ ˜H(t) can be calculated directly δ ˜Hext(r, t) = � l e−iνt � eil·rvext(r, l) + � ij eil·Ri |˜pi⟩ Dext ij (l) ⟨˜pj| � , (28) with vext(r, l) = − Bα(l)σαesign(νω)ig·r, Dext ij (l) =⟨φi|eil·(r−Ri)vext(r, l)|φj⟩ − ⟨˜φi|eil·(r−Ri)vext(r, l)|˜φj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (29) The contribution to δ ˜H(t) from the induced densities has a similar expression as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (28) and can be calculated via an explicit finite difference, as ˜H is a functional of ˜n, ˆn and ρij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The first-order densities are found to be δ˜n(r, l) = � nk θnk tr{˜u† nk(r)σ˜u(1) nk(r, l) + ˜u(1),† nk (r, −l)σ˜unk(r)}, δˆn(r, l) = � i,j eil(Ri−r)δϱij(l)Qij(r), δϱij(l) = � nk θnk[⟨˜unk|˜pik⟩⟨˜pjk+l|˜u(1) nk(l)⟩ + ⟨˜u(1) nk(−l)|˜pik−l⟩⟨˜pjk|˜unk⟩], (30) where we define |˜pik⟩ = e−ik·(r−Ri)|˜pi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' To linear order in external fields, the first order effective local potential is given by δ˜veff(l) ≈ ˜veff[˜n + ˆn + δ˜n(l) + δˆn(l)] − ˜veff[˜n + ˆn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (31) Similarly, the first-order nonlocal potentials can be approximated as δ ˆDij(l) ≈ � eil·(r−Ri) 1 2 tr{σαδ˜veff(r, l)}Qα ij(r)dr, δ ˜Dij(l) ≈ ˜Dij(ϱ + δϱ(l)) − ˜Dij(ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (32) 6 Though introduced as forward differences in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (31) and (32), these quantities are evaluated using 4th-order centered finite differences, with a step length of a thousandth the density variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' With the above results, the final Sternheimer equations become � νSk+l + εnkSk+l − ˜Hk+l � |˜u(1) nk(l)⟩ = δ ˜Hk(l)|˜unk⟩, (33) with Sk+l = 1 + � ij |˜pik+l⟩ qij ⟨˜pjk+l| , ˜Hk+l = −1 2∆k+l + ˜veff + � ij |˜pik+l⟩ Dij ⟨˜pjk+l| , δ ˜Hk(l) = vext(l) + δ˜veff(l) + � ij |˜pik+l⟩ [Dext ij (l) + δ ˆDij(l) + δ ˜Dij(l)] ⟨˜pjk| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (34) Here ˜unk and ˜u(1) nk(l) are the cell-periodic parts of corresponding pseudo wavefunctions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The pseudized Sternheimer equations in ±(ω, q) channels are solved separately in each iteration using a variant of residual minimization method with a direct inversion in the iterative subspace (RMM-DIIS), [44, 45] which is already implemented in VASP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The L¨owdin perturbation theory is also performed to correct the first-order wavefunctions in the subspace of occupied states and low-lying excitations to speed up convergence |˜u(1) nk(l)⟩ → |˜u(1) nk(l)⟩ − � n′ |˜un′k+l⟩⟨˜un′k+l|Sk+l|˜u(1) nk(l)⟩ + � n′ |˜un′k+l⟩⟨˜un′k+l|δ ˜H|˜unk⟩ ν + εnk − εn′k+l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (35) In the last equation above, the summations on n′ run over the occupied bands plus a few empty bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Apparently, solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (33) requires a k-grid supple- mented by two additional grids shifted by ±q when q itself is not on the k-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Doing so, however, not only increases the computational burden but also, more seri- ously, obliterates the exact cancellation of the contribu- tion of the occupied manifold to the density change due to a k-grid discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The latter can be eas- ily avoided by employing a pair of grids with a q shift, which also reduces the calculation partly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (33) is solved on the k-grid in +q channel, and on the k + q grid in −q channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' It is observed that in this dual grid setup, the above cancellation is well preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The xc potentials in ALDA are functionals of real- valued densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Thus, calculating δ˜veff(l) like in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (31) requires caution as δ˜n(l) and δˆn(l) are usually com- plex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In fact, the real and imaginary part of δ˜veff(l) are calculated separately, Fδ˜veff(l) ≈ ˜veff[˜n+ˆn+Fδ˜n(l)+Fδˆn(l)]−˜veff[˜n+ˆn] (36) where F = Re, Im takes the real or imaginary part, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In the case of nonlocal potential, δ ˜Dij(l) and δϱij(l) are first decomposed into two independent Her- mitian matrices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Hermitian part and anti-Hermitian part multiplied by −i), and then finite differenced sepa- rately in an analogous fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Symmetry reduction is also performed in our imple- mentation, where the summation over k points in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (30) is restricted to the symmetry-irreducible part of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The symmetry group here is the subgroup of the magnetic group of the studied crystal in which the external electromagnetic fields in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (9) is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' APPLICATION TO SPIN-WAVE SPECTRUM CALCULATION Our implementation enables computing the linear sus- ceptibilities χαβ with the self-consistent inclusion of the interaction kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Directly inverting χαβ yields the di- electric tensor, which is composed of the usual charge sector ϵ00, spin sector ϵαβ, and the spin-charge sector ϵ0β, each embodying unique physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Computing χαβ then can have diverse applications in evaluating materi- als properties pertaining to both charge and spin fluctu- ations, or in subsequent many-body calculations beyond the Kohn-Sham mean fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' One immediate applica- tion that has received considerable attention is the cal- culation of spin-wave excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [9–16] According to the fluctuation-dissipation theorem, the spin-spin correlation function, directly accessible by various spin-sensitive in- elastic scattering probes, [46–48] is related to the imagi- nary part of the linear susceptibilities, S+−(p, ω) = Im χ+−(g, g, ω, q) 1 − e−ℏω/kBT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (37) 7 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Reported implementations of DFPT for spin wave spectra calculations by solving the Sternheimer equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Authors (Year) Potential Basis set Software Savrasov (1998) [9] Full potential LMTO1 LMTO Magnons Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (2018) [12] NCPP2, USPP3 Planewave QE4 Gorni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (2018) [13] NCPP Planewave QE Tancogne-Dejean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (2020) [14] NCPP Real space Octopus Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (2020) [15] Full potential LAPW5 Elk 1Linear muffin-tin orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2Norm-conserving pseudopotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 3Ultrasoft pseudopotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 4Quantum ESPRESSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 5Linearised augmented-plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Although for magnetic systems dominated by local mo- ments the magnon can be described effectively by lo- calized spin models, this method is subject to debate when delocalization sets in, and ultimately of question- able validity for itinerant magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In these latter cases, which include a wide range of magnetic mate- rials, the DFPT route becomes invaluable for comput- ing the spin wave spectra ab initio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' To our knowledge, there have been just a handful of works devoted to im- plementing DFPT scheme for this purpose by solving the Sternheimer equations, as summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In these efforts, implementations are limited to the full- potential, [9, 15] or norm-conserving and ultrasoft pseu- dopotentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [12–14] In this section, we present an initial application of our implementation of DFPT in the PAW framework to the calculations of spin-wave spectra for a couple of mag- netic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For ferromagnets with the spin polar- ized along z direction, transverse magnetic field can be applied by choosing B = (0, 1, −i, 0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (9), from which χ+− is calculated directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In these cases, the only remaining symmetry operation keeping the crys- tal and the transverse magnetic field unchanged is iden- tity transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Thus, there is no room for sym- metry reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For notationaly convenience, we define χ+−(p, ω) ≡ χ+−(g, g, ω, q) as p = q + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Half-metallic chromium dioxide As shown in the inset in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2, chromium dioxide, CrO2, is a ferromagnetic half-metallic oxide with a ru- tile crystal structure, where each Cr atom is situated at the center of an octahedral cage formed by oxygen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [49, 50] Widely used as a magnetic recording ma- terial, CrO2 also has various potential applications in spintronics and magnetoelectronics [51, 52] due to its half-metallic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The experimental lattice parameters a = b = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='4218 ˚A and c = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='9182 ˚A [50] are used in our calcu- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The planewave energy cutoff is set to be 500 eV and a 13 × 13 × 20 Γ-centered mesh of k-points is FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Spin-resolved densities of states of CrO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The spin- flip gap is found to be about 310 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Inset: the crystal structure of CrO2, highlighting the CrO6 octahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The spin-resolved density of states of CrO2 is com- puted from the collinear spin-polarized calculation and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2, where the half-metallic band structure is clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The magnetic moment of Cr is found to be 2 µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' We then turn to the noncollinear calculations, and compute the transverse spin susceptibility χ+−(p, ω) along [100] and [001] directions for ω ≤ 400 meV on 10 meV intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The broadening parameter η introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (15) is set to be 50 meV in the calculations in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 3(a) shows the computed Imχ+−(p, ω), without spin-orbit interaction, along two p paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In general, χ+−(p, ω) is not periodic in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Then the branch in the first Brillouin Zone is composed of acoustic magnon modes, while the branch in the second Brillouin zone op- tical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The profile of the magnon peak at a given p is nearly perfect Lorentzian over the entire energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The extracted half width at half maximum ηp are almost a constant and equal to the artificial broadening param- eter η, indicating that the Landau damping in CrO2 is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' This is expected given that half-metallic CrO2 has a large spin-flip gap around 310 meV, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' up down N(e) (States/eV/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') 5 3 2 0 28 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (a) Imχ+−(p, ω) of CrO2 along [100] and [001] di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The black vertical line indicates the center of the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (b) The folded magnon energy dispersion ωq of CrO2 in the first Brillouin zone, extracted from Im χ+− shown in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The inset shows the quadratic fits to ωq at small q along [100] and [001] directions, respectively, for cal- culations without Hubbard U correction and with U eff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The squares are the calculated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The location of the maxima of magnon peaks, ωp, are then recorded and folded to the first Brillouin zone (ωq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 3(b), we find one acoustic magnon branch and one optical branch, consistent with the fact that the unit cell in CrO2 contains two mag- netic Cr atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' There is no magnon gap at the Brillouin zone boundaries, which is result of the n glide symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The energy of long-wavelength acoustic magnons is quadratic in q with a gapless Goldstone mode, ωq = D∥(q2 x + q2 y) + Dzq2 z, as expected for a ferromagnet with spin rotation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The spin stiffness coefficients are found to be D∥ = 82 meV·˚A2 along [100] and Dz = 92 meV·˚A2 along [001] directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' From this, we can estimate the average spin stiffness coefficient to be 85 meV·˚A2, which is close to the experimental measured result (∼ 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 meV·˚A2 [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For comparison, additional electron correlation is in- cluded statically within the LSDA+U formalism [54] in both the ground state calculation and subsequent DFPT calculations, with U eff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [55] As shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 3(b), the gapless Goldstone mode is ob- tained again, accompanied by an average spin stiffness FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (a) Imχ+−(p = 0, ω) as a function of ω for different λ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' λ = 1 corresponds to the actual strength of spin-orbit coupling in CrO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The squares are the calculated data and the solid lines are the fits with Lorentzian line shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (b) The Goldstone gap as a function of λ2 in CrO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The solid line is the linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' coefficient D = 391 meV·˚A2, almost five times as large as the one without Hubbard U correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The energy of magnon in CrO2 seems to be highly overestimated in LSDA+U calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As a further test, we examine the Goldstone gap as a result of breaking the spin rotation symmetry by in- troducing spin-orbit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The atomic spin-orbit interaction for Cr is fairly weak (on the order of tens of meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Since the gap in the Goldstone magnon is second order in the spin-orbit coupling, it is small for CrO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In order to visualize the effect of spin-orbit coupling, we in- troduce a parameter λ to artificially tune its strength (or speed of light), as in H = H0 + λHsoc, where λ = 1 cor- responds to the actual strength of spin-orbit coupling in CrO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The calculated Imχ+−(p = 0, ω) as a function of ω for different λ values are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Apparently there is blue shift of the Goldstone mode with increas- ing λ, indicating the emergence of a Goldstone gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The Goldstone gap indeed shows a quadratic dependence on λ, as demonstrated by the gap-vs-λ2 plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The extrapolated Goldstone gap in CrO2 is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='1 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Heusler intermetallic Cu2MnAl The ternary intermetallic Cu2MnAl is a Mn-based full-Heusler alloy with the L21 structure type (see in- set in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The experimental lattice parameter for the conventional cubic cell (space group Fm¯3m) is a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='968 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [56] Cu2MnAl is ferromagnetic below the relatively high Curie temperature (603 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [56] Apart from being regarded as a prototype for understanding the electronic correlations in Heusler intermetallics, [57] Cu2MnAl is also being used as a neutron polarizer and monochromator material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [58, 59] A planewave energy cutoff of 350 eV and a 15×15×15 Γ-centered k-grid are used in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Spin- orbit coupling is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The spin-resolved den- sities of states of Cu2MnAl computed from the collinear (a) X102 400 4 300 3 Imx+-(p, w) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') (meV) 200 2 3 100 1 0 (2,0,0) (0,0,0) (0,0,2) (b) 400 350 300 250 80 (meV) U 0eV 口 200 60 U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='1 eV 40 20中 口 100 0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='3, 0, 0) (0, 0, 0) (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='2) 50 0 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5, 0, 0) (0, 0, 0) (0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5)(a) X102 (b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 40 >=0 Imx+-(p = O, w) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') =1 — >= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 >=5 30 日 >= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 neV) =10 中 >= 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 00 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 0 0 20 40 0 52 102 152 w (meV) 129 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Spin-resolved densities of states of Cu2MnAl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In- set: the crystal structure of full Heusler Cu2MnAl, showing a conventional cubic unit cell for the L21 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' spin-polarized calculation confirms the ferromagnetism of Cu2MnAl, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The magnetic moment is carried primarily by Mn atoms and computed to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='4 µB/Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The transverse spin susceptibility χ+−(p, ω) along [100], [110] and [111] crystallographic directions is then computed in noncollinear calculations for ω ≤ 300 meV on 5 meV intervals, with a broadening parameter η of 50 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 6(a) shows the computed magnon spectral func- tion Imχ+−(p, ω) along the three principal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The acoustic magnon branch is seen clearly only at low energies near the Brillouin zone center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The spectral peaks of these low-energy modes can be adequately fit- ted with the Lorentzian lineshape as in the CrO2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The dispersion of the long-wavelength modes is quadratic and isotropic, as demonstrated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A spin stiffness coefficient D = 268 meV·˚A2 is procured from the quadratic fit, which is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 times larger than the experimental value of 175 meV·˚A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [60] Notably, at higher energies and near Brillouin zone boundaries, the magnon peaks become fuzzier and broader, attesting to substantial Landau damping in this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' In stark contrast to the CrO2 case with almost no Landau damping, the coupling to the Stoner contin- uum in Cu2MnAl bestows a magnon peak at a given p an asymmetric profile that defies a Lorentzian fit, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The stronger Landau damping in this sys- tem is consistent with the absence of the spin-flip gap as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Viewing the coupling to the Stoner continuum as a Fano-type resonance, we superimpose a linear function on the Lorentzian to describe the asym- metric line shape, as A(p, ω) = apηp (ω − ωp)2 + η2p + ξp(ω − ωp) (38) with ωp, ap, ηp and ξp as fitting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As it turns out, this simple modification leads to sat- isfactory fitting for the entire spectrum, as evidenced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The extracted magnon dispersion is then shown FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (a) Imχ+−(p, ω) of Cu2MnAl along [100], [110] and [111] directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Reciprocal lattice vectors of conventional cubic cell are adopted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (b,c) Imχ+−(p, ω) as a function of ω for p = (1, 0, 0), (1, 1, 0) and (1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The squares are the calculated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The solid lines are the fits with (b) symmetric or (c) asymmetric Lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 7(a), which coincides with the calculated results of Buczek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [10, 61] and agrees well with the ex- perimental observations along [100] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [60] Along the [110] and [111] directions where the Landau damp- ing seems more pronounced, our computed dispersion shows significant discrepancy from the experimental one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For low-energy modes, ηp is dominated by the artificial broadening parameter η and the asymmetry is small, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' With increasing energies, however, the broadening quickly exceeds η and the asymmetry be- comes pronounced, especially near Brillouin zone bound- aries, both providing quantitative characterization of the Landau damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' SUMMARY AND OUTLOOK In conclusion, we report an implementation of the DFPT method in the PAW framework, which is capable of computing the full linear susceptibilities of real mate- rials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A nested iterative procedure is employed to self- consistently solve the Sternheimer equations, to procure linear susceptibilities along with the first-order wavefunc- tions and densities in monochromatic and periodic exter- nal electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The time cost of each DFPT calculation (given an external field direction, momentum and frequency) is comparable to that of a corresponding Kohn-Sham DFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' up down Mn N(e) (States/eV/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') Al 5 3 2 1 0 2(a) ×102 300 3 250 Imx+-(p, w) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') 200 2 (meV) 150 3 100 50 0 (0,0,0) (1,0,0)/(1,1,0) (0,0,0) (1,1,1) (b) (c) ×102 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 Imx+-(p, w) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') Imx+-(p,w) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=') (1,0,0) (1,0,0) (1,1,0) (1,1,0) (1,1,1) (1,1,1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 0 0 0 100 200 300 0 100 200 300 (meV) w (meV) 310 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (a) The magnon energy dispersion ωp of Cu2MnAl along [100], [110] and [111] directions, extracted from the asymmetric Lorentzian fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' △ is the experimental data of Tajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [60] Inset is the quadratic fit to the long- wavelength acoustic magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' (b) Broadening ηp and asym- metry ξp of the magnon peaks in Cu2MnAl along [100], [110] and [111] directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The error bars show 95% confidence bounds on the fitting parameters from the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As a demonstration, we compute the spin wave spec- tra for CrO2 and Cu2MnAl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Gapless magnon disper- sion is obtained for both materials from the calculations without spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The spin stiffness coeffi- cient extracted from the quadratic fit is in agreement with experimental value for CrO2 but 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content='5 times larger for Cu2MnAl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' The Landau damping in CrO2 is insignifi- cant due to its half-metallic nature, while in Cu2MnAl is remarkable at high energies and can be quantified with a simple asymmetric Lorentzian fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' LSDA+U method as well as the effect of spin-orbit coupling are examined in CrO2, from which the former highly overestimates the magnon energy, while the latter gives rise to a Goldstone gap quadratic in spin-orbit coupling strength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' There is clearly room for future developments, to make the current implementation more efficient and versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' From an algorithm viewpoint, the occupied subspace is not projected out in Sternheimer equations in the cur- rent implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As the contribution to the first- order wavefunctions from the occupied states does not contribute to the first-order densities, projecting out the occupied subspace [8] can potentially improve the effi- ciency and stability of the nested iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As an addi- tional benefit of the projection, it also renders the prin- ciple integrals explicit and amenable to analytic tech- niques, which can further reduce the number of k-points required and improve efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Alternative iterative techniques should be tested in general, for both inner and outer loops, especially in conjunction with the pro- jection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' From a physics viewpoint, a few tasks are on immediate agenda and new possibilities are clearly on the horizon, beyond the initial demonstrations presented herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' For the spin-wave spectral functions, it will be valuable to compare the computed spectra with experimental results for more materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A particularly interesting comparison can be made between the dispersion relations obtained ab initio from our DFPT implementation and those from Heisenberg models parametrized from constrained DFT energies on the basis of the magnetic force theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [62– 64] Such comparisons should be examined in detail for materials in the localized and the itinerant limits, as well as for the continuum falling in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Further system- atic studies for the gradient correction (as in generalized gradient approximations) and for the Hubbard correction in LSDA+U method can reveal the effect of correlation on the spin-wave spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' As the first-order wavefunc- tions are also produced in our code, it is also tempting to evaluate other physical properties, related to density and current responses, such as the magnetoelectric coupling and related transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' A particular connec- tion may be made by observing that W = f + fχf (39) is the screened kernel, which now includes the charge, spin and cross screening effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' This will enable an- alyzing the many-electron effects in magnetic materials with strong spin-orbit coupling, and potentially evalu- ating novel bound states from the screened charge/spin interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge the financial support from the Na- tional Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 11725415), the National Key R&D Program of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2018YFA0305601 and 2021YFA1400100), and the Innovation Program for Quantum Science and Technology (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' 2021ZD0302600).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfVfcN/content/2301.00143v1.pdf'} +page_content=' Kubo, Statistical-mechanical theory of irreversible processes.' metadata={'source': 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Kokkotas3,5 +1Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy +2INFN, Sezione di Roma, Piazzale Aldo Moro 2, 00185, Roma, Italy +3Theoretical Astrophysics, IAAT, University of T¨ubingen, 72076 T¨ubingen, Germany +4Physics Department, University of Trento, Via Sommarive 14, 38123 Trento, Italy and +5Section of Astrophysics, Astronomy, and Mechanics, Department of Physics, +University of Athens, Panepistimiopolis Zografos GR15783, Athens, Greece +The growing capacity of gravitational-wave astronomy and black-hole imaging will soon enable +us to emphatically decide if astrophysical compact objects in galactic centers are black holes. Sgr +A*, one of the most prolific astronomical radio sources in our galaxy, is the focal point for tests of +general relativity. Current mass and spin constraints predict that Sgr A* is supermassive and slowly +rotating, thus can be conservatively modeled as a Schwarzschild black hole. Nevertheless, the well- +established presence of accretion disks and astrophysical environments around supermassive compact +objects can significantly deform their geometry and complicate their observational scientific yield. +Here, we study extreme-mass-ratio binaries comprised of a minuscule secondary object inspiraling +onto a supermassive Zipoy-Voorhees compact object; the simplest exact solution of general relativity +that describes a static, spheroidal deformation of Schwarzschild spacetime. We examine geodesics +of prolate and oblate deformations for generic orbits and reevaluate the non-integrability of Zipoy- +Voorhees spacetime through the existence of resonant islands in the orbital phase space. By including +radiation loss with post-Newtonian techniques, we evolve stellar-mass secondary objects around a +supermassive Zipoy-Voorhees primary and find clear imprints of non-integrability in these systems. +The peculiar structure of the primary, allows for, not only typical single crossings of transient +resonant islands, that are well-known for non-Kerr objects, but also inspirals that transverse through +several islands, in a brief period of time, that lead to multiple glitches in the gravitational-wave +frequency evolution of the binary. The detectability of glitches with future spaceborne detectors can, +therefore, narrow down the parameter space of exotic solutions that can, otherwise, cast identical +shadows with black holes. +I. +INTRODUCTION +The Schwarzschild spacetime [1] is unequivocally the +simplest and most remarkable black hole (BH) solution +of general relativity. +It describes a vacuum compact +object with an event horizon and a static, spherically- +symmetric exterior. The remarkable symmetry proper- +ties of Schwarzschild geometry places it at the top of +Einstein field equation solutions in what regards its sim- +plicity and singular externally-observable property; the +gravitational mass. +Its ultimate successor, the Kerr metric [2], is undoubt- +edly the most successful and astrophysically-relevant so- +lution of the vacuum field equations that describes a spin- +ning, stationary and axisymmetric BH with an oblate, +spheroidal external geometry. Despite the fact that Kerr +BHs possess less symmetries than Schwarzschild space- +times, they compensate by including spin; a very crucial +aspect of most astrophysical compact objects, and fur- +ther form an integrable (separable) system of equations +of motion for massless and massive particles due to the +existence of the Carter constant [3]. The above state- +ment of integrability translates to the absence of chaos +in geodesics around Kerr BHs, and is traced trivially to +Schwarzschild spacetime [4]. +From an astrophysical perspective, a significant volume +that surrounds BHs in the Universe consists of plasma, +accretion disks, matter configurations and halos that ex- +tend significantly far away. Thus, it is hard for one to +realize BHs in pure vacuum, especially those occupying +the centers of galaxies, where a perplex and highly dy- +namical environment is present. +The historic observations of shadows from supermas- +sive compact objects in the center of M87* galaxy [5] +and Sgr A* in our galaxy [6] has opened a new realm of +observational yield with BH imaging. An accretion disk +can, in principle, deform the surrounding geometry of a +BH so that it continuously deviates from a Schwarzschild +or Kerr description, causing degeneracies between a mul- +titude of other exact, though more exotic, solutions that +may mimic the observed shadow of supermassive com- +pact objects. +To study potential degeneracies present in current elec- +tromagnetic observations, the literature usually operates +in spacetime deviations from the Kerr description, known +as bumpy/parameterized [7–15] or non-Kerr compact ob- +jects [16–22]. A distinctive feature of some of these ob- +jects is the absence of Carter symmetry, which leads to +chaotic phenomena in particle-dynamics. +In what re- +gards Sgr A*, though, spin is not as crucial as the devi- +ation from spherical symmetry itself, since current con- +straints on its spin conclude that it is less than 10% of +its maximal allowed [23, 24]. Therefore, a plethora of Sgr +A*-like objects in the Universe can be sufficiently mod- +eled as Schwarzschild (or slowly-rotating Kerr) BHs or +exotic BH mimickers. +arXiv:2301.11483v1 [gr-qc] 27 Jan 2023 + +2 +Gravitational-wave +(GW) +astrophysics +has +been +proven to be an extraordinary tool to break the afore- +mentioned degeneracies, thus synergies between GW +and shadow observations are necessary in order to +search for the ultimate spacetime description of known +astrophysical compact objects. +To that end, +the +LIGO/Virgo/Kagra collaboration has been flooding our +databases with numerous GW detections from coalesc- +ing compact objects [25]. Such detectors, although ex- +tremely successful so far (at the level of groundbreaking), +have the unfortunate attribute of being plagued by plan- +etary noise, since they are placed on Earth, and present +unavoidable limitations due to their scale, that play a +crucial role in their target span and precision. +The Laser Interferometer Space Antenna (LISA) [26] is +a space-borne GW detector that will open new realms in +GW astrophysics, due to its unprecedented level of accu- +racy [27–30]. It will target, in particular, mHz sources of +GWs which are undetectable with current ground-based +detectors. +One of the prime objectives of LISA (and +other space-based programs [31–33]) is the detection of +GWs from extreme-mass-ratio inspirals (EMRIs) [34, 35], +which involve a primary supermassive compact object, +such as those lurking in galactic cores, and a secondary +stellar-mass compact object. Environmental effects and +spacetime deformations in EMRIs should, therefore, be +taken into account in order to maximize the scientific +yield from these sources [36–60]. +Fortunately or not, it is atypical to find integrable +spacetimes, especially when complex astrophysical envi- +ronments are involved or the primary is an exotic com- +pact object [61]. +To that end, a significant spacetime +degeneracy is present in Sgr A* and M87*. Current in- +vestigations have concluded that an abundance of exotic +objects can cast shadows that are practically indistin- +guishable (in specified regions of their parameter space) +from those of Schwarzschild (or Kerr) BHs [62–69], there- +fore assuming that Sgr A* and M87* are BHs, only from +their shadow silhouette, can lead to significant misinter- +pretation. Indeed, we need to invoke geodesics [70], ac- +cretion disk analyses [71] and GW ringdown tests [72–84] +in order to understand if supermassive compact objects +are typical BHs or exotic in nature. +In this study, we will investigate the simplest defor- +mation of Schwarzschild geometry, the Zipoy-Voorhees +(ZV) metric [85, 86] (also known as the γ-metric), that +describes a vacuum, static, and spheroidal solution of +Einstein equations which is continuously connected to +Schwarzschild by a deformation parameter δ (in our con- +vention). +The ZV metric, presenting a spheroidal de- +formation of an otherwise static geometry can pose as +a good model for a static compact object surrounded +by a compact environment or an accretion disk, such as +those residing in galactic centers. Recent shadow inves- +tigations [64] have shown that when the deformation pa- +rameter δ > 1 then very precise measurements will be +needed in order to rule out an exotic compact object de- +scribed by this geometry. A peculiarity of this solution is +the appearance of a curvature singularity at its surface, +thus characterizing it as a naked singularity. +Another +important feature for our analysis is that the ZV met- +ric has a non-zero mass quadrupole and non-integrable +geodesics [87–89], despite earlier claims of integrability +[90, 91]. Since non-integrable EMRIs present very dis- +tinctive characteristics in phase space, such as prolonged +resonant islands where geodesics share the same rational +ratio of orbital frequencies [22, 92–96] and discontinuities +in the GW frequencies during island crossings (‘glitches’) +[97, 98], here we examine if similar effects are present in +ZV EMRIs, and in particular GW glitches from crossings +of subdominant resonant islands. +We confirm that the conclusions of Refs. [87–89] are +correct, that is the ZV metric is non-integrable due to +the existence of chaotic layers of plunging geodesics and +a series of resonant islands of stability. We further choose +a primary supermassive compact object described by the +ZV metric and evolve EMRIs with stellar-mass secon- +daries (with fixed mass ratio) to assess the effect of non- +integrability at the orbital and waveform level. We find +plateaus in the dissipative evolution of the ratio of radial +and polar frequencies, that designate the crossing of res- +onant islands, and subsequently observe glitches in the +GW frequency evolution of the EMRI. Due to the atyp- +ical structure of the ZV primary, a variety of successive +resonant islands accumulate close to the edge of bound +geodesics. +By evolving EMRIs through successive resonances, +with generic initial conditions, we find that consecutive +glitches can appear in short timescales (of order of sev- +eral days) in the GW frequency evolution of the inspiral. +Since typical glitches experienced by non-Kerr EMRIs, +with the same mass ratio as the ZV one, are usually sep- +arated by months or even years of dissipative evolution +[97, 98], the detection of multiple GW glitches in brief pe- +riods of time may demonstrate that very slowly-rotating +supermassive compact objects are not Schwarzschild (or +slowly-rotating Kerr) BHs. These findings will contribute +in placing tighter constraints on exotic geometries, such +as naked singularities, and narrow down the parameter +space of viable BH mimicker primaries that can imitate +the shadow of supermassive BHs. +In what follows we use geometrized units so that the +gravitational constant and speed of light satisfy G = c = +1. +II. +THE ZIPOY-VOORHEES METRIC +The ZV spacetime [85, 86] describes a two parameter +family of exact vacuum solutions to the Einstein equa- +tions that are static, axisymmetric and asymptotically +flat. The line element in Erez-Rosen coordinates [99, 100] +reads +ds2 = − F(r)dt2 + F −1(r) +� +G(r, θ)dr2 + H(r, θ)dθ2 ++(r2 − 2kr) sin2 θdφ2� +, +(1) + +3 +with k = M/δ, where M is the gravitational mass of +the object and δ the deformation parameter, while the +functions involved in the metric tensor components are +F(r) = +� +1 − 2k +r +�δ +, +G(r, θ) = +� +r2 − 2kr +r2 − 2kr + k2 sin2 θ +�δ2−1 +, +(2) +H(r, θ) = +� +r2 − 2kr +�δ2 +� +r2 − 2kr + k2 sin2 θ +�δ2−1 . +From Eqs. +(1), (2), it is straightforward to obtain +the Schwarzschild limit when δ = 1. +Since at the +Schwarzschild limit, k = M, δ can be interpreted as a +measure for how much more (or less) mass M = kδ the +ZV object has when compared to a Schwarzschild BH. +Subsequently, the deformation δ captures the oblateness +of the compact object. When δ > 1 the ZV geometry de- +scribes a spacetime around the central object that is more +oblate than a Schwarzschild BH, while when 0 < δ < 1 +the central object is more prolate. For δ = 0 (which iden- +tically means M = 0) we obtain the Minkowski space- +time. +According to the no-hair theorem, when 0 < δ ̸= 1 +holds the event horizon is broken; a true curvature sin- +gularity appear at r = 2k [101], besides the typical one +at r = 0, and the ZV metric describes a naked singular- +ity [102–105]. Interestingly, when spherical symmetry is +broken the geometry obtains a non-zero quadrupole mo- +ment M2 = δ(1 − δ2)M 3/3 [106, 107] which eventually +leads to the Carter constant (or any other higher order +Killing tensor) being broken [87–89]. +III. +ORBITAL DYNAMICS +Regardless of its peculiar causal structure, the ZV pri- +mary we will focus on should possess as many ‘good’ fea- +tures as those of astrophysical compact objects. It has +been shown that the line element (1) has an innermost +stable circular orbit (ISCO) when δ > 1/ +√ +5 at [71, 88] +rISCO = k +� +1 + 3δ + +� +5δ2 − 1 +� +. +(3) +Furthermore, if δ ≥ 1/2 then the geometry has both +an ISCO and a photon sphere (PS), where unstable null +geodesics accumulate [108, 109], at [71] +rPS = k(1 + 2δ). +(4) +Notice that at the Schwarzschild limit where δ = 1, rPS = +3M and rISCO = 6M as expected. +Hereafter, we will +focus on spacetime deformations that are larger than 1/2 +in order to have an exotic central object with a PS and +an ISCO. +A. +Geodesics +A first-order approximation to EMRI evolution can +be accomplished through geodesics of a point-particle of +mass µ which plays the role of the secondary orbiting +around the primary supermassive compact object. +The geodesic equations read +¨xκ + Γκ +λν ˙xλ ˙xν = 0, +(5) +where Γκ +λν are the Christoffel symbols of spacetime, xκ +is the four-position vector, ˙xκ is the four-velocity vector +and the overdot denotes differentiation with respect to +proper time τ. +Stationary and axisymmetric spacetimes, such as Eq. +(1), possess metric tensor components that are t- and +φ-independent. Therefore, they admit at least two con- +served quantities (due to stationarity and axisymmetry) +throughout geodesic evolution, namely the energy E and +z-component of the orbital angular momentum Lz +E/µ = F(r)˙t, +Lz/µ = +� +r2 − 2kr +� +sin2 θ +F(r) +˙φ. +(6) +The t- and φ-momenta can be expressed with respect to +the conserved quantities and the non-zero metric tensor +components. Together with the conservation of the rest +mass µ of the secondary, (preservation of four-velocity) +which leads to gλν ˙xλ ˙xν = −1, the geodesics of test par- +ticles present three constants of motion. Specifically, the +conservation of the secondary’s four-velocity leads to a +constraint for bound orbits +˙r2 + H(r, θ) +G(r, θ) +˙θ2 + Veff = 0, +(7) +where the effective potential Veff has the form +Veff ≡ +1 +grr +� +1 + E2 +gtt ++ L2 +z +gφφ +� +, += +F(r) +G(r, θ) +� +1 − E2 +F(r) + +F(r)L2 +z +(r2 − 2kr) sin2 θ +� +. +(8) +The curve defined when Veff = 0. i.e. the curve of zero +velocity (CZV), can be used in order to choose proper +initial conditions that lead to bound orbits in the external +vicinity of the primary. +Bound geodesic motion can, generically, be charac- +terized by three orbital frequencies. +These frequencies +are associated with the radial rate of transition between +the periapsis and apoapsis of the geodesic (ωr), the +rate of longitudinal oscillations through the equatorial +plane (ωθ) and the revolution around the primary (ωφ). +Generic trajectories with irrational ratios of orbital fre- +quencies span on two-dimensional tori and fill them com- +pletely. To the contrary, when the ratio of two orbital +frequencies is a rational number then the geodesic is pe- +riodic (or resonant) and returns to its initial position af- +ter a number of oscillations defined by the multiplicity + +4 +of the resonance [4]. Such orbits are special in the sense +that they are not phase-space filling and therefore, can +directly affect the evolution of EMRIs when encountered +[22, 92, 93, 97, 98, 110–122]. +B. +Inspirals +To construct the inspiral trajectory we numerically in- +tegrate the coupled system of r, θ equations, after uti- +lizing Eqs. (6), augmented with post-Newtonian (PN) +fluxes for the energy and angular momentum, respec- +tively [123–126]. This treatment, though approximate, +takes into account the dominant contribution of the sec- +ondary’s radiative backreaction to the spacetime geome- +try, at second PN order, and results to an adiabatic evo- +lution of the EMRI through time-dependent shifts onto +the energy and z-component of angular momentum of +the secondary. +Since the inspiral evolves very slowly, +the orbit is treated, in small timescales, as a geodesic, +while for long timescales the trajectory is driven adia- +batically through successively damped geodesics. +This +method, known as the hybrid kludge scheme, has been +shown to perform very well when compared to Teukolsky- +based Kerr waveforms for EMRIs [127]. +Notice that the ZV metric has a non-trivial multipo- +lar structure due to the deformation parameter δ, and +in particular a non-zero mass quadrupole tensor, as op- +posed to that of Schwarzschild BHs. At second PN order, +the kludge scheme [124] involves the mass quadrupole +moment M2 (for Kerr), thus to construct a more appro- +priate inspiral around ZV compact objects we apply a +modification to the fluxes (see [92, 93, 97, 98, 128, 129]) +in order to include the quadrupole moment M2 of the ZV +metric, which represents the effect of δ on the evolution +of E, Lz, and set the spin parameter to zero. +The adiabatic approximation, together with the flux +augmentation, employed here has recently been found +to provide results qualitatively equivalent to evolutions +with instantaneous self-force in non-Kerr electromagnetic +analogues, which indicates that the methods we use can +in principle describe resonance-crossings with sufficient +accuracy in EMRIs [122]. Nevertheless, more accurate +inspirals can be built by directly solving the wave equa- +tion resulting from metric perturbations and calculating +the GW emission at the object and infinity, though this +is a much more time consuming task. +We assume linear variations of the momenta as in [97, +98, 130] +E1 = E0 +µ + +�dE +dt +����� +0 +Nr Tr, +(9) +Lz,1 = Lz,0 +µ ++ +�dLz +dt +����� +0 +Nr Tr, +(10) +where +E0, Lz,0 +are +the +initial +energy +and +z- +component of the angular momentum, +respectively, +and +⟨dE/dt⟩|0, ⟨dLz/dt⟩|0 +are +the +radiation +fluxes +calculated at the beginning of the inspiral, through the +equations in [98, 126]. Tr is the time that the orbit takes +to travel from the periapsis to apoapsis and back, while +the fluxes (9) and (10) are updated every Nr cycles for +the whole EMRI evolution. +C. +Detecting chaos +To understand the phase space structure of orbits +around the ZV primary we can employ well-known tools +in order to gain further intuition regarding orbital phe- +nomena and chaotic imprints. A typical example is the +Poincar´e surface of section which is constructed by suc- +cessive intersections of geodesics, with varying initial con- +ditions, on a surface of section (here, we choose the equa- +torial plane) with strictly positive (or strictly negative) +direction of intersection. The structure of the Poincar´e +map can instantly reveal if ergodic chaos is present, +through disorganized intersections, or indirect imprints +of non-integrability with the appearance of resonant is- +lands, that encapsulate stable periodic points [4], and ex- +ist due to non-integrability, in accord to the Kolmogorov- +Arnold-Moser (KAM) and Poincar´e-Birkhoff theorems +[131–133]. Since the ZV metric does not have a Carter- +like constant these features are present in its orbital phase +space [88]. +Another tool to detect chaos is the rotation number. +We calculate it by tracking the angle ϑ between two suc- +cessive intersections on the Poincar´e map relative to the +fixed central point of the map which corresponds to a +circular orbit that intersects the surface of section ex- +actly at the same point. The rotation number is defined +as the accumulation of many angles measured between +consecutive intersections as [4] +νϑ = +1 +2πN +N +� +i=1 +ϑi, +(11) +for which when N → ∞ (with N the number of angles +measured), Eq. (11) converges to the radial and polar +orbital frequency ratio νϑ = ωr/ωθ. Calculating consecu- +tive rotation numbers for different geodesics, by smoothly +varying one of the parameters or initial conditions of the +system while keeping the rest fixed, leads to a rotation +curve. Integrable systems demonstrate monotonous rota- +tion curves. On the other hand, non-integrable systems +possess transient plateaus with a non-zero width when +geodesics occupy resonant islands. This designates a cru- +cial aspect of resonant islands, that is when a geodesic is +inside the island it shares the same rational ratio ωr/ωθ +with the stable periodic point which leads to the plateau +formation; a phenomenon that does not appear in inte- +grable systems whatsoever, even though resonances still +exist but occupy only a single point in phase space. 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+● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +5 +6 +7 +8 +9 +10 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +r/M +r +νϑ=1/4 +νϑ=2/7 +νϑ=1/3 +νϑ=1/2 +FIG. 1. +Poincar´e map of a test-particle secondary with +µ = 1M⊙ orbiting around a ZV supermassive object with +M = 106M⊙ and δ = 1.5. +The secondary’s conserved en- +ergy and angular momentum as E/µ = 0.95, Lz/µ = 3M, +respectively. Black curves that surround the central point of +the map designate intersections through the equatorial plane +of generic orbits, while colored curves designate intersections +that belong to different resonant islands of stability. +onant islands, emanate. Nevertheless, by changing the +initial conditions the orbit can be driven through the is- +land and give rise to a typical plateau. +Rotation curves are not only a tool that is used for +geodesics but can also be employed in dissipative sce- +narios, where the rotation number evolves with respect +to time. +In the case of an EMRI, one can use se- +lected timesteps of the inspiral as initial conditions for +a geodesic evolution. Through the non-dissipative tra- +jectory, the Poincar´e map and eventually the rotation +number of each timestep can be calculated in order to +plot a series of rotation numbers as the EMRI evolves +with time. The same attributes hold here as well, namely +monotonous dissipative rotation curves for integrable sys- +tems and appearance of plateaus for non-integrable EM- +RIs. +For more information regarding integrability and +chaos in EMRIs see the following series of works and ref- +erences therein [4, 22, 88, 92–94, 97, 98, 134, 135]. +IV. +GEODESIC AND INSPIRAL EVOLUTION +By solving the coupled r, θ second-order ordinary dif- +ferential equations, together with the first-order decou- +pled equations for t and φ from Eqs. (6), we obtain bound +orbits that reside inside CZVs and never plunge nor es- +cape to infinity. To check the precision of our geodesics +we evolve the constraint equation (7) for ∼ 104 revo- +lutions and find that it is satisfied within one part in +108 − 1010 depending on the initial conditions and defor- +mation of spacetime. +To guarantee numerical accuracy for inspirals, we cal- +culate the 4-velocity in each update of the fluxes and +check its conservation along a geodesic evolution with +initial conditions the energy, z-component of angular mo- +mentum, position and velocity at every update timestep. +For all simulations presented hereafter, the constraint is +satisfied to within a part in ∼ 108 for the first 104 cross- +ings through the equatorial plane. +As a first step, we reproduce some qualitative features +of the ZV metric, namely its non-integrability [88], which +will be later used to choose initial conditions in order to +evolve EMRIs. +In Fig. +1 we show the Poincar´e map +for a ZV central object with δ = 1.5, meaning that +the we utilized an oblate deformation with respect to +Schwarzschild. We first observe a central point on the +map. Since the map captures intersections of geodesics, +the central point designates an orbit that is circular and +always cuts the surface of section at the same (r, ˙r) po- +sition. Around the center, various black curves appear +(known as KAM curves) that are formed through suc- +cessive intersections of generic orbits with varying ini- +tial position r(0)/M. A Schwarzschild BH (or in general +any spacetime with integrable geodesics) would exhibit a +Poincar´e map with KAM curves that only surround the +central point of the map. The fact that our object does +not have a fourth constant of motion (non-integrable), +leads to the formation of nested islands around stable +resonant points (see [22] for a zoom into the encapsu- +lated structure of these islands). +With a pedantic search on the available parameter +space, we can easily spot four resonant islands with dif- +ferent multiplicity, which we designate in Fig. +1 with +colored curves. Notice how those do not surround the +central point, but rather the stable points in phase space +where the true resonant orbits emanate, and that the +multiplicity defined by the denominator of the reso- +nance (the longitudinal oscillations through the equato- +rial plane) corresponds to the number of islands. It is +noteworthy that the resonances appearing here, besides +the 1/2-resonant island, are not those that strongly af- +fect an inspiral, especially when assessing their impact in +EMRI parameter estimation [110, 118, 119], yet we will +later see that they can also significantly contribute to it +if the spacetime is non-integrable, mainly because they +accumulate before plunge. +The rotation curves presented in Fig. 2 promote the +previous discussion perfectly. Various plateaus and in- +flections appear right where we expect the rotation num- +ber to have a rational ratio. The plateaus clearly des- +ignate that any geodesic residing in a resonant island +shares the same rational ratio of orbital frequencies with +the center of the island, where stable periodic orbits em- +anate. Even when an inflection point shows up at the +rotation curve, meaning that the geodesic lies between +the tips of two resonant islands where unstable periodic +orbits exist, with a certain change in the initial velocity +˙r(0), we can access the island as shown in Fig. 3. For +completeness, we have considered both oblate (δ = 1.5) + +6 +5 +5.5 +6 +6.5 +7 +7.5 +8 +8.5 +9 +9.5 +0.2 +0.3 +0.4 +0.5 +0.6 +r/M +νϑ +δ=1.5 (oblate ZV) +δ=1 (Schwarzschild) +δ=0.5 (prolate ZV) +4.75 +4.80 +4.85 +4.90 +4.95 +0.25 +0.27 +0.29 +0.31 +0.33 +r/M +νϑ +νϑ=1/4 +νϑ=2/7 +νϑ=1/3 +FIG. 2. Left: Rotation curves for geodesics of a test particle with µ = 1M⊙, E/µ = 0.95, Lz/µ = 3M and ˙r(0) = 0 around a +ZV object with M = 106M⊙ with varying deformation δ. For completeness, we present the rotation curve for a Schwarzschild +BH with the same parameters as above and δ = 1. Right: Zoom into a region of interest for the δ = 1.5 case. The horizontal +colored lines designate where resonances appear. +4.797 +4.799 +4.801 +4.803 +4.805 +4.807 +0.284 +0.285 +0.286 +0.287 +0.288 +r/M +νϑ +νϑ=2/7 +4.878 +4.88 +4.882 +4.884 +4.886 +0.284 +0.285 +0.286 +0.287 +0.288 +r/M +νϑ +νϑ=2/7 +FIG. 3. +Left: Zoom into the inflection point for the 2/7-resonant island shown on the right subfigure of Fig. +2. +initial +conditions and parameter as the same as Fig. +2. +Right: Rotation curve for geodesics of a test particle with µ = 1M⊙, +E/µ = 0.95, Lz/µ = 3M and ˙r(0) = 0.01 around a ZV object with M = 106M⊙ with δ = 1.5. +and prolate (δ = 0.5) deformations1 and encounter simi- +lar effects. +For the rest of the discussion we will focus on oblate +deformations for the following reasons: (i) oblate defor- +mations are usually the ones enabling the strongest ef- +fects of non-integrability and chaos (see [22, 93, 94, 97, +98, 121, 129]), (ii) as seen in Fig. +2, an oblate defor- +mation drives the resonances closer to the central object +thus we expect amplified chaotic effects and (iii) δ > 1 is +an optimal choice to imitate BHs since for these cases the +ZV object possesses a PS, an ISCO and produces similar +shadows [64]. +1 In Fig. +2 we also present a typical rotation curve for +Schwarzschild spacetime. +So far, the discussion involved zeroth-order approxima- +tions of EMRIs since there was no radiation loss. Turn- +ing on the fluxes (rates of change of orbital energy and +angular momentum) (9) and (10) leads to a numerical +integration that is more intricate but the results are very +interesting since the secondary inspirals adiabatically to- +wards the primary due to fluxes through GWs. Fig. 4 +presents a typical inspiral of a secondary that transverses +the 1/3-resonant island. The initial conditions are not +fine tuned so what is presented here is a generic feature. +For this inspiral, we have updated the fluxes Nr = 800 +times, every Tr = 200M. Since the revolution period in +this case is ∼ 90M, the total evolution time of the EMRI +is ttotal = 1.6×105M ∼ 1800 revolutions (roughly 9 days +for the mass of the primary considered). +The fluxes undergo a rather peculiar behavior at first +glance. At a certain time, they become more negative + +7 +3.6 +4 +4.4 +4.8 +5.2 +-6.72 +-6.71 +-6.7 +-6.69 +-6.68 +-6.67 +t (days) +dE/dt (× 10-16) +3.6 +4 +4.4 +4.8 +5.2 +5.6 +0.3322 +0.3327 +0.3332 +0.3337 +0.3342 +t (days) +νϑ +νϑ=1/3 +FIG. 4. Left: Evolution of energy flux dE/dt for an inspiraling secondary with µ = 1M⊙ and initial parameters E0/µ = 0.95, +Lz,0/µ = 3M, r(0) = 4.985M, θ(0) = π/2, ˙r(0) = 0, where ˙θ(0) is defined by the constraint equation (7), around a ZV primary +with M = 106M⊙ and δ = 1.5. Right: Evolution of the rotation number of the same inspiral as in the left panel. The horizontal +solid red line designates the 1/3-resonant island. For both plots, the vertical dashed red line corresponds the time when the +secondary passes through the center of the island. +2 +4 +6 +8 +10 +12 +-7.57 +-7.7 +-7.83 +-7.96 +-8.09 +-8.21 +t (days) +dE/dt (× 10-16) +2 +4 +6 +8 +10 +12 +0.25 +0.26 +0.27 +0.28 +0.29 +t (days) +νϑ +νϑ=2/7 +νϑ=1/4 +FIG. 5. Left: Evolution of energy flux dE/dt for an inspiraling secondary with µ = 1M⊙ and initial parameters E0/µ = 0.95, +Lz,0/µ = 3M, r(0) = 4.824M, θ(0) = π/2, ˙r(0) = 0, where ˙θ(0) is defined by the constraint equation (7), around a ZV primary +with M = 106M⊙ and δ = 1.5. Right: Evolution of the rotation number of the same inspiral as in the left panel. The horizontal +solid colored lines designates the 2/7 (green) and 1/4-resonant islands (blue). For both plots, the vertical dashed colored lines +corresponds the time when the secondary passes through the center of the corresponding islands. +abruptly. +This instant of time designates the entry of +the secondary into the 1/3 plateau of the island (see +right panel in Fig. 4). Right after the entry, the rate +of change of orbital energy reaches a local maximum, at +a special instant of time which is shown with a verti- +cal dashed line, beyond which it continues to decline due +to the inspiral. This behavior is not at all peculiar, but +rather has a true physical meaning. Regardless of the fact +that the secondary transverses the island while sharing +the same rotation number throughout it, the secondary +only reaches the stable periodic point at the center of +the island on a certain time designated with the verti- +cal dashed line. This is when the orbit becomes exactly +periodic. Periodic orbits are the closest to circular ones. +Since circular orbits emit monochromatic radiation at a +single frequency (twice the revolution frequency) their +energy emission is minimized (thus the rates of change of +orbital energy and angular momentum are maximized). +Since a resonance emits quasi-monochromatic radiation, +at the time that the inspiral passes through the perfect +resonance, the fluxes are also maximized. The exit from +the island takes place when the flux value at the entry is +met again. The secondary can spend ∼ 100 cycles in per- +fect resonance which translates to roughly half a day or +∼ 5% of the whole EMRI evolution. The picture is qual- +itatively the same for the flux of angular momentum. +Fig. 5 presents an even more intriguing EMRI evolu- +tion around the ZV primary, that undergoes two consec- + +8 +2 +2.5 +3 +3.5 +4 +4.5 +5 +-7.6 +-7.63 +-7.66 +-7.69 +t (days) +dE/dt (× 10-16) +11.2 +11.4 +11.6 +11.8 +-8.09 +-8.11 +-8.13 +-8.15 +-8.17 +t (days) +dE/dt (× 10-16) +FIG. 6. Left: Zoom into the evolution of energy flux dE/dt from Fig. 5 through the 2/7-resonant island. Right: Same as the +left panel for the 1/4-resonant island crossing. +utive island crossings during a single evolution, namely +the 2/7 and 1/4 islands. Initial conditions are slightly +(but not strongly) fine tuned so what is presented here is +quasi-generic, in a sense that it can occur for a small +but non-zero set of initial conditions. +For this inspi- +ral, we have updated the fluxes Nr = 1100 times, ev- +ery Tr = 200M. +Since the revolution period in this +case is ∼ 80M, the total evolution time of the EMRI +is ttotal = 2.2 × 105M ∼ 2800 revolutions (roughly 12 +days for the mass of the primary considered). +The time that the secondary spends in the first island, +in perfect resonance, is ∼ 200 cycles which translates to +roughly a day. After exiting the first island, the orbit en- +ters shortly after a subsequent resonant island and occu- +pies it for another ∼ 50 cycles which translate to one fifth +of a day. In total the inspirals experiences 250 cycles of +perfect resonance (without taking into account pre- and +post-resonant effects) which correspond to ∼ 10% of the +whole evolution. Similar analyses for non-Kerr EMRIs +with the same mass ratios used here (µ/M = 10−6) ex- +perience 250 cycles in the most prominent 2/3-resonant +island [97, 98], though due to the much slower inspiral +(resonances appear further from the central object) this +only corresponds to ∼ 3% of the whole evolution spent in +a single island. Practically, the fact that a multitude of +islands gather close to the plunge gives us the ability to +probe subdominant resonances consecutively, in a short +period of time, and have an EMRI that experiences a +significant fraction of its evolution in resonance. +Zooming in on the left plot of Fig. +5 (see Fig. +6), +we observe a complete agreement with what is demon- +strated in Fig. 4. Especially for the 2/7-resonant island +(left plot in Fig. 6), the initial condition chosen here al- +lows for the secondary to remain in it for a substantial +amount of time that leads to a significant maximization +of the energy flux. This means that the orbit enters deep +into the island and probes the stable periodic point for +a significant amount of time till it exits. The initial con- +dition may look special, since a typical crossing would +not maximize fluxes considerably, nevertheless, plateaus +on dissipative rotation curves are still generic for a wide +range of initial conditions. +V. +GRAVITATIONAL WAVES +In this section we approximate the dominant GW emis- +sion of an inspiraling stellar-mass secondary around an +oblate ZV supermassive compact object and search for +imprints of non-integrability in the waveforms produced +by such EMRIs when detected by a LISA-like interfer- +ometer. For the GW modeling, we use the quadrupole +approximation described below. +A. +Quadrupole formula +The radiative component of the metric perturbation in- +troduced by the secondary at luminosity distance d from +the source T can be read at the transverse and traceless +gauge as +hTT +ij += 2 +d +d2Qij +dt2 , +(12) +where Qij +is the symmetric and trace-free (STF) +quadrupole tensor +Qij = +�� +xixjT tt(t, xi) d3x +�STF +, +(13) +with t being the coordinate time measured at very large +distances from the detector. The source term of the sec- +ondary (which is treated as a point particle through a +delta function) is +T tt(t, xi) = µδ(3) � +xi − Zi(t) +� +, +(14) + +9 +FIG. 7. Left: Frequency evolution of an EMRI through a consecutive crossing of resonant islands with parameters and initial +conditions as in Figs. 5 and 6. The approximate GWs are detected at luminosity distance d = 100 Mpc. The spectrogram +presents the effect on a single harmonic of the waveform when it crosses the 2/7-resonant island. +Right: Continuation of +frequency evolution of the same EMRI with initial conditions as in the left panel. The spectrogram presents the effect on the +waveform when it crosses the 1/4-resonant island. +where Z(t) = (x(t), y(t), z(t)) the position vector in +pseudo-Cartesian coordinates and +x(t) = r(t) sin θ(t) cos φ(t), +(15) +y(t) = r(t) sin θ(t) sin φ(t), +(16) +z(t) = r(t) cos θ(t), +(17) +the trajectory components with respect to flat spheri- +cal coordinates, under the assumption that our space- +borne detector is positioned at infinity. Even though we +have identified the Schwarzschild-like coordinates (r, θ, φ) +of the secondary’s trajectory with flat-space coordinates, +known in the literature as the “particle-on-a-string” ap- +proximation, and we assume a finite luminosity distance +d from the source, such prescription is not strictly valid. +Nevertheless, it has been found to work very well when +generating EMRI waveforms in GR [127]. +GWs can be projected onto two polarizations, + and +×, with the introduction of two unit vectors, p and q, +which are defined with respect to a third unit vector n +that points from the source to the detector. The triplet +of unit vectors (p, q, n) is chosen so that they form an +orthonormal basis. The polarization tensor components +read +ϵij ++ = pipj − qiqj, +ϵij +× = piqj + pjqi, +(18) +and allow for the metric perturbation to be written as +hij(t) = ϵij ++h+(t) + ϵij +×h×(t), +(19) +with +h+(t) = 1 +2ϵij ++hij(t), +h×(t) = 1 +2ϵij +×hij(t). +(20) +The GW polarization components can then be described +in terms of the position, Zi(t), velocity, vi(t) = dZi/dt, +and acceleration ai(t) = d2Zi/dt2 vectors as [130] +h+,×(t) = 2µ +d ϵ+,× +ij +� +ai(t)Zj(t) + vi(t)vj(t) +� +. +(21) +LISA’s response to an incident GW is rather complicated +and depends on the antennae response patterns (see [125, +136] for the full equations). Here we assume a detector +that lies at a luminosity distance d with orientation n = +(0, 0, 1) with respect to the source, and utilize +hα(t) = +√ +3 +2 +� +F + +α (t)h+(t) + F × +α (t)h×(t) +� +, +(22) +where α = (I, II) is an index representing the different +antenna pattern functions F + +α , F × +α which can be found +in Refs. [125, 136, 137]. For phenomenological purposes +we will use a single-channel approximation, that is set +α = I in Eq. (22), since it is enough to accommodate +the fundamental parts of gravitational radiation emitted +by the EMRI, i.e. its phasing. +B. +Gravitational-wave frequency evolution and +cumulative glitches +To comprehend how a ZV EMRI imprints non- +integrable effects, such as plateaus, onto its emitted wave- +form, we calculate the GW in the Einstein-quadrupole +approximation for the particular inspiral outlined in Sec. +IV that crosses two consecutive islands, namely the 2/7 + +35 +70 +2.60 +30 +60 +2.80 +2.58 +intensity (×10-16) +25 +50 +2.56 +2.75 +20 +40 +2.54 +15 +30 +2.70 +2.52 +10 +20 +5 +2.50 +10 +2.65 +2 +3 +10 +12 +1 +4 +5 +8 +9 +11 +time (days) +time (days)10 +and 1/4-resonances. We focus solely on this example in +order to demonstrate that even subdominant resonances +affect EMRI evolution, when the spacetime presents +chaotic features. +We perform a Fourier transform on the extracted wave- +form from the inspiral and plot a density spectrogram +that displays the evolution of one of the harmonics with +respect to time. We use the same methodology for the +spectrogram as in [97, 98], i.e. we cut the waveform in +time segments with a particular window size and offset +and perform consecutive Fourier transforms, in order to +overcome the uncertainty between frequency and time +resolution. Here, since the inspiral is evolving fast and +the island crossings have different timescales, we use dis- +tinct window size and offsets for the chunks of evolu- +tion around the two resonant islands. This is necessary +for demonstration purposes since the first resonance is +crossed much slower than the second one. +In Fig. +7 we show the spectrogram of the GW ex- +tracted from the EMRI that successively transverses two +resonant islands of stability. In both cases, when the is- +land is met the waveform frequency evolution loses mono- +tonicity. The resonant crossings manifest into the GW +with either a plateau-like pattern (left plot in Fig. 7) or +a rapid glitch2 (right plot in Fig. 7). Both instances des- +ignate that the system is non-integrable, especially the +encounter of the first island that shares a striking similar- +ity with plateaus appearing in rotation curves. However, +this is an outcome of short-term occupancy in the islands +and altogether a smaller deformation of spacetime, with +respect to those presented in non-Kerr EMRIs [97, 98]. If +the inspiral is given more time in the island, the manifes- +tations should resemble more those of discontinuous GW +glitches. Yet, the fact that resonant island crossings are +imprinted in the GW of a ZV EMRI, in a short period of +time, can serve as a ‘smoking gun’ for a non-integrable +BH mimicker in the center of our galaxy, since these phe- +nomena do not appear in Schwarzschild EMRIs that have +monotonous rotation curves and spectrograms. +VI. +CONCLUDING REMARKS +Probing the spacetime around supermassive compact +objects with EMRIs is one of the prime targets for space- +borne detectors like LISA. Compact objects are either +spinning and/or surrounded by astrophysical environ- +ments, especially those residing in active galactic nuclei. +Therefore, spherical symmetry is rather fragile and is +usually broken. When there are not enough spacetime +symmetries to guarantee the integrability of geodesics +around these objects, then LISA may be able to detect +2 Unfortunately, the inspiral enters into a region of phase space +right after the crossing of the 1/4-resonance where pure chaos +is present and the Fourier peaks become continuous so we could +not evolve the system for more time. +particular phenomenological imprints of such manifesta- +tions like GW glitches around expected transient orbital +resonances [97, 98] that differ significantly from instru- +mental glitches [138]. +We showed that deformations from spherical symme- +try which keep the spacetime static, described by the +ZV geometry, have the potential to exhibit GW glitches +when involved as primary objects in EMRIs, due to the +non-integrability of test-particle dynamics. These phe- +nomena not only appear for single resonant island cross- +ings, such as those occurring in non-Kerr EMRIs, but also +are present in a cumulative and short-timescale manner, +where the secondary transverses two (and possibly more) +subdominant resonant islands. Note though that this can +also occur in non-Kerr EMRIs if long-lasting evolutions +are to be performed, though the effect of subdominant +resonances are usually suppressed [97, 98]. +When supermassive compact objects, such as those +residing in galactic centers, have multiple interpreta- +tions due to the plethora of objects that can cast sim- +ilar shadows, GW astronomy with LISA can, in princi- +ple, distinguish between models that are integrable or +not through the detection of glitches in gravitational +waveforms. Synergies between shadow observations and +space-borne detectors can, therefore, narrow down the +parameter space of solutions describing supermassive ob- +jects in galactic centers, and in particular can emphati- +cally decide if M87* and Sgr A* are described by Kerr +and Schwarzschild geometries, respectively, unless the +EMRI includes multiple [40, 139] or spinning [120, 140] +secondaries. +Our analysis only deals with the phenomenological im- +prints of non-integrability. Nevertheless, if one wants to +utilize the aforementioned phenomenology in practice, +a consistent glitch modeling analysis for non-integrable +EMRIs should be carried out, in a systematic way in or- +der to understand, first, if these phenomena are clearly +detectable with space interferometers, second, to which +extent they affect parameter estimation, and third, to +which degree these effects differ from standard transient +resonances experienced by integrable EMRIs which are +already sufficiently modeled with PN techniques [118, +119] and gravitational self-force [110, 111, 117, 141, 142]. +ACKNOWLEDGMENTS +The authors warmly thank Prof. +Theocharis Apos- +tolatos for critical comments and helpful discussions. +This work was supported by the DAAD program for +the “promotion of the exchange and scientific cooper- +ation between Greece and Germany IKYDAAD 2022” +(57628320). 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Grav. 26, 213001 (2009), +arXiv:0908.1664 [gr-qc]. + diff --git a/V9FJT4oBgHgl3EQfOizn/content/tmp_files/load_file.txt b/V9FJT4oBgHgl3EQfOizn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2455e4031b0488bdbadf5f9aebd5d89f49ffce64 --- /dev/null +++ b/V9FJT4oBgHgl3EQfOizn/content/tmp_files/load_file.txt @@ -0,0 +1,6738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf,len=6737 +page_content='Geodesics and gravitational waves in chaotic extreme-mass-ratio inspirals: the curious case of Zipoy-Voorhees black-hole mimickers Kyriakos Destounis1,2,3, Giulia Huez3,4 and Kostas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Kokkotas3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 1Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Sapienza Universit`a di Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Piazzale Aldo Moro 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 00185,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Italy 2INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Sezione di Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Piazzale Aldo Moro 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 00185,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Italy 3Theoretical Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' IAAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' University of T¨ubingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 72076 T¨ubingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Germany 4Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' University of Trento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Via Sommarive 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 38123 Trento,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Italy and 5Section of Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' and Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' University of Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Panepistimiopolis Zografos GR15783,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Athens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Greece The growing capacity of gravitational-wave astronomy and black-hole imaging will soon enable us to emphatically decide if astrophysical compact objects in galactic centers are black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Sgr A*, one of the most prolific astronomical radio sources in our galaxy, is the focal point for tests of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Current mass and spin constraints predict that Sgr A* is supermassive and slowly rotating, thus can be conservatively modeled as a Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Nevertheless, the well- established presence of accretion disks and astrophysical environments around supermassive compact objects can significantly deform their geometry and complicate their observational scientific yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Here, we study extreme-mass-ratio binaries comprised of a minuscule secondary object inspiraling onto a supermassive Zipoy-Voorhees compact object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' the simplest exact solution of general relativity that describes a static, spheroidal deformation of Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We examine geodesics of prolate and oblate deformations for generic orbits and reevaluate the non-integrability of Zipoy- Voorhees spacetime through the existence of resonant islands in the orbital phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' By including radiation loss with post-Newtonian techniques, we evolve stellar-mass secondary objects around a supermassive Zipoy-Voorhees primary and find clear imprints of non-integrability in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The peculiar structure of the primary, allows for, not only typical single crossings of transient resonant islands, that are well-known for non-Kerr objects, but also inspirals that transverse through several islands, in a brief period of time, that lead to multiple glitches in the gravitational-wave frequency evolution of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The detectability of glitches with future spaceborne detectors can, therefore, narrow down the parameter space of exotic solutions that can, otherwise, cast identical shadows with black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' INTRODUCTION The Schwarzschild spacetime [1] is unequivocally the simplest and most remarkable black hole (BH) solution of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' It describes a vacuum compact object with an event horizon and a static, spherically- symmetric exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The remarkable symmetry proper- ties of Schwarzschild geometry places it at the top of Einstein field equation solutions in what regards its sim- plicity and singular externally-observable property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' the gravitational mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Its ultimate successor, the Kerr metric [2], is undoubt- edly the most successful and astrophysically-relevant so- lution of the vacuum field equations that describes a spin- ning, stationary and axisymmetric BH with an oblate, spheroidal external geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Despite the fact that Kerr BHs possess less symmetries than Schwarzschild space- times, they compensate by including spin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' a very crucial aspect of most astrophysical compact objects, and fur- ther form an integrable (separable) system of equations of motion for massless and massive particles due to the existence of the Carter constant [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The above state- ment of integrability translates to the absence of chaos in geodesics around Kerr BHs, and is traced trivially to Schwarzschild spacetime [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' From an astrophysical perspective, a significant volume that surrounds BHs in the Universe consists of plasma, accretion disks, matter configurations and halos that ex- tend significantly far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Thus, it is hard for one to realize BHs in pure vacuum, especially those occupying the centers of galaxies, where a perplex and highly dy- namical environment is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The historic observations of shadows from supermas- sive compact objects in the center of M87* galaxy [5] and Sgr A* in our galaxy [6] has opened a new realm of observational yield with BH imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' An accretion disk can, in principle, deform the surrounding geometry of a BH so that it continuously deviates from a Schwarzschild or Kerr description, causing degeneracies between a mul- titude of other exact, though more exotic, solutions that may mimic the observed shadow of supermassive com- pact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' To study potential degeneracies present in current elec- tromagnetic observations, the literature usually operates in spacetime deviations from the Kerr description, known as bumpy/parameterized [7–15] or non-Kerr compact ob- jects [16–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' A distinctive feature of some of these ob- jects is the absence of Carter symmetry, which leads to chaotic phenomena in particle-dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In what re- gards Sgr A*, though, spin is not as crucial as the devi- ation from spherical symmetry itself, since current con- straints on its spin conclude that it is less than 10% of its maximal allowed [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Therefore, a plethora of Sgr A*-like objects in the Universe can be sufficiently mod- eled as Schwarzschild (or slowly-rotating Kerr) BHs or exotic BH mimickers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='11483v1 [gr-qc] 27 Jan 2023 2 Gravitational-wave (GW) astrophysics has been proven to be an extraordinary tool to break the afore- mentioned degeneracies, thus synergies between GW and shadow observations are necessary in order to search for the ultimate spacetime description of known astrophysical compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' To that end, the LIGO/Virgo/Kagra collaboration has been flooding our databases with numerous GW detections from coalesc- ing compact objects [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Such detectors, although ex- tremely successful so far (at the level of groundbreaking), have the unfortunate attribute of being plagued by plan- etary noise, since they are placed on Earth, and present unavoidable limitations due to their scale, that play a crucial role in their target span and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The Laser Interferometer Space Antenna (LISA) [26] is a space-borne GW detector that will open new realms in GW astrophysics, due to its unprecedented level of accu- racy [27–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' It will target, in particular, mHz sources of GWs which are undetectable with current ground-based detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' One of the prime objectives of LISA (and other space-based programs [31–33]) is the detection of GWs from extreme-mass-ratio inspirals (EMRIs) [34, 35], which involve a primary supermassive compact object, such as those lurking in galactic cores, and a secondary stellar-mass compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Environmental effects and spacetime deformations in EMRIs should, therefore, be taken into account in order to maximize the scientific yield from these sources [36–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Fortunately or not, it is atypical to find integrable spacetimes, especially when complex astrophysical envi- ronments are involved or the primary is an exotic com- pact object [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' To that end, a significant spacetime degeneracy is present in Sgr A* and M87*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Current in- vestigations have concluded that an abundance of exotic objects can cast shadows that are practically indistin- guishable (in specified regions of their parameter space) from those of Schwarzschild (or Kerr) BHs [62–69], there- fore assuming that Sgr A* and M87* are BHs, only from their shadow silhouette, can lead to significant misinter- pretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Indeed, we need to invoke geodesics [70], ac- cretion disk analyses [71] and GW ringdown tests [72–84] in order to understand if supermassive compact objects are typical BHs or exotic in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In this study, we will investigate the simplest defor- mation of Schwarzschild geometry, the Zipoy-Voorhees (ZV) metric [85, 86] (also known as the γ-metric), that describes a vacuum, static, and spheroidal solution of Einstein equations which is continuously connected to Schwarzschild by a deformation parameter δ (in our con- vention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The ZV metric, presenting a spheroidal de- formation of an otherwise static geometry can pose as a good model for a static compact object surrounded by a compact environment or an accretion disk, such as those residing in galactic centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Recent shadow inves- tigations [64] have shown that when the deformation pa- rameter δ > 1 then very precise measurements will be needed in order to rule out an exotic compact object de- scribed by this geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' A peculiarity of this solution is the appearance of a curvature singularity at its surface, thus characterizing it as a naked singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Another important feature for our analysis is that the ZV met- ric has a non-zero mass quadrupole and non-integrable geodesics [87–89], despite earlier claims of integrability [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since non-integrable EMRIs present very dis- tinctive characteristics in phase space, such as prolonged resonant islands where geodesics share the same rational ratio of orbital frequencies [22, 92–96] and discontinuities in the GW frequencies during island crossings (‘glitches’) [97, 98], here we examine if similar effects are present in ZV EMRIs, and in particular GW glitches from crossings of subdominant resonant islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We confirm that the conclusions of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' [87–89] are correct, that is the ZV metric is non-integrable due to the existence of chaotic layers of plunging geodesics and a series of resonant islands of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We further choose a primary supermassive compact object described by the ZV metric and evolve EMRIs with stellar-mass secon- daries (with fixed mass ratio) to assess the effect of non- integrability at the orbital and waveform level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We find plateaus in the dissipative evolution of the ratio of radial and polar frequencies, that designate the crossing of res- onant islands, and subsequently observe glitches in the GW frequency evolution of the EMRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Due to the atyp- ical structure of the ZV primary, a variety of successive resonant islands accumulate close to the edge of bound geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' By evolving EMRIs through successive resonances, with generic initial conditions, we find that consecutive glitches can appear in short timescales (of order of sev- eral days) in the GW frequency evolution of the inspiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since typical glitches experienced by non-Kerr EMRIs, with the same mass ratio as the ZV one, are usually sep- arated by months or even years of dissipative evolution [97, 98], the detection of multiple GW glitches in brief pe- riods of time may demonstrate that very slowly-rotating supermassive compact objects are not Schwarzschild (or slowly-rotating Kerr) BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' These findings will contribute in placing tighter constraints on exotic geometries, such as naked singularities, and narrow down the parameter space of viable BH mimicker primaries that can imitate the shadow of supermassive BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In what follows we use geometrized units so that the gravitational constant and speed of light satisfy G = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' THE ZIPOY-VOORHEES METRIC The ZV spacetime [85, 86] describes a two parameter family of exact vacuum solutions to the Einstein equa- tions that are static, axisymmetric and asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The line element in Erez-Rosen coordinates [99, 100] reads ds2 = − F(r)dt2 + F −1(r) � G(r, θ)dr2 + H(r, θ)dθ2 +(r2 − 2kr) sin2 θdφ2� , (1) 3 with k = M/δ, where M is the gravitational mass of the object and δ the deformation parameter, while the functions involved in the metric tensor components are F(r) = � 1 − 2k r �δ , G(r, θ) = � r2 − 2kr r2 − 2kr + k2 sin2 θ �δ2−1 , (2) H(r, θ) = � r2 − 2kr �δ2 � r2 − 2kr + k2 sin2 θ �δ2−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (1), (2), it is straightforward to obtain the Schwarzschild limit when δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since at the Schwarzschild limit, k = M, δ can be interpreted as a measure for how much more (or less) mass M = kδ the ZV object has when compared to a Schwarzschild BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Subsequently, the deformation δ captures the oblateness of the compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' When δ > 1 the ZV geometry de- scribes a spacetime around the central object that is more oblate than a Schwarzschild BH, while when 0 < δ < 1 the central object is more prolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For δ = 0 (which iden- tically means M = 0) we obtain the Minkowski space- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' According to the no-hair theorem, when 0 < δ ̸= 1 holds the event horizon is broken;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' a true curvature sin- gularity appear at r = 2k [101], besides the typical one at r = 0, and the ZV metric describes a naked singular- ity [102–105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Interestingly, when spherical symmetry is broken the geometry obtains a non-zero quadrupole mo- ment M2 = δ(1 − δ2)M 3/3 [106, 107] which eventually leads to the Carter constant (or any other higher order Killing tensor) being broken [87–89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' ORBITAL DYNAMICS Regardless of its peculiar causal structure, the ZV pri- mary we will focus on should possess as many ‘good’ fea- tures as those of astrophysical compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' It has been shown that the line element (1) has an innermost stable circular orbit (ISCO) when δ > 1/ √ 5 at [71, 88] rISCO = k � 1 + 3δ + � 5δ2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (3) Furthermore, if δ ≥ 1/2 then the geometry has both an ISCO and a photon sphere (PS), where unstable null geodesics accumulate [108, 109], at [71] rPS = k(1 + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (4) Notice that at the Schwarzschild limit where δ = 1, rPS = 3M and rISCO = 6M as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Hereafter, we will focus on spacetime deformations that are larger than 1/2 in order to have an exotic central object with a PS and an ISCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Geodesics A first-order approximation to EMRI evolution can be accomplished through geodesics of a point-particle of mass µ which plays the role of the secondary orbiting around the primary supermassive compact object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The geodesic equations read ¨xκ + Γκ λν ˙xλ ˙xν = 0, (5) where Γκ λν are the Christoffel symbols of spacetime, xκ is the four-position vector, ˙xκ is the four-velocity vector and the overdot denotes differentiation with respect to proper time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Stationary and axisymmetric spacetimes, such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (1), possess metric tensor components that are t- and φ-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Therefore, they admit at least two con- served quantities (due to stationarity and axisymmetry) throughout geodesic evolution, namely the energy E and z-component of the orbital angular momentum Lz E/µ = F(r)˙t, Lz/µ = � r2 − 2kr � sin2 θ F(r) ˙φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (6) The t- and φ-momenta can be expressed with respect to the conserved quantities and the non-zero metric tensor components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Together with the conservation of the rest mass µ of the secondary, (preservation of four-velocity) which leads to gλν ˙xλ ˙xν = −1, the geodesics of test par- ticles present three constants of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Specifically, the conservation of the secondary’s four-velocity leads to a constraint for bound orbits ˙r2 + H(r, θ) G(r, θ) ˙θ2 + Veff = 0, (7) where the effective potential Veff has the form Veff ≡ 1 grr � 1 + E2 gtt + L2 z gφφ � , = F(r) G(r, θ) � 1 − E2 F(r) + F(r)L2 z (r2 − 2kr) sin2 θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (8) The curve defined when Veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' the curve of zero velocity (CZV), can be used in order to choose proper initial conditions that lead to bound orbits in the external vicinity of the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Bound geodesic motion can, generically, be charac- terized by three orbital frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' These frequencies are associated with the radial rate of transition between the periapsis and apoapsis of the geodesic (ωr), the rate of longitudinal oscillations through the equatorial plane (ωθ) and the revolution around the primary (ωφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Generic trajectories with irrational ratios of orbital fre- quencies span on two-dimensional tori and fill them com- pletely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' To the contrary, when the ratio of two orbital frequencies is a rational number then the geodesic is pe- riodic (or resonant) and returns to its initial position af- ter a number of oscillations defined by the multiplicity 4 of the resonance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Such orbits are special in the sense that they are not phase-space filling and therefore, can directly affect the evolution of EMRIs when encountered [22, 92, 93, 97, 98, 110–122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Inspirals To construct the inspiral trajectory we numerically in- tegrate the coupled system of r, θ equations, after uti- lizing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (6), augmented with post-Newtonian (PN) fluxes for the energy and angular momentum, respec- tively [123–126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This treatment, though approximate, takes into account the dominant contribution of the sec- ondary’s radiative backreaction to the spacetime geome- try, at second PN order, and results to an adiabatic evo- lution of the EMRI through time-dependent shifts onto the energy and z-component of angular momentum of the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since the inspiral evolves very slowly, the orbit is treated, in small timescales, as a geodesic, while for long timescales the trajectory is driven adia- batically through successively damped geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This method, known as the hybrid kludge scheme, has been shown to perform very well when compared to Teukolsky- based Kerr waveforms for EMRIs [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Notice that the ZV metric has a non-trivial multipo- lar structure due to the deformation parameter δ, and in particular a non-zero mass quadrupole tensor, as op- posed to that of Schwarzschild BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' At second PN order, the kludge scheme [124] involves the mass quadrupole moment M2 (for Kerr), thus to construct a more appro- priate inspiral around ZV compact objects we apply a modification to the fluxes (see [92, 93, 97, 98, 128, 129]) in order to include the quadrupole moment M2 of the ZV metric, which represents the effect of δ on the evolution of E, Lz, and set the spin parameter to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The adiabatic approximation, together with the flux augmentation, employed here has recently been found to provide results qualitatively equivalent to evolutions with instantaneous self-force in non-Kerr electromagnetic analogues, which indicates that the methods we use can in principle describe resonance-crossings with sufficient accuracy in EMRIs [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Nevertheless, more accurate inspirals can be built by directly solving the wave equa- tion resulting from metric perturbations and calculating the GW emission at the object and infinity, though this is a much more time consuming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We assume linear variations of the momenta as in [97, 98, 130] E1 = E0 µ + �dE dt ����� 0 Nr Tr, (9) Lz,1 = Lz,0 µ + �dLz dt ����� 0 Nr Tr, (10) where E0, Lz,0 are the initial energy and z- component of the angular momentum, respectively, and ⟨dE/dt⟩|0, ⟨dLz/dt⟩|0 are the radiation fluxes calculated at the beginning of the inspiral, through the equations in [98, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Tr is the time that the orbit takes to travel from the periapsis to apoapsis and back, while the fluxes (9) and (10) are updated every Nr cycles for the whole EMRI evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Detecting chaos To understand the phase space structure of orbits around the ZV primary we can employ well-known tools in order to gain further intuition regarding orbital phe- nomena and chaotic imprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' A typical example is the Poincar´e surface of section which is constructed by suc- cessive intersections of geodesics, with varying initial con- ditions, on a surface of section (here, we choose the equa- torial plane) with strictly positive (or strictly negative) direction of intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The structure of the Poincar´e map can instantly reveal if ergodic chaos is present, through disorganized intersections, or indirect imprints of non-integrability with the appearance of resonant is- lands, that encapsulate stable periodic points [4], and ex- ist due to non-integrability, in accord to the Kolmogorov- Arnold-Moser (KAM) and Poincar´e-Birkhoff theorems [131–133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since the ZV metric does not have a Carter- like constant these features are present in its orbital phase space [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Another tool to detect chaos is the rotation number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We calculate it by tracking the angle ϑ between two suc- cessive intersections on the Poincar´e map relative to the fixed central point of the map which corresponds to a circular orbit that intersects the surface of section ex- actly at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The rotation number is defined as the accumulation of many angles measured between consecutive intersections as [4] νϑ = 1 2πN N � i=1 ϑi, (11) for which when N → ∞ (with N the number of angles measured), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (11) converges to the radial and polar orbital frequency ratio νϑ = ωr/ωθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Calculating consecu- tive rotation numbers for different geodesics, by smoothly varying one of the parameters or initial conditions of the system while keeping the rest fixed, leads to a rotation curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Integrable systems demonstrate monotonous rota- tion curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' On the other hand, non-integrable systems possess transient plateaus with a non-zero width when geodesics occupy resonant islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This designates a cru- cial aspect of resonant islands, that is when a geodesic is inside the island it shares the same rational ratio ωr/ωθ with the stable periodic point which leads to the plateau formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' a phenomenon that does not appear in inte- grable systems whatsoever, even though resonances still exist but occupy only a single point in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In- flection points also appear when trajectories pass through the intersection of resonant islands where unstable peri- odic points reside and chaotic layers,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='●● ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='●● 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='06 r/M r\uf110 νϑ=1/4 νϑ=2/7 νϑ=1/3 νϑ=1/2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Poincar´e map of a test-particle secondary with µ = 1M⊙ orbiting around a ZV supermassive object with M = 106M⊙ and δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The secondary’s conserved en- ergy and angular momentum as E/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='95, Lz/µ = 3M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Black curves that surround the central point of the map designate intersections through the equatorial plane of generic orbits, while colored curves designate intersections that belong to different resonant islands of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' onant islands, emanate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Nevertheless, by changing the initial conditions the orbit can be driven through the is- land and give rise to a typical plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Rotation curves are not only a tool that is used for geodesics but can also be employed in dissipative sce- narios, where the rotation number evolves with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In the case of an EMRI, one can use se- lected timesteps of the inspiral as initial conditions for a geodesic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Through the non-dissipative tra- jectory, the Poincar´e map and eventually the rotation number of each timestep can be calculated in order to plot a series of rotation numbers as the EMRI evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The same attributes hold here as well, namely monotonous dissipative rotation curves for integrable sys- tems and appearance of plateaus for non-integrable EM- RIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For more information regarding integrability and chaos in EMRIs see the following series of works and ref- erences therein [4, 22, 88, 92–94, 97, 98, 134, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' GEODESIC AND INSPIRAL EVOLUTION By solving the coupled r, θ second-order ordinary dif- ferential equations, together with the first-order decou- pled equations for t and φ from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (6), we obtain bound orbits that reside inside CZVs and never plunge nor es- cape to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' To check the precision of our geodesics we evolve the constraint equation (7) for ∼ 104 revo- lutions and find that it is satisfied within one part in 108 − 1010 depending on the initial conditions and defor- mation of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' To guarantee numerical accuracy for inspirals, we cal- culate the 4-velocity in each update of the fluxes and check its conservation along a geodesic evolution with initial conditions the energy, z-component of angular mo- mentum, position and velocity at every update timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For all simulations presented hereafter, the constraint is satisfied to within a part in ∼ 108 for the first 104 cross- ings through the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' As a first step, we reproduce some qualitative features of the ZV metric, namely its non-integrability [88], which will be later used to choose initial conditions in order to evolve EMRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 1 we show the Poincar´e map for a ZV central object with δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5, meaning that the we utilized an oblate deformation with respect to Schwarzschild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We first observe a central point on the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since the map captures intersections of geodesics, the central point designates an orbit that is circular and always cuts the surface of section at the same (r, ˙r) po- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Around the center, various black curves appear (known as KAM curves) that are formed through suc- cessive intersections of generic orbits with varying ini- tial position r(0)/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' A Schwarzschild BH (or in general any spacetime with integrable geodesics) would exhibit a Poincar´e map with KAM curves that only surround the central point of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The fact that our object does not have a fourth constant of motion (non-integrable), leads to the formation of nested islands around stable resonant points (see [22] for a zoom into the encapsu- lated structure of these islands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' With a pedantic search on the available parameter space, we can easily spot four resonant islands with dif- ferent multiplicity, which we designate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 1 with colored curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Notice how those do not surround the central point, but rather the stable points in phase space where the true resonant orbits emanate, and that the multiplicity defined by the denominator of the reso- nance (the longitudinal oscillations through the equato- rial plane) corresponds to the number of islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' It is noteworthy that the resonances appearing here, besides the 1/2-resonant island, are not those that strongly af- fect an inspiral, especially when assessing their impact in EMRI parameter estimation [110, 118, 119], yet we will later see that they can also significantly contribute to it if the spacetime is non-integrable, mainly because they accumulate before plunge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The rotation curves presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2 promote the previous discussion perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Various plateaus and in- flections appear right where we expect the rotation num- ber to have a rational ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The plateaus clearly des- ignate that any geodesic residing in a resonant island shares the same rational ratio of orbital frequencies with the center of the island, where stable periodic orbits em- anate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Even when an inflection point shows up at the rotation curve, meaning that the geodesic lies between the tips of two resonant islands where unstable periodic orbits exist, with a certain change in the initial velocity ˙r(0), we can access the island as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For completeness, we have considered both oblate (δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5) 6 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 r/M νϑ δ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 (oblate ZV) δ=1 (Schwarzschild) δ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 (prolate ZV) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='80 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='85 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='90 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='33 r/M νϑ νϑ=1/4 νϑ=2/7 νϑ=1/3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Left: Rotation curves for geodesics of a test particle with µ = 1M⊙, E/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='95, Lz/µ = 3M and ˙r(0) = 0 around a ZV object with M = 106M⊙ with varying deformation δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For completeness, we present the rotation curve for a Schwarzschild BH with the same parameters as above and δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right: Zoom into a region of interest for the δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The horizontal colored lines designate where resonances appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='797 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='799 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='801 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='803 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='805 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='288 r/M νϑ νϑ=2/7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='878 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='882 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='884 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='288 r/M νϑ νϑ=2/7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Left: Zoom into the inflection point for the 2/7-resonant island shown on the right subfigure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' initial conditions and parameter as the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right: Rotation curve for geodesics of a test particle with µ = 1M⊙, E/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='95, Lz/µ = 3M and ˙r(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='01 around a ZV object with M = 106M⊙ with δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' and prolate (δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5) deformations1 and encounter simi- lar effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For the rest of the discussion we will focus on oblate deformations for the following reasons: (i) oblate defor- mations are usually the ones enabling the strongest ef- fects of non-integrability and chaos (see [22, 93, 94, 97, 98, 121, 129]), (ii) as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2, an oblate defor- mation drives the resonances closer to the central object thus we expect amplified chaotic effects and (iii) δ > 1 is an optimal choice to imitate BHs since for these cases the ZV object possesses a PS, an ISCO and produces similar shadows [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 1 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2 we also present a typical rotation curve for Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' So far, the discussion involved zeroth-order approxima- tions of EMRIs since there was no radiation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Turn- ing on the fluxes (rates of change of orbital energy and angular momentum) (9) and (10) leads to a numerical integration that is more intricate but the results are very interesting since the secondary inspirals adiabatically to- wards the primary due to fluxes through GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 4 presents a typical inspiral of a secondary that transverses the 1/3-resonant island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The initial conditions are not fine tuned so what is presented here is a generic feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For this inspiral, we have updated the fluxes Nr = 800 times, every Tr = 200M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since the revolution period in this case is ∼ 90M, the total evolution time of the EMRI is ttotal = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6×105M ∼ 1800 revolutions (roughly 9 days for the mass of the primary considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The fluxes undergo a rather peculiar behavior at first glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' At a certain time, they become more negative 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='72 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='71 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='69 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='67 t (days) dE/dt (× 10-16) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='3322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='3327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='3332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='3337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='3342 t (days) νϑ νϑ=1/3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Left: Evolution of energy flux dE/dt for an inspiraling secondary with µ = 1M⊙ and initial parameters E0/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='95, Lz,0/µ = 3M, r(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='985M, θ(0) = π/2, ˙r(0) = 0, where ˙θ(0) is defined by the constraint equation (7), around a ZV primary with M = 106M⊙ and δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right: Evolution of the rotation number of the same inspiral as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The horizontal solid red line designates the 1/3-resonant island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For both plots, the vertical dashed red line corresponds the time when the secondary passes through the center of the island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 2 4 6 8 10 12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='57 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='83 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='96 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='21 t (days) dE/dt (× 10-16) 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='29 t (days) νϑ νϑ=2/7 νϑ=1/4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Left: Evolution of energy flux dE/dt for an inspiraling secondary with µ = 1M⊙ and initial parameters E0/µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='95, Lz,0/µ = 3M, r(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='824M, θ(0) = π/2, ˙r(0) = 0, where ˙θ(0) is defined by the constraint equation (7), around a ZV primary with M = 106M⊙ and δ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right: Evolution of the rotation number of the same inspiral as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The horizontal solid colored lines designates the 2/7 (green) and 1/4-resonant islands (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For both plots, the vertical dashed colored lines corresponds the time when the secondary passes through the center of the corresponding islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This instant of time designates the entry of the secondary into the 1/3 plateau of the island (see right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right after the entry, the rate of change of orbital energy reaches a local maximum, at a special instant of time which is shown with a verti- cal dashed line, beyond which it continues to decline due to the inspiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This behavior is not at all peculiar, but rather has a true physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Regardless of the fact that the secondary transverses the island while sharing the same rotation number throughout it, the secondary only reaches the stable periodic point at the center of the island on a certain time designated with the verti- cal dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This is when the orbit becomes exactly periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Periodic orbits are the closest to circular ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since circular orbits emit monochromatic radiation at a single frequency (twice the revolution frequency) their energy emission is minimized (thus the rates of change of orbital energy and angular momentum are maximized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since a resonance emits quasi-monochromatic radiation, at the time that the inspiral passes through the perfect resonance, the fluxes are also maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The exit from the island takes place when the flux value at the entry is met again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The secondary can spend ∼ 100 cycles in per- fect resonance which translates to roughly half a day or ∼ 5% of the whole EMRI evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The picture is qual- itatively the same for the flux of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 5 presents an even more intriguing EMRI evolu- tion around the ZV primary, that undergoes two consec- 8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='5 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='66 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='69 t (days) dE/dt (× 10-16) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='17 t (days) dE/dt (× 10-16) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Left: Zoom into the evolution of energy flux dE/dt from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 5 through the 2/7-resonant island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right: Same as the left panel for the 1/4-resonant island crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' utive island crossings during a single evolution, namely the 2/7 and 1/4 islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Initial conditions are slightly (but not strongly) fine tuned so what is presented here is quasi-generic, in a sense that it can occur for a small but non-zero set of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For this inspi- ral, we have updated the fluxes Nr = 1100 times, ev- ery Tr = 200M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Since the revolution period in this case is ∼ 80M, the total evolution time of the EMRI is ttotal = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='2 × 105M ∼ 2800 revolutions (roughly 12 days for the mass of the primary considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The time that the secondary spends in the first island, in perfect resonance, is ∼ 200 cycles which translates to roughly a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' After exiting the first island, the orbit en- ters shortly after a subsequent resonant island and occu- pies it for another ∼ 50 cycles which translate to one fifth of a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In total the inspirals experiences 250 cycles of perfect resonance (without taking into account pre- and post-resonant effects) which correspond to ∼ 10% of the whole evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Similar analyses for non-Kerr EMRIs with the same mass ratios used here (µ/M = 10−6) ex- perience 250 cycles in the most prominent 2/3-resonant island [97, 98], though due to the much slower inspiral (resonances appear further from the central object) this only corresponds to ∼ 3% of the whole evolution spent in a single island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Practically, the fact that a multitude of islands gather close to the plunge gives us the ability to probe subdominant resonances consecutively, in a short period of time, and have an EMRI that experiences a significant fraction of its evolution in resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Zooming in on the left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 5 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 6), we observe a complete agreement with what is demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Especially for the 2/7-resonant island (left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 6), the initial condition chosen here al- lows for the secondary to remain in it for a substantial amount of time that leads to a significant maximization of the energy flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This means that the orbit enters deep into the island and probes the stable periodic point for a significant amount of time till it exits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The initial con- dition may look special, since a typical crossing would not maximize fluxes considerably, nevertheless, plateaus on dissipative rotation curves are still generic for a wide range of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' GRAVITATIONAL WAVES In this section we approximate the dominant GW emis- sion of an inspiraling stellar-mass secondary around an oblate ZV supermassive compact object and search for imprints of non-integrability in the waveforms produced by such EMRIs when detected by a LISA-like interfer- ometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For the GW modeling, we use the quadrupole approximation described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Quadrupole formula The radiative component of the metric perturbation in- troduced by the secondary at luminosity distance d from the source T can be read at the transverse and traceless gauge as hTT ij = 2 d d2Qij dt2 , (12) where Qij is the symmetric and trace-free (STF) quadrupole tensor Qij = �� xixjT tt(t, xi) d3x �STF , (13) with t being the coordinate time measured at very large distances from the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The source term of the sec- ondary (which is treated as a point particle through a delta function) is T tt(t, xi) = µδ(3) � xi − Zi(t) � , (14) 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Left: Frequency evolution of an EMRI through a consecutive crossing of resonant islands with parameters and initial conditions as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The approximate GWs are detected at luminosity distance d = 100 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The spectrogram presents the effect on a single harmonic of the waveform when it crosses the 2/7-resonant island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Right: Continuation of frequency evolution of the same EMRI with initial conditions as in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The spectrogram presents the effect on the waveform when it crosses the 1/4-resonant island.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' where Z(t) = (x(t), y(t), z(t)) the position vector in pseudo-Cartesian coordinates and x(t) = r(t) sin θ(t) cos φ(t), (15) y(t) = r(t) sin θ(t) sin φ(t), (16) z(t) = r(t) cos θ(t), (17) the trajectory components with respect to flat spheri- cal coordinates, under the assumption that our space- borne detector is positioned at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Even though we have identified the Schwarzschild-like coordinates (r, θ, φ) of the secondary’s trajectory with flat-space coordinates, known in the literature as the “particle-on-a-string” ap- proximation, and we assume a finite luminosity distance d from the source, such prescription is not strictly valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Nevertheless, it has been found to work very well when generating EMRI waveforms in GR [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' GWs can be projected onto two polarizations, + and ×, with the introduction of two unit vectors, p and q, which are defined with respect to a third unit vector n that points from the source to the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The triplet of unit vectors (p, q, n) is chosen so that they form an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The polarization tensor components read ϵij + = pipj − qiqj, ϵij × = piqj + pjqi, (18) and allow for the metric perturbation to be written as hij(t) = ϵij +h+(t) + ϵij ×h×(t), (19) with h+(t) = 1 2ϵij +hij(t), h×(t) = 1 2ϵij ×hij(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (20) The GW polarization components can then be described in terms of the position, Zi(t), velocity, vi(t) = dZi/dt, and acceleration ai(t) = d2Zi/dt2 vectors as [130] h+,×(t) = 2µ d ϵ+,× ij � ai(t)Zj(t) + vi(t)vj(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (21) LISA’s response to an incident GW is rather complicated and depends on the antennae response patterns (see [125, 136] for the full equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Here we assume a detector that lies at a luminosity distance d with orientation n = (0, 0, 1) with respect to the source, and utilize hα(t) = √ 3 2 � F + α (t)h+(t) + F × α (t)h×(t) � , (22) where α = (I, II) is an index representing the different antenna pattern functions F + α , F × α which can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' [125, 136, 137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' For phenomenological purposes we will use a single-channel approximation, that is set α = I in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' (22), since it is enough to accommodate the fundamental parts of gravitational radiation emitted by the EMRI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' its phasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Gravitational-wave frequency evolution and cumulative glitches To comprehend how a ZV EMRI imprints non- integrable effects, such as plateaus, onto its emitted wave- form, we calculate the GW in the Einstein-quadrupole approximation for the particular inspiral outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' IV that crosses two consecutive islands, namely the 2/7 35 70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='60 30 60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='80 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='58 intensity (×10-16) 25 50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='75 20 40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='54 15 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='52 10 20 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='50 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='65 2 3 10 12 1 4 5 8 9 11 time (days) time (days)10 and 1/4-resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We focus solely on this example in order to demonstrate that even subdominant resonances affect EMRI evolution, when the spacetime presents chaotic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We perform a Fourier transform on the extracted wave- form from the inspiral and plot a density spectrogram that displays the evolution of one of the harmonics with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We use the same methodology for the spectrogram as in [97, 98], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' we cut the waveform in time segments with a particular window size and offset and perform consecutive Fourier transforms, in order to overcome the uncertainty between frequency and time resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Here, since the inspiral is evolving fast and the island crossings have different timescales, we use dis- tinct window size and offsets for the chunks of evolu- tion around the two resonant islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This is necessary for demonstration purposes since the first resonance is crossed much slower than the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 7 we show the spectrogram of the GW ex- tracted from the EMRI that successively transverses two resonant islands of stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' In both cases, when the is- land is met the waveform frequency evolution loses mono- tonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' The resonant crossings manifest into the GW with either a plateau-like pattern (left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 7) or a rapid glitch2 (right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Both instances des- ignate that the system is non-integrable, especially the encounter of the first island that shares a striking similar- ity with plateaus appearing in rotation curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' However, this is an outcome of short-term occupancy in the islands and altogether a smaller deformation of spacetime, with respect to those presented in non-Kerr EMRIs [97, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' If the inspiral is given more time in the island, the manifes- tations should resemble more those of discontinuous GW glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Yet, the fact that resonant island crossings are imprinted in the GW of a ZV EMRI, in a short period of time, can serve as a ‘smoking gun’ for a non-integrable BH mimicker in the center of our galaxy, since these phe- nomena do not appear in Schwarzschild EMRIs that have monotonous rotation curves and spectrograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' CONCLUDING REMARKS Probing the spacetime around supermassive compact objects with EMRIs is one of the prime targets for space- borne detectors like LISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Compact objects are either spinning and/or surrounded by astrophysical environ- ments, especially those residing in active galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Therefore, spherical symmetry is rather fragile and is usually broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' When there are not enough spacetime symmetries to guarantee the integrability of geodesics around these objects, then LISA may be able to detect 2 Unfortunately, the inspiral enters into a region of phase space right after the crossing of the 1/4-resonance where pure chaos is present and the Fourier peaks become continuous so we could not evolve the system for more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' particular phenomenological imprints of such manifesta- tions like GW glitches around expected transient orbital resonances [97, 98] that differ significantly from instru- mental glitches [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' We showed that deformations from spherical symme- try which keep the spacetime static, described by the ZV geometry, have the potential to exhibit GW glitches when involved as primary objects in EMRIs, due to the non-integrability of test-particle dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' These phe- nomena not only appear for single resonant island cross- ings, such as those occurring in non-Kerr EMRIs, but also are present in a cumulative and short-timescale manner, where the secondary transverses two (and possibly more) subdominant resonant islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Note though that this can also occur in non-Kerr EMRIs if long-lasting evolutions are to be performed, though the effect of subdominant resonances are usually suppressed [97, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' When supermassive compact objects, such as those residing in galactic centers, have multiple interpreta- tions due to the plethora of objects that can cast sim- ilar shadows, GW astronomy with LISA can, in princi- ple, distinguish between models that are integrable or not through the detection of glitches in gravitational waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Synergies between shadow observations and space-borne detectors can, therefore, narrow down the parameter space of solutions describing supermassive ob- jects in galactic centers, and in particular can emphati- cally decide if M87* and Sgr A* are described by Kerr and Schwarzschild geometries, respectively, unless the EMRI includes multiple [40, 139] or spinning [120, 140] secondaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Our analysis only deals with the phenomenological im- prints of non-integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Nevertheless,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' if one wants to utilize the aforementioned phenomenology in practice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' a consistent glitch modeling analysis for non-integrable EMRIs should be carried out,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' in a systematic way in or- der to understand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' first,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' if these phenomena are clearly detectable with space interferometers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' to which extent they affect parameter estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' and third,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' to which degree these effects differ from standard transient resonances experienced by integrable EMRIs which are already sufficiently modeled with PN techniques [118,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 119] and gravitational self-force [110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 117,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 141,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' 142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors warmly thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' Theocharis Apos- tolatos for critical comments and helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' This work was supported by the DAAD program for the “promotion of the exchange and scientific cooper- ation between Greece and Germany IKYDAAD 2022” (57628320).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' are grateful for hospitality provided by the Department of Physics of the Univer- sity of Athens, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' acknowledges financial sup- port provided under the European Union’s H2020 ERC, Starting Grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' DarkGRA–757480 and the MIUR PRIN and FARE programmes (GW-NEXT, CUP: 11 B84I20000100001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FJT4oBgHgl3EQfOizn/content/2301.11483v1.pdf'} +page_content=' [1] K.' metadata={'source': 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Wan1†, Yun Lai2, C.T. Chan3, and Meng Xiao1, 4* +1Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education and School of Physics and +Technology, Wuhan University, Wuhan, China +2National Laboratory of Solid State Microstructures, School of Physics, and Collaborative Innovation Center +of Advanced Microstructures, Nanjing University, Nanjing, 210093, China +3Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, +Hong Kong, China +4Wuhan Institute of Quantum Technology, Wuhan 430206, China +Corresponding E-mail: † ddwan@whu.edu.cn; * phmxiao@whu.edu.cn + +A boundary mode localized on one side of a finite-size lattice can tunnel to the opposite side +which results in unwanted couplings. Conventional wisdom tells that the tunneling probability +decays exponentially with the size of the system which thus requires many lattices before +eventually becoming negligibly small. Here we show that the tunneling probability for some +boundary modes can apparently vanish at specific wave vectors. Meanwhile, the number of wave +vectors where tunneling probability vanishes equals the number of lattices perpendicular to the +boundary. Thus, similar to bound states in the continuum, a boundary mode can be completely +trapped within very few lattices whereat the bulk band gap is not even well-defined. Our idea is +proven analytically, and experimentally validated in a dielectric photonic crystal. This feature +allows for the extreme flexibility in tunning the hopping between localized states or channels, +which facilitates unprecedented manipulation of light such as integrating multiple waveguides +without crosstalk and photonic non-abelian braiding. + + + +2 + +The spectrum of a system typically consists of continuous spectra and discrete spectra (left panel of +Fig. 1a). Conventional wisdom says that eigenvalue spectrum of bound states is discrete, while the +eigenvalue spectrum of unbound states forms a continuum. For electronic systems, the usual +convention is that negative energy states are bound and are hence discrete, while positive energy states +form a continuum. For light and sound waves, all eigen-frequencies are positive and as such, discrete +states form due to the boundary condition imposed by a barrier, which is a material that forbid wave +propagation (e.g. having a “band gap” 1, 2). A typical example is the discrete states that exist in a cavity. +Discrete states can remain a bound state in the sense that there is no “free state” continuum it can +couple with. The discrete state can be perfectly confined by the barrier if the width of the barrier is +infinite (Fig. 1a-II). When the width of this barrier is finite, there is some probability that the state can +tunnel through the barrier and becomes a resonance state (Fig. 1a-III). If a state's energy lies inside the +continuous spectrum, it will unavoidably couple with states in the continuum and become a resonance +state. As an exception to this rule, bound states in the continuum (BICs) can be spatially bound with +energy inside the continuous spectrum (Fig. 1a-I)3-9. Here, parallel to BICs, we show another +counterintuitive concept: a state can get completely trapped (infinite Q factor if no intrinsic loss) by a +band gap material with a finite and very small thickness. The solid black line in Fig. 1a-IV sketches +one such state when the number of lattices is 𝑁𝑦 = 4 (𝑁𝑦 can be even smaller as will be shown later) +whereat the bulk gap is not well-defined. + +A state being completely trapped indicates that there is no probability for the state to tunnel through +the band gap material. Considering a symmetric double-well configuration as shown in Fig. 1a-V, then +a state localized on the left-hand-side well (blue line) cannot tunnel into the state on the right-hand- +side well (red line). Or equivalently, there is no coupling (no hopping) between the two states in Fig. +1a-V. We note that the probability of tunneling is a crucial factor for quantum information processing +as it determines the lifetime of the trapped states10-13. Control over the hopping coefficient also enables +manipulating interaction between different trapped states to generate various entangled quantum +states14-16. Meanwhile, fine-tuning of the coupling coefficient is a key requirement in programmable +photonic simulators17, non-abelian braiding of photons18 and quantum computers19, especially when +nanostructures are considered. + +3 + + +With one additional direction, the two states in Fig. 1a-V will extend into two waveguide modes as +sketched in Fig. 1b. Coupling between waveguide modes introduces crosstalk which limits the +integration of multiple waveguide channels into a compact device20. With the recent explosive growth +of research in topological physics21-24, topological boundary modes and hinge modes associated with +nontrivial bulk topologies have attracted a lot of attention due to their robustness against disorder and +fabrication imperfections25-29. However, these boundary and hinge modes also suffer from the +tunneling effect when the width of the system is not large enough30-33. So, even topological boundary +modes can be gapped in the presence of cross coupling. On the other hand, controlling the tunneling +of topological boundary modes or hinge modes can achieve new versatile controllability, such as spin +flipping34, electrical switching35, etc. + +Here, we consider the coupling between two boundary modes on a two-dimensional (2D) strip +geometry. Different from the prevailing understanding that the coupling vanishes only when the width +of the strip is large enough, we discover that the coupling can vanish (i.e., no tunneling) at specific +wave vectors (nodes) for a narrow strip with very few lattices. Interestingly, the number of nodes +equals the number of lattices perpendicular to the boundary. Moreover, similar as BICs, one of the +boundary modes with these specific wave vectors exhibit infinite Q factor even if wave leakage is +introduced at one side. Using a Hamiltonian model, we analytically solved this system and proved the +existence of this novel feature. Furthermore, we find this unique feature presents in many systems such +as 2D plasmonic spheres array and 2D photonic crystals (PCs). We further verify this feature +experimentally within a 2D PC, showing both the mode dispersions and the capability of tuning the +magnitude of the hopping. + +Our system is sketched in Fig. 1b, which is periodic along the x-direction and finite along the y- +direction. Each unit cell contains one atom or meta-atom which can support multiple modes such as +S , +xP , +yP , etc. These modes interact and evolve into bands and the physics can be captured + +4 + +succinctly using the usual tight-binding description. Inside a band gap, the system may exhibit +boundary modes if the parameters are appropriately chosen, as shown schematically in Fig. 1b with +the red and blue shaded regions denoting the mode profiles. These boundary modes can either be +Shockley states36, Tamm states37 or originate from topological reasons21, 22, 24, 25. In Fig. 1c, we sketch +the typical consequences of the coupling between these two boundary modes. Here the shaded areas +represent the projection of bulk bands, and the blue and red lines represent the dispersion of the +boundary modes. If no symmetry forbids coupling, they will couple whenever they cross each other, +constantly forming mini-gaps. + +In contrast to Fig. 1c, we show that the coupling can vanish at some specific nodes as pictorially shown +in Fig. 1d. Due to the mirror symmetry 𝑚𝑦, the boundary modes couple to form even and odd modes, +as shown respectively by the red and blue lines. They twist with each other and form several nodal +points. We emphasis that 𝑚𝑦 alone cannot necessarily guarantee the presence of these nodal points. +As will be shown more rigorously later, there is no coupling between boundary modes at these nodal +points. Intriguingly, we find that the number of nodes equals the number of lattices along the y- +direction ( +y +N ). Meanwhile, the coupling between boundary modes here is size dependent and is thus a +peculiar type of finite size effects (FSEs).30, 32, 38 We note that anomalous FSEs has been noted for +helical boundary modes in topological insulator where the strength of the FSE decrease non- +monotonically with the size32, 33, 39. The oscillation length therein is hundreds or thousands of lattice +constants which is thus significantly different from our case as the coupling here oscillates at the lattice +scale. + +First, we provide a tight-binding model Hamiltonian that exhibits the salient features outlined in Fig. +1d. We derive this Hamiltonian with the coupled dipole equation40. We assume each site supports two +dipoles +xP and +yP . For simplicity, we assume that other excitations are either far away in energy or +are orthogonal to +xP and +yP (e.g., the +zP dipole). The hopping from a dipole +jP at lattice site +j +R +to the dipole +iP at +i +R is given by41 + +5 + +𝑮⃡ (𝒓) = +𝑡 +𝑟5 [3(𝑷𝑗 ∙ 𝒓)𝒓 − 𝑟2𝑷𝑗] + + + + + + + + (1) +with +j +i += +− +r +R +R , and here t is introduced only to ensure that the unit of hopping is energy. For +convenience, we set +1 +t = and r is in a unit of the lattice constant a. We truncate the hopping to the +next nearest neighbor which are the minimal interactions needed to explain the physics presented in +Fig. 1d. The momentum space Hamiltonian for a periodic system is thus +( +) +( +) +4cos +cos +2 +cos +/ +2 +3sin +sin +/ +2 +3sin +sin +/ +2 +2cos ++cos +4 +cos +/ +2 +x +y +x +x +y +x +y +x +y +x +k +k +k +k +k +H +k +k +k +k +k + + +− +− +− + + +=  + + + +− +− ++ + + + +(2) +The bands are nondegenerate except at Γ and 𝑀 as protected by the +4v +C symmetry (See +Supplementary Materials Sec. I). We note that though we derive the Hamiltonian with the coupled +dipole equation, a similar tight-binding Hamiltonian can also describe electronic systems of the same +symmetry if p-orbitals ( +xP and +yP ) are dominant in the energy range of interest. + +As we are interested in a strip geometry which is periodic along the x-direction and finite along the y- +direction, we provide the projected band structure of the periodic system along the +xk direction as +shown in Figs. 2a-d with the light gray background (the projected bulk band continuum) as a reference. +Here we can focus on the +0 +xk  + region as the +0 +xk  + region is simply related by time-reversal +symmetry. Now we have a band gap region between the two bands and the band gap closes only at +0 +xk = + and +/ +xk +a + += + . The bulk polarization 𝑝𝑦 = ∫ +−𝑖⟨𝑢𝑖|𝜕𝑘𝑦|𝑢𝑖⟩𝑑𝑘𝑦 +𝐵𝑍 + for any value of +xk is +quantized as  , and hence a semi-infinite system possesses a topological boundary mode which is +located inside the band gap with dispersion connecting the two band edges42. (See Supplementary +Materials Sec. I) + +Then we proceed to investigate the peculiar coupling feature for a finite number of lattice sites ( +y +N ) +along the y-direction. First, we start with an extreme case where +1 +y +N = as shown in Fig. 2a. This + +6 + +case corresponds to an infinite chain of dipoles. In this case, +xP and +yP dipoles decouple and exhibit +positive and negative dispersions, respectively.43 These two bands cross once at +/ 2 +xk +a + += +. When +2 +y +N = +, i.e., two coupled infinite chains, +xP and +yP dipole couple with each other and there is no +pure band with only +xP or +yP component. Instead, one can label the band with either mirror +symmetric (even) or antisymmetric (odd). As shown in Fig. 2b, there are now 4 bands in total and two +of them appear inside the bulk gap region, with one even state (red) and one odd state (blue) for the +yP component. Interestingly, now the red and blue bands cross with each other twice which is also the +value of +y +N . As we further increase +y +N to 3 and 4 as shown in Figs. 2c and 2d, respectively, the gray +bands start filling up the projection of the bulk periodic bands, and the red and blue bands approach +the dispersion of the boundary modes for a semi-infinite system. Still, the number of nodes is always +equal to +y +N . + +To prove that the number of nodes between the even and odd mode is equal to +y +N , we analytically +solve the system. (Proof in Supplementary Materials Sec. II). We first obtain the eigen energy and +eigenstates of the boundary mode for a semi-infinite system [equation (S1-S7)]. Then with perturbation +theory to the first order, we obtain that the hopping strength +E + between the two boundary modes +localized on opposite boundaries as given by equation (S12). Thus, to the first-order approximation, +the frequencies of the even and odd boundary modes are +e +E +E ++  and +e +E +E +−  +, respectively. When +0 +E + += , the hopping strength is zero, and then there is a nodal point on the even and odd boundary +bands. It can be proved that +E + is an oscillation function of +xk and exhibits +y +N zeros for +(0, +/ ) +xk +a + + +. (Detail proof is also provided in Supplementary Materials Sec. II). To check that the +zeros of +E + indeed predict the number of nodes, we provide the locations of boundary band nodes +(red +) and the zeros of +E + + (blue square) in Fig. 2e. The number of the red + exactly equals the +number of the blue square for every +y +N , and the difference comes from the first-order approximation +we take. + + +7 + +The physics of such a peculiar feature goes beyond the tight-binding model as will be demonstrated +later with full wave simulations. As our system only requires the symmetry imposed by the +xP and +yP modes in a square lattice, the physics discussed here should be quite universal. Possible platforms +are photonic crystals, phononic crystals, cold atoms, and 2D materials with properly chosen orbitals. +As an example, we show that this peculiar coupling feature exists in a 2D array of plasmonic spheres +in Supplementary Material Sec. III. Different from the tight-binding model, the coupling between +plasmonic spheres extends to infinity. + +In addition, the physics discussed here also exists in 2D dielectric PCs where the band dispersion +originates from dipolar modes arranged in a square lattice. Previous studies showed that for a 2D PC +consisting of dielectric cylinders in air and considering the polarization with the electric field along +the cylinders (TM), the typical three lowest order modes are one monopole and two in-plane dipoles44. +As shown in Supplementary Material Sec. IV, when the frequency of the doubly-degenerate dipole +mode is lower than the monopole mode at the  point, there are two bands consisting of nearly pure +in-plane dipole modes (ranging from 9.7GHz to 13.3GHz in Fig. S6). Such a system exhibits the same +coupling feature induced twisting boundary modes with nodes, i.e., the odd and even boundary modes +cross with each other +y +N times over the positive first BZ. + +Furthermore, the vanishing of coupling is robust even if the monopole mode is hybridized with the +two in-plane dipole modes. The reason lies in the fact that the nodes between the odd and even +boundary modes are protected by the mirror symmetry and the hybridization with +z +M does not break +this mirror symmetry. As long as the nodes between the two boundary modes do not annihilate (the +perturbation introduced by +z +M is not strong enough), the number of nodes would be the same. Such +a feature then further releases the constraints on the observation. In the following, we provide an +experimental demonstration within a 2D dielectric PC in the microwave regime in the presence of +hybridization with +z +M . Figure 3a sketches the experimental setup, where we consider a mirror- + +8 + +symmetrical strip geometry of square lattice PCs. Aluminum plates are placed at the upper, lower, front, +and rear edges as PEC boundaries. The PEC boundary at the upper side is tightly attached to the +dielectric cylinders, allowing only TM modes. In the experiment, we need to move the upper PEC +boundary so as to measure the electric field distribution. This experimental setup, however, +unavoidably introduces a sub-millimeter (~0.5 mm) air gap between the cylinders and the upper PEC +boundary. Such an air gap has limited impacts on the dispersions of the band of interest. (See analysis +in Supplementary Materials Sec. V). For a better characterization of the boundary modes, we also keep +a small air gap (d = 4 mm) between the side PEC boundaries and the PC42. The presence of this air gap +only shifts the dispersion of the boundary modes a little bit while the interested effect is not affected, +which verifies once again the robustness of this FSE. (Supplementary Material Sec. V) + +We start with the measurement of the band dispersion. Figure 3b shows the projected band dispersion +for a semi-infinite system, wherein the gray areas represent the projection of the bulk bands while the +cyan line denotes the boundary mode inside the bulk band gap at around 12 GHz. Here, the frequencies +of the monopole mode and the dipole modes at the  point are 12.3 GHz and 12.7 GHz, respectively, +and thus the lower two bands are not pure in-plane dipole modes. We excite the sample at one side of +the sample, measure the field distributions and then apply Fourier transform to obtain their dispersions. +Figures 3c-f show the measured band dispersions (color code) together with the simulated band +dispersion (lines) for +y +N =2, 3, 4, and 5, respectively. The measured dispersion agrees well with the +simulations. As expected, the number of boundary band nodes equals +y +N for all the samples we have +measured and thus the twisting of boundary modes and the number of nodes is confirmed +experimentally. + +The boundary modes can be utilized as waveguide channels to guide waves. In our system, the +tunneling probability vanishes at these nodes, and the boundary mode is completely trapped to one +side. Similar to BICs, these trapped boundary modes should exhibit an infinite Q factor if the system +has no intrinsic loss. As shown in Fig. 4a, we replace one of the PEC boundaries with air while keeping + +9 + +all the other parameters the same as in Fig. 3. Figures 4b-d show the band structure and the Q factor +of the remaining boundary mode for +y +N =2, 3, and 4, wherein the colored line marks the boundary +band and the solid black lines denote the bulk bands. The bulk bands and one of the boundary bands +located near the remaining PEC boundary shift slightly, while the other boundary mode couples with +light cone and thus is not shown here. As a reference, the red and blue dash lines denote the boundary +mode dispersion in Fig. 3. The area outside the light cone (lower right shaded region) and above the +first-order diffraction limit (upper right shaded region) are not shown. It is clear that the Q factors of +the boundary modes are infinite at these nodes since they cannot tunnel through and thus leak outside. +In other words, the boundary modes at nodes can be trapped completely within very few lattices at +which the bulk band gap is not even well-defined. In contrast, those modes not at nodes host a finite +Q factor and become leaky modes. We also show the eigenfield distribution of one boundary mode at +nodes in Fig. 4a for +y +N = 3, where we can see the fields vanish in the air (almost completely white +inside the air region). More information about the eigenfields is provided in Supplementary Materials +Sec. VI. + +It is difficult to verify the infinity Q factor of the boundary modes at nodes in our experimental setup +due to the large intrinsic loss of material in microwaves. As an alternative way, we keep the two-side +PEC configuration and demonstrate the hopping between these two boundary modes. The hopping +between two boundary modes on opposite sides vanishes at these nodes and thus each boundary mode +can be confined to one side of the waveguide as it propagates. Otherwise, the field energy oscillates +between two boundaries alternatively. In general, the coupling strength between two boundary modes +is a function +xk as shown in Fig. S10a in Supplementary Material Sec. VII. In the following, we +confirm this peculiar coupling property experimentally with the same PC sample with +y +N = 4. There +are four nodes on the twisting bands of boundary modes and we choose the one at 11.80 GHz +( +0.4 +/ +xk +a + += +). Meanwhile, we also perform the experiments at 11.63 GHz where the coupling strength +is finite to see the field oscillation between two boundaries. + + +10 + +Figure 5a shows a photo of the experiment setup. Same as before, the sample will be covered with +another PEC top layer in the experiments and simulations. A point source, labeled by the red star, is +placed at the lower-left corner to excite predominantly the boundary mode localized on the lower +boundary. The measured and simulated electric field distributions inside the waveguide (bounded by +the black dashed lines) are shown in Figs. 5b and 5c. In the simulations, the relative permittivity of the +cylinders is set as 𝜀𝑟 = 9.0+0.02i with an imaginary part to simulate the inevitable loss in the +experiments. As shown in Fig. 5b, at a frequency of 11.63 GHz (not one of the nodal points), the +stimulated boundary mode couples to another boundary after propagating tens of lattices and then +recouples to the original boundary. In contrast, when the frequency matches the frequency of a node +(11.80 GHz), the excited boundary mode stays on the lower side while propagating to the right. The +field oscillation period is inversely proportional to the coupling strength between the boundary modes: +the smaller the coupling strength, the longer the oscillation period. The oscillation period is infinite at +a node frequency. Thus just a few lattices are able to prohibit the interaction between adjacent boundary +guiding modes effectively. Such a non-coupling feature facilitates the integration of multiple +waveguide channels into a compact device. + +In summary, we analytically solved a next-nearest-neighbor hopping model to investigate the +vanishing of coupling between boundary modes. The boundary modes of a finite-width strip twist +around each other, intersecting at nodes and the number of nodes equals the number of lattices across +the strip. This physics is ubiquitous and appears in systems such as the plasmonic sphere array, +dielectric PCs, and cold atoms. It leads to a filtering effect where only the components with wave +vectors matching those of the nodes survive (see Supplementary Material Sec. VIII). We +experimentally demonstrate this peculiar feature in PCs by measuring the mode dispersion and +mapping the electric field distributions of the boundary modes. Our work points to the possibility of +getting rid of the formation of gaps due to cross-coupling with properly chosen orbitals and lattice, and +thus opens a feasible way for integrating multiple waveguides into an extremely compact device. + +11 + +Acknowledgements +This work is supported by the National Natural Science Foundation of China (Grants No. 12274330, +No. 12274332, No. 11904264 and No. 11904265). D.W. is also supported by the Hubei Provincial +Natural Science Foundation (Grant No. 2020CFB670) and the Knowledge Innovation Program of +Wuhan-Shuguang (Grant No. 2022010801020125). C.T.C is supported by Research Grants Council +(RGC) Hong Kong through Grant AoE/P-502/20 and Croucher Foundation (CAS20SC01). Y.L. is +supported by the National Natural Science Foundation of China (Grants No. 12174188 and No. +11974176). + +Author contributions +D.W. and M.X. initiated and supervised the project. T.L. did the simulations, designed the samples and +performed the experiments. 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Mater. 10, 582-586 +(2011). + +15 + +Figures: + + + +Fig. 1 | Boundary modes can be trapped by a band gap material with a few lattices. a, Illustration +of a bound state trapped by a few lattices. b, The coupling of two boundary modes (denoted by the red +and blue shaded regions) localized on opposite sides of a strip geometry of a square lattice with +xP +and +yP orbitals. c-d, Sketches of the band dispersion for two different systems, where the coupling +between boundary modes is finite for all +xk in (c) and vanishes at several nodes for certain +xk s in +(d). + + +a +Spectrum +Mode profile +continuous +Bound statein +thecontinuum +potential barrier +Regular +potential +boundstate +barrier +discretized +Resonancestate +(Tunneling) +Trapped +bound state +(No tunneling) +b +d +E4 +E +AE(kt) +-Node +k16 + + + + +Fig. 2 | Band structure of the minimal model. a-d, The band structures for different +y +N s, where the +gray areas represent the projection of bulk bands, and the red (blue) curves are even (odd) modes with +energy predominantly localized at the boundaries. e, The locations of nodes on the boundary modes +(red plus signs) and the zeros of +E + [blue square symbols, defined in equation (S12)] for different +y +N s. + + + +a +二 +C +N =3 +e +4 +4 +20 +0 +15. +4 +_4 +10 +b +N +d +4 +4 +0 +5 +4 +-4 +0 +0.0 +0.5 +0 +k.a/2元 +0 +0.5 +0.5 +k.a/2元 +k,a/2元17 + + + + +Fig. 3 | Measured band dispersions for +y +N =2, 3, 4, and 5. a, Illustration of the experimental setup +for the case of +y +N =3. Here light yellow represents the cylinders of the PC and light gray denotes the +surrounding PECs. b, The projected band structure (gray) with dispersion of the boundary mode (cyan) +for a semi-infinite system along the +xk direction. Here we also numerically calculate the Zak phase +of the lower band, which is a constant + for all the +xk . c-f, Measured (color code) and simulated +(lines) band structures for +y +N =2, 3, 4, 5, respectively. Here the green lines represent bulk modes, and +the red and blue lines for even and odd boundary modes, respectively. In the simulations, a tiny air gap +of thickness 0.4mm, 0.3mm, 0.5mm, 0.5mm between the cylinders and the top PEC cover is introduced +for +y +N =2, 3, 4, 5, respectively. The number of lattices in the x-direction is kept at +x +N = 57 which is +enough to map out the mode dispersions. The lattice constant of the PC, the height, radius and relative +permittivity of the cylinder are a = 14 mm, h = 8 mm, r = 3 mm, and  r = 9.0, respectively. A small +air gap d = 4mm is kept between the side PEC boundaries and the PC. + + +a +c +N. =2 +N. =4 +Frequency (GHz) +12 +PEC +11 +min +max +10 +b +d +14 +N, =3 +N, =5 +12 +(ZH) +Frequency +12 +1 +Zakphase=元 +10 +10 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +k.a/2元 +k,a/2元 +k,a/2元18 + + + + +Fig. 4 | Infinite Q modes at the nodes. a, A one-side open configuration with +y +N = 3. Here one of +the PEC boundaries is replaced with air. The color represents the amplitude of the electric field +distribution of the boundary mode at +0.27 +/ +xk +a + += + (the wave vector of the first node for +y +N = 3). b- +d, The bulk bands (solid black lines), boundary band (colored line), and the corresponding Q factor of +boundary mode for +y +N =2, 3, and 4. As a reference, the red and blue dash lines denote the boundary +mode dispersion in Fig. 3. The lower shadow area is outside the light cone, and the upper shadow area +is the higher-order diffraction domain. Except for the one-side open boundary, all the other parameters +are the same as Fig. 3. + + + +a +b +N.=2 +Q-factor +8 +Air +106 +105 +104 +103 +102 +PEC +101 +10 +0 +C +d +N.=3 +N..=4 +Q-factor +8 +12 +106 +105 +104 +11 +103 +102 +10 +10 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +k.α/2元 +k.α /2元19 + + + + +Fig. 5 | Coupling between boundary modes for +y +N = 4. a, A photo of the experimental sample with +x +N = 57, +y +N = 4, and the red star denotes the position of the point source. The region between the +black dashed lines was scanned. b, Electric field distributions at a non-node frequency of 11.63 GHz, +whereat the coupling strength is finite. c, Electric fields at a node frequency (11.80 GHz in the +experiments and 11.82GHz in the simulations), whereat the coupling strength vanishes. The relative +permittivity of the cylinders is set as 𝜀𝑟 = 9.0+0.02i to simulate the inevitable loss in the experiments. +All the other parameters of the system are the same as Fig. 3. + + +a +PEC +PEC +max +min +b +Sim. +Exp. +C +Sim. +Exp. \ No newline at end of file diff --git a/VNE1T4oBgHgl3EQfvAVK/content/tmp_files/load_file.txt b/VNE1T4oBgHgl3EQfvAVK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41ee4b627dfe7fa0a35b42c65c9961f35031ac30 --- /dev/null +++ b/VNE1T4oBgHgl3EQfvAVK/content/tmp_files/load_file.txt @@ -0,0 +1,828 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf,len=827 +page_content='1 Trapped boundary modes without a well-defined bulk gap Tao Liu1, Kai Bai1, Yicheng Zhang1, Duanduan Wan1†, Yun Lai2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Chan3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' and Meng Xiao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 4* 1Key Laboratory of Artificial Micro- and Nano-structures of Ministry of Education and School of Physics and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Wuhan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' China 2National Laboratory of Solid State Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' and Collaborative Innovation Center of Advanced Microstructures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' China 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The Hong Kong University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Clear Water Bay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Kowloon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' China 4Wuhan Institute of Quantum Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Wuhan 430206,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' China Corresponding E-mail: † ddwan@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' * phmxiao@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='cn A boundary mode localized on one side of a finite-size lattice can tunnel to the opposite side which results in unwanted couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Conventional wisdom tells that the tunneling probability decays exponentially with the size of the system which thus requires many lattices before eventually becoming negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here we show that the tunneling probability for some boundary modes can apparently vanish at specific wave vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Meanwhile, the number of wave vectors where tunneling probability vanishes equals the number of lattices perpendicular to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Thus, similar to bound states in the continuum, a boundary mode can be completely trapped within very few lattices whereat the bulk band gap is not even well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Our idea is proven analytically, and experimentally validated in a dielectric photonic crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' This feature allows for the extreme flexibility in tunning the hopping between localized states or channels, which facilitates unprecedented manipulation of light such as integrating multiple waveguides without crosstalk and photonic non-abelian braiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2 The spectrum of a system typically consists of continuous spectra and discrete spectra (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Conventional wisdom says that eigenvalue spectrum of bound states is discrete, while the eigenvalue spectrum of unbound states forms a continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' For electronic systems, the usual convention is that negative energy states are bound and are hence discrete, while positive energy states form a continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' For light and sound waves, all eigen-frequencies are positive and as such, discrete states form due to the boundary condition imposed by a barrier, which is a material that forbid wave propagation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' having a “band gap” 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' A typical example is the discrete states that exist in a cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Discrete states can remain a bound state in the sense that there is no “free state” continuum it can couple with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The discrete state can be perfectly confined by the barrier if the width of the barrier is infinite (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' When the width of this barrier is finite, there is some probability that the state can tunnel through the barrier and becomes a resonance state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=" If a state's energy lies inside the continuous spectrum, it will unavoidably couple with states in the continuum and become a resonance state." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As an exception to this rule, bound states in the continuum (BICs) can be spatially bound with energy inside the continuous spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-I)3-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here, parallel to BICs, we show another counterintuitive concept: a state can get completely trapped (infinite Q factor if no intrinsic loss) by a band gap material with a finite and very small thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The solid black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-IV sketches one such state when the number of lattices is 𝑁𝑦 = 4 (𝑁𝑦 can be even smaller as will be shown later) whereat the bulk gap is not well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' A state being completely trapped indicates that there is no probability for the state to tunnel through the band gap material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Considering a symmetric double-well configuration as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-V, then a state localized on the left-hand-side well (blue line) cannot tunnel into the state on the right-hand- side well (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Or equivalently, there is no coupling (no hopping) between the two states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We note that the probability of tunneling is a crucial factor for quantum information processing as it determines the lifetime of the trapped states10-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Control over the hopping coefficient also enables manipulating interaction between different trapped states to generate various entangled quantum states14-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Meanwhile, fine-tuning of the coupling coefficient is a key requirement in programmable photonic simulators17, non-abelian braiding of photons18 and quantum computers19, especially when nanostructures are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3 With one additional direction, the two states in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1a-V will extend into two waveguide modes as sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Coupling between waveguide modes introduces crosstalk which limits the integration of multiple waveguide channels into a compact device20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' With the recent explosive growth of research in topological physics21-24, topological boundary modes and hinge modes associated with nontrivial bulk topologies have attracted a lot of attention due to their robustness against disorder and fabrication imperfections25-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' However, these boundary and hinge modes also suffer from the tunneling effect when the width of the system is not large enough30-33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' So, even topological boundary modes can be gapped in the presence of cross coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' On the other hand, controlling the tunneling of topological boundary modes or hinge modes can achieve new versatile controllability, such as spin flipping34, electrical switching35, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here, we consider the coupling between two boundary modes on a two-dimensional (2D) strip geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Different from the prevailing understanding that the coupling vanishes only when the width of the strip is large enough, we discover that the coupling can vanish (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', no tunneling) at specific wave vectors (nodes) for a narrow strip with very few lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Interestingly, the number of nodes equals the number of lattices perpendicular to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Moreover, similar as BICs, one of the boundary modes with these specific wave vectors exhibit infinite Q factor even if wave leakage is introduced at one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Using a Hamiltonian model, we analytically solved this system and proved the existence of this novel feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Furthermore, we find this unique feature presents in many systems such as 2D plasmonic spheres array and 2D photonic crystals (PCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We further verify this feature experimentally within a 2D PC, showing both the mode dispersions and the capability of tuning the magnitude of the hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Our system is sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1b, which is periodic along the x-direction and finite along the y- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Each unit cell contains one atom or meta-atom which can support multiple modes such as S , xP , yP , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' These modes interact and evolve into bands and the physics can be captured 4 succinctly using the usual tight-binding description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Inside a band gap, the system may exhibit boundary modes if the parameters are appropriately chosen, as shown schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1b with the red and blue shaded regions denoting the mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' These boundary modes can either be Shockley states36, Tamm states37 or originate from topological reasons21, 22, 24, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1c, we sketch the typical consequences of the coupling between these two boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here the shaded areas represent the projection of bulk bands, and the blue and red lines represent the dispersion of the boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' If no symmetry forbids coupling, they will couple whenever they cross each other, constantly forming mini-gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In contrast to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1c, we show that the coupling can vanish at some specific nodes as pictorially shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Due to the mirror symmetry 𝑚𝑦, the boundary modes couple to form even and odd modes, as shown respectively by the red and blue lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' They twist with each other and form several nodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We emphasis that 𝑚𝑦 alone cannot necessarily guarantee the presence of these nodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As will be shown more rigorously later, there is no coupling between boundary modes at these nodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Intriguingly, we find that the number of nodes equals the number of lattices along the y- direction ( y N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Meanwhile, the coupling between boundary modes here is size dependent and is thus a peculiar type of finite size effects (FSEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='30, 32, 38 We note that anomalous FSEs has been noted for helical boundary modes in topological insulator where the strength of the FSE decrease non- monotonically with the size32, 33, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The oscillation length therein is hundreds or thousands of lattice constants which is thus significantly different from our case as the coupling here oscillates at the lattice scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' First, we provide a tight-binding model Hamiltonian that exhibits the salient features outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We derive this Hamiltonian with the coupled dipole equation40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We assume each site supports two dipoles xP and yP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' For simplicity, we assume that other excitations are either far away in energy or are orthogonal to xP and yP (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', the zP dipole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The hopping from a dipole jP at lattice site j R to the dipole iP at i R is given by41 5 𝑮⃡ (𝒓) = 𝑡 𝑟5 [3(𝑷𝑗 𝒓)𝒓 − 𝑟2𝑷𝑗] (1) with j i = − r R R , and here t is introduced only to ensure that the unit of hopping is energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' For convenience, we set 1 t = and r is in a unit of the lattice constant a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We truncate the hopping to the next nearest neighbor which are the minimal interactions needed to explain the physics presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The momentum space Hamiltonian for a periodic system is thus ( ) ( ) 4cos cos 2 cos / 2 3sin sin / 2 3sin sin / 2 2cos +cos 4 cos / 2 x y x x y x y x y x k k k k k H k k k k k \uf0e6 \uf0f6 − − − \uf0e7 \uf0f7 = \uf0e7 \uf0f7 \uf0e7 \uf0f7 − − + \uf0e8 \uf0f8 (2) The bands are nondegenerate except at Γ and 𝑀 as protected by the 4v C symmetry (See Supplementary Materials Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We note that though we derive the Hamiltonian with the coupled dipole equation, a similar tight-binding Hamiltonian can also describe electronic systems of the same symmetry if p-orbitals ( xP and yP ) are dominant in the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As we are interested in a strip geometry which is periodic along the x-direction and finite along the y- direction, we provide the projected band structure of the periodic system along the xk direction as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2a-d with the light gray background (the projected bulk band continuum) as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here we can focus on the 0 xk \uf03e region as the 0 xk \uf03c region is simply related by time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Now we have a band gap region between the two bands and the band gap closes only at 0 xk = and / xk a \uf070 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The bulk polarization 𝑝𝑦 = ∫ −𝑖⟨𝑢𝑖|𝜕𝑘𝑦|𝑢𝑖⟩𝑑𝑘𝑦 𝐵𝑍 for any value of xk is quantized as \uf070 , and hence a semi-infinite system possesses a topological boundary mode which is located inside the band gap with dispersion connecting the two band edges42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' (See Supplementary Materials Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' I) Then we proceed to investigate the peculiar coupling feature for a finite number of lattice sites ( y N ) along the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' First, we start with an extreme case where 1 y N = as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' This 6 case corresponds to an infinite chain of dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In this case, xP and yP dipoles decouple and exhibit positive and negative dispersions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='43 These two bands cross once at / 2 xk a \uf070 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' When 2 y N = , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', two coupled infinite chains, xP and yP dipole couple with each other and there is no pure band with only xP or yP component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Instead, one can label the band with either mirror symmetric (even) or antisymmetric (odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2b, there are now 4 bands in total and two of them appear inside the bulk gap region, with one even state (red) and one odd state (blue) for the yP component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Interestingly, now the red and blue bands cross with each other twice which is also the value of y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As we further increase y N to 3 and 4 as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2c and 2d, respectively, the gray bands start filling up the projection of the bulk periodic bands, and the red and blue bands approach the dispersion of the boundary modes for a semi-infinite system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Still, the number of nodes is always equal to y N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' To prove that the number of nodes between the even and odd mode is equal to y N , we analytically solve the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' (Proof in Supplementary Materials Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We first obtain the eigen energy and eigenstates of the boundary mode for a semi-infinite system [equation (S1-S7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Then with perturbation theory to the first order, we obtain that the hopping strength E \uf044 between the two boundary modes localized on opposite boundaries as given by equation (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Thus, to the first-order approximation, the frequencies of the even and odd boundary modes are e E E + \uf044 and e E E − \uf044 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' When 0 E \uf044 = , the hopping strength is zero, and then there is a nodal point on the even and odd boundary bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' It can be proved that E \uf044 is an oscillation function of xk and exhibits y N zeros for (0, / ) xk a \uf070 \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' (Detail proof is also provided in Supplementary Materials Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' To check that the zeros of E \uf044 indeed predict the number of nodes, we provide the locations of boundary band nodes (red +) and the zeros of E \uf044 (blue square) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The number of the red + exactly equals the number of the blue square for every y N , and the difference comes from the first-order approximation we take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 7 The physics of such a peculiar feature goes beyond the tight-binding model as will be demonstrated later with full wave simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As our system only requires the symmetry imposed by the xP and yP modes in a square lattice, the physics discussed here should be quite universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Possible platforms are photonic crystals, phononic crystals, cold atoms, and 2D materials with properly chosen orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As an example, we show that this peculiar coupling feature exists in a 2D array of plasmonic spheres in Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Different from the tight-binding model, the coupling between plasmonic spheres extends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In addition, the physics discussed here also exists in 2D dielectric PCs where the band dispersion originates from dipolar modes arranged in a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Previous studies showed that for a 2D PC consisting of dielectric cylinders in air and considering the polarization with the electric field along the cylinders (TM), the typical three lowest order modes are one monopole and two in-plane dipoles44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As shown in Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' IV, when the frequency of the doubly-degenerate dipole mode is lower than the monopole mode at the \uf047 point, there are two bands consisting of nearly pure in-plane dipole modes (ranging from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='7GHz to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3GHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Such a system exhibits the same coupling feature induced twisting boundary modes with nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', the odd and even boundary modes cross with each other y N times over the positive first BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Furthermore, the vanishing of coupling is robust even if the monopole mode is hybridized with the two in-plane dipole modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The reason lies in the fact that the nodes between the odd and even boundary modes are protected by the mirror symmetry and the hybridization with z M does not break this mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As long as the nodes between the two boundary modes do not annihilate (the perturbation introduced by z M is not strong enough), the number of nodes would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Such a feature then further releases the constraints on the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In the following, we provide an experimental demonstration within a 2D dielectric PC in the microwave regime in the presence of hybridization with z M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Figure 3a sketches the experimental setup, where we consider a mirror- 8 symmetrical strip geometry of square lattice PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Aluminum plates are placed at the upper, lower, front, and rear edges as PEC boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The PEC boundary at the upper side is tightly attached to the dielectric cylinders, allowing only TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In the experiment, we need to move the upper PEC boundary so as to measure the electric field distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' This experimental setup, however, unavoidably introduces a sub-millimeter (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 mm) air gap between the cylinders and the upper PEC boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Such an air gap has limited impacts on the dispersions of the band of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' (See analysis in Supplementary Materials Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' For a better characterization of the boundary modes, we also keep a small air gap (d = 4 mm) between the side PEC boundaries and the PC42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The presence of this air gap only shifts the dispersion of the boundary modes a little bit while the interested effect is not affected, which verifies once again the robustness of this FSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' (Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' V) We start with the measurement of the band dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Figure 3b shows the projected band dispersion for a semi-infinite system, wherein the gray areas represent the projection of the bulk bands while the cyan line denotes the boundary mode inside the bulk band gap at around 12 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here, the frequencies of the monopole mode and the dipole modes at the \uf047 point are 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3 GHz and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='7 GHz, respectively, and thus the lower two bands are not pure in-plane dipole modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We excite the sample at one side of the sample, measure the field distributions and then apply Fourier transform to obtain their dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Figures 3c-f show the measured band dispersions (color code) together with the simulated band dispersion (lines) for y N =2, 3, 4, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The measured dispersion agrees well with the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As expected, the number of boundary band nodes equals y N for all the samples we have measured and thus the twisting of boundary modes and the number of nodes is confirmed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The boundary modes can be utilized as waveguide channels to guide waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In our system, the tunneling probability vanishes at these nodes, and the boundary mode is completely trapped to one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Similar to BICs, these trapped boundary modes should exhibit an infinite Q factor if the system has no intrinsic loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 4a, we replace one of the PEC boundaries with air while keeping 9 all the other parameters the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Figures 4b-d show the band structure and the Q factor of the remaining boundary mode for y N =2, 3, and 4, wherein the colored line marks the boundary band and the solid black lines denote the bulk bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The bulk bands and one of the boundary bands located near the remaining PEC boundary shift slightly, while the other boundary mode couples with light cone and thus is not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As a reference, the red and blue dash lines denote the boundary mode dispersion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The area outside the light cone (lower right shaded region) and above the first-order diffraction limit (upper right shaded region) are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' It is clear that the Q factors of the boundary modes are infinite at these nodes since they cannot tunnel through and thus leak outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In other words, the boundary modes at nodes can be trapped completely within very few lattices at which the bulk band gap is not even well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In contrast, those modes not at nodes host a finite Q factor and become leaky modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We also show the eigenfield distribution of one boundary mode at nodes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 4a for y N = 3, where we can see the fields vanish in the air (almost completely white inside the air region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' More information about the eigenfields is provided in Supplementary Materials Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' It is difficult to verify the infinity Q factor of the boundary modes at nodes in our experimental setup due to the large intrinsic loss of material in microwaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As an alternative way, we keep the two-side PEC configuration and demonstrate the hopping between these two boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The hopping between two boundary modes on opposite sides vanishes at these nodes and thus each boundary mode can be confined to one side of the waveguide as it propagates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Otherwise, the field energy oscillates between two boundaries alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In general, the coupling strength between two boundary modes is a function xk as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' S10a in Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In the following, we confirm this peculiar coupling property experimentally with the same PC sample with y N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' There are four nodes on the twisting bands of boundary modes and we choose the one at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='80 GHz ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4 / xk a \uf070 = ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Meanwhile, we also perform the experiments at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='63 GHz where the coupling strength is finite to see the field oscillation between two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 10 Figure 5a shows a photo of the experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Same as before, the sample will be covered with another PEC top layer in the experiments and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' A point source, labeled by the red star, is placed at the lower-left corner to excite predominantly the boundary mode localized on the lower boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The measured and simulated electric field distributions inside the waveguide (bounded by the black dashed lines) are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 5b and 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In the simulations, the relative permittivity of the cylinders is set as 𝜀𝑟 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='02i with an imaginary part to simulate the inevitable loss in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 5b, at a frequency of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='63 GHz (not one of the nodal points), the stimulated boundary mode couples to another boundary after propagating tens of lattices and then recouples to the original boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In contrast, when the frequency matches the frequency of a node (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='80 GHz), the excited boundary mode stays on the lower side while propagating to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The field oscillation period is inversely proportional to the coupling strength between the boundary modes: the smaller the coupling strength, the longer the oscillation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The oscillation period is infinite at a node frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Thus just a few lattices are able to prohibit the interaction between adjacent boundary guiding modes effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Such a non-coupling feature facilitates the integration of multiple waveguide channels into a compact device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In summary, we analytically solved a next-nearest-neighbor hopping model to investigate the vanishing of coupling between boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The boundary modes of a finite-width strip twist around each other, intersecting at nodes and the number of nodes equals the number of lattices across the strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' This physics is ubiquitous and appears in systems such as the plasmonic sphere array, dielectric PCs, and cold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' It leads to a filtering effect where only the components with wave vectors matching those of the nodes survive (see Supplementary Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' VIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' We experimentally demonstrate this peculiar feature in PCs by measuring the mode dispersion and mapping the electric field distributions of the boundary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Our work points to the possibility of getting rid of the formation of gaps due to cross-coupling with properly chosen orbitals and lattice, and thus opens a feasible way for integrating multiple waveguides into an extremely compact device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 11 Acknowledgements This work is supported by the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 12274330, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 12274332, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 11904264 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 11904265).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' is also supported by the Hubei Provincial Natural Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2020CFB670) and the Knowledge Innovation Program of Wuhan-Shuguang (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2022010801020125).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='C is supported by Research Grants Council (RGC) Hong Kong through Grant AoE/P-502/20 and Croucher Foundation (CAS20SC01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' is supported by the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 12174188 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 11974176).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Author contributions D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' initiated and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' did the simulations, designed the samples and performed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', Lai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', Hang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', Zheng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=', Chan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 10, 582-586 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 15 Figures: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 1 | Boundary modes can be trapped by a band gap material with a few lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a, Illustration of a bound state trapped by a few lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' b, The coupling of two boundary modes (denoted by the red and blue shaded regions) localized on opposite sides of a strip geometry of a square lattice with xP and yP orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' c-d, Sketches of the band dispersion for two different systems, where the coupling between boundary modes is finite for all xk in (c) and vanishes at several nodes for certain xk s in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a Spectrum Mode profile continuous Bound statein thecontinuum potential barrier Regular potential boundstate barrier discretized Resonancestate (Tunneling) Trapped bound state (No tunneling) b d E4 E AE(kt) Node k16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 2 | Band structure of the minimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a-d, The band structures for different y N s, where the gray areas represent the projection of bulk bands, and the red (blue) curves are even (odd) modes with energy predominantly localized at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' e, The locations of nodes on the boundary modes (red plus signs) and the zeros of E \uf044 [blue square symbols, defined in equation (S12)] for different y N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a 二 C N =3 e 4 4 20 0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 4 _4 10 b N d 4 4 0 5 4 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 0 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='a/2元 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='a/2元 k,a/2元17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3 | Measured band dispersions for y N =2, 3, 4, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a, Illustration of the experimental setup for the case of y N =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here light yellow represents the cylinders of the PC and light gray denotes the surrounding PECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' b, The projected band structure (gray) with dispersion of the boundary mode (cyan) for a semi-infinite system along the xk direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here we also numerically calculate the Zak phase of the lower band, which is a constant for all the xk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' c-f, Measured (color code) and simulated (lines) band structures for y N =2, 3, 4, 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here the green lines represent bulk modes, and the red and blue lines for even and odd boundary modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' In the simulations, a tiny air gap of thickness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4mm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3mm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5mm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5mm between the cylinders and the top PEC cover is introduced for y N =2, 3, 4, 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The number of lattices in the x-direction is kept at x N = 57 which is enough to map out the mode dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The lattice constant of the PC, the height, radius and relative permittivity of the cylinder are a = 14 mm, h = 8 mm, r = 3 mm, and \uf065 r = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' A small air gap d = 4mm is kept between the side PEC boundaries and the PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a c N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' =2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' =4 Frequency (GHz) 12 PEC 11 min max 10 b d 14 N, =3 N, =5 12 (ZH) Frequency 12 1 Zakphase=元 10 10 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='a/2元 k,a/2元 k,a/2元18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 4 | Infinite Q modes at the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a, A one-side open configuration with y N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Here one of the PEC boundaries is replaced with air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The color represents the amplitude of the electric field distribution of the boundary mode at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='27 / xk a \uf070 = (the wave vector of the first node for y N = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' b- d, The bulk bands (solid black lines), boundary band (colored line), and the corresponding Q factor of boundary mode for y N =2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' As a reference, the red and blue dash lines denote the boundary mode dispersion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The lower shadow area is outside the light cone, and the upper shadow area is the higher-order diffraction domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' Except for the one-side open boundary, all the other parameters are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a b N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='=2 Q factor 8 Air 106 105 104 103 102 PEC 101 10 0 C d N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='=3 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='.=4 Q factor 8 12 106 105 104 11 103 102 10 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='5 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='α/2元 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='α /2元19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 5 | Coupling between boundary modes for y N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a, A photo of the experimental sample with x N = 57, y N = 4, and the red star denotes the position of the point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The region between the black dashed lines was scanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' b, Electric field distributions at a non-node frequency of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='63 GHz, whereat the coupling strength is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' c, Electric fields at a node frequency (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='80 GHz in the experiments and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='82GHz in the simulations), whereat the coupling strength vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' The relative permittivity of the cylinders is set as 𝜀𝑟 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content='02i to simulate the inevitable loss in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' All the other parameters of the system are the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNE1T4oBgHgl3EQfvAVK/content/2301.03394v1.pdf'} +page_content=' a PEC PEC max min b Sim.' 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100644 index 0000000000000000000000000000000000000000..fecff3695ee9a2c78922166dcda5e5e8e5e0e769 --- /dev/null +++ b/XNE5T4oBgHgl3EQfCQ64/content/tmp_files/2301.05394v1.pdf.txt @@ -0,0 +1,396 @@ +arXiv:2301.05394v1 [math.AC] 13 Jan 2023 +EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE +MODULE OVER SYMBOLIC REES ALGEBRAS +CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR +Abstract. Let A be a symbolic (or an extended symbolic) Rees algebra +(need not be Noetherian) of dimension d. Let P be a finitely generated +projective A-module of rank ≥ d. Then P has a unimodular element. This +improves the classical result of Serre for the mentioned class of algebras. +1. Introduction +Let R be a commutative ring and P a finitely generated projective R-module. An +element p ∈ P is said to be a unimodular if there exists φ ∈ Hom(P, R) such that +φ(p) = 1, in other words, P splits off a free summand of rank one, i.e. P ∼= Q ⊕ R +for some projective R-module Q. If R is a Noetherian commutative ring of dimension +d, then a classical result of Serre [21] asserts that every finitely generated projective R- +modules of rank > d has a unimodular element. In fact, this is the best possible result in +general as a common example which can be found easily in the literature is the tangent +bundle over the real algebraic sphere. Therefore if the rank(P) ≤ d, then the question +that P has a unimodular element, is subtle. In the case rank(P) = dim(R), where R is +an affine algebra over an algebraically closed field, there is a well developed obstruction +theory which is due to Murthy [12, Theorem 3.7]. (For a proof of the hypotheses in [12, +Theorem 3.7], see [10, Corollary 1.5]). +Other than Murthy’s works, there are various fascinating obstruction theory in the +literature when rank(P) ≤ dim(R). +For this, we recommend the reader to look at +MathSciNet citations of Murthy’s paper [12]. We do not deal with such a fancy theory +here as our paper is in the pursuit of discovering commutative rings where such an +obstruction does not exist. +In this paper, we show that symbolic Ress algebra and extended symbolic Rees alge- +bras (see Definition 2.6) are examples of commutative rings where such an obstruction +does not exist. The importance of symbolic Rees algebra first comes from the coun- +terexample of Hilbert’s 14th problem. Rees [16] provided the first counterexample to +Zariski’s version of Hilbert’s 14th problem by proving an ideal J of R whose symbolic +Rees algebra Rs(J) is not finitely generated. We prove the following result about an +existence of a unimodular element in a projective module over a symbolic Rees algebra +or over an extended symbolic Rees algebra. +Theorem 1.1. (Theorem 3.5) Let R be a commutative noetherian domain of dimension +d and I an ideal of R. Let A = Rs(I) or Rs(I, x−1) (symbolic or extended symbolic +Rees algebra)(see Definition 2.6) and P a finitely generated projective A-module of rank +≥ d+1. Then P has a unimodular element, i.e. P ∼= Q⊕A for some projective A-module +Q. +The similar results are obtained for polynomial rings by Plumstead [13], for Lau- +rent polynomial rings by Mandal [11] and for overring of polynomial rings by Rao [14]. +For Rees algebras, the analogous conjectures are considered in [15]. The above results +2020 Mathematics Subject Classification. Primary 13C10; Secondary 19A13. +Key words and phrases. projective modules, unimodular element, symbolic Rees algebra. +1 + +2 +CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR +are generalized to polynomial and Laurent polynomial rings for several variables by +Bhatwadekar–Roy [3] and Bhatwadekar–Lindel–Rao [2]. +For monoid extension, exis- +tence of unimodular problem is considered by Sarwar [19] [20], Keshari–Sarwar [8], and +Keshari–Mathew [9]. +Let R be a commutative Noetherian domain of Krull dimension d. Then the Rees +algebras over R are Noetherian but symbolic Rees algebras need not be Noetherian (for +example, see [17]). However we can prove the Theorem 1.1 without using the assumption +that symbolic Rees algebras are Noetherian. +This is possible because of Heitmann’s +result [6] over non-Noetherian rings. +For an extension of Heitmann’s result, see the +results of Gupta [5]. Our method is to use standard sheaf patching techniques. More +precisely, we find two covers, then we prove the results separately on each cover, finally +we patch/glue to get the results over the original space. A similar technique is used +in [15]. +However there to prove the result in one of the two covers, they have used +Plumstead’s generalized dimension function. +Here we prove the similar result using +Heitmann’s result as symbolic Rees algebras need not be Noetherian. +In section 2, +we have recalled some basic definitions and results which will be used throughout the +paper, also we have studied the Krull dimension of (extended) symbolic Rees algebras +via valuation dimension. Theorem 3.5 is proved in section 3. +2. Recollection of basic definitions and results +2.1. Notation. Throughout the paper, we assume that the rings are commutative with +the unity and the modules are finitely generated. Also, we assume that projective module +have the constant rank function. +Now we recall some basic notions. For a ring R, let dimR denote the Krull dimension +of R. Let Ass (R) denote the set of associated prime ideals of R, and ht(I) denote the +height of the ideal I. +Definition 2.1 (Unimodular Element). Let R be a ring R and M an R-module. For +m ∈ M, we define an ideal OM(m) = {ϕ(m) : ϕ ∈ M∗ = HomR(M, R)} of R which is +called the order ideal. The element m in M is called a unimodular element if OM(m) = R, +i.e. there exists a surjective R-linear map ϕ ∈ M∗ such that ϕ(m) = 1. Let Um (M) +denote the set of all unimodular elements in M. +Valuation dimension. The notion of the valuation dimension was introduced by +Jaffard [7]. Let A be an integral domain, an overring B of A is a subring of the field of +fractions K of A that contains A, i.e. A ⊆ B ⊆ K. +Valuation overring of A means an overring of A which is a valuation ring. +Definition 2.2 (Valuation Dimension). Let R be an integral domain, the valuation +dimension of R is the supremum of dimensions of the valuation overring of R. Valuation +dimension of R is denoted by dimvR, i.e. +dimvR = Sup {dimV : V is a valuation overring of R}. +Theorem 2.3. [1, Theorem 0.1] Let R be an integral domain which is not a field, K +is a field of fraction of R. Let F denote the algebraic extension of K, and d a positive +integer. Then we say, R have finite valuation dimension d (write dimvR = d) if the +following equivalent conditions are satisfied: +(1) Each valuation overring of R in F has dimension ≤ d and there exists a valuation +overring of R in F of dimension d. +(2) Each overring of R in F has dimension ≤ d and there exists a overring of R in +F of dimension d. +(3) dimR[x1, . . . , xd] = 2d. + +EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE +3 +(4) dimR[x1, . . . , xn] = n + d if n ≥ d − 1. +We say that dimvR = ∞ if there exists no d satisfying the conditions 1 to 4. For the +sake of completeness, each field is assigned valuation dimension 0. +Clearly dimR ≤ dimvR for every domain R and if B is an overring of A, then dimvB ≤ +dimvA. +2.2. Recollection of Rees algebra and Symbolic Rees Algebras. Let R be a +commutative ring and I an ideal of R. The Rees algebra (also known as blow-up algebra) +of R with respect to I as a subring of R[x] define to be +R[Ix] = { +n +� +i=0 +aixi : ai ∈ Ii} = +� +n≥0 +Inxn ⊆ R[x]. +The extended Rees algebra of R with respect to I as a subring of R[x, x−1] define to be +R[Ix, x−1] = { +n +� +i=−n +aixi : ai ∈ Ii} = +� +n∈Z +Inxn ⊆ R[x, x−1]. +where for n ≤ 0, In = R by convention. +The following theorem is for the Krull dimension of Rees algebra and extended Rees +algebra (see [22, Theorem 5.1.4]). +Theorem 2.4. Let R be a Noetherian ring and I an ideal of R. If dimR is finite, then +(1) dimR[Ix] = dimR + 1, if I ⊈ p for some p ∈ Spec R with dim(R/p) = dimR and +dimR[Ix] = dimR, otherwise. +(2) dimR[Ix, x−1] = dimR + 1. +Corollary 2.5. Let R be a Noetherian domain of dimension d and (0) ̸= I an ideal of +R. Then dimvR[Ix] = d + 1. +For further properties of Rees algebras R[Ix] and extended Rees algebras R[Ix, x−1], +one can see [22]. +Definition 2.6. [Symbolic Power, Symbolic Rees Algebra] Let R be a commutative +Noetherian ring and I an ideal of R. The n-th symbolic power of I is an ideal of R +defined by +I(n) = +� +p∈Ass (R/I) +(InRp ∩ R). +In general, given an ideal I of R the ordinary power In do not coincide with the +symbolic power I(n). In particular, if I = p ∈ Spec (R), then p(n) = pnRp ∩ R and if I is +a maximal ideal m, then m(n) = mn. Clearly from the definition I(1) = I, In ⊆ I(n) and +I(n+1) ⊆ I(n) for all n ≥ 1. +Let R be a commutative noetherian ring and I an ideal of R. The symbolic Rees +algebra of R with respect to I is a graded algebra defined as +Rs(I) = { +n +� +i=0 +aixi : ai ∈ I(i)} = +� +n≥0 +I(n)xn ⊆ R[x]. +Sometimes it is also known as symbolic blow-up algebra. In particular, for all n ≥ 2 if +I(n) = In, then the symbolic Rees algebra Rs(I) coincide with the Rees algebra R[Ix]. +One can define the extended symbolic Rees algebra of an ideal I of R as an extension +of Rs(I). Let us denote Rs(I, x−1) as extended symbolic Rees algebra define to be +Rs(I, x−1) = { +n +� +i=−n +aixi : ai ∈ I(i)} = +� +n∈Z +I(n)xn ⊆ R[x, x−1]. + +4 +CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR +where for n ≤ 0, I(n) = R by convention. +For further properties of Rs(I) and Rs(I, x−1), one can see [4], [22]. +Lemma 2.7. Let R be a commutative Noetherian domain of dimension d. Then +dimRs(I) = +� +d, +if I = (0); +d + 1, +otherwise. +Proof. For I = (0), there is nothing to prove. So we can assume I ̸= (0). By Corollary +2.5, dimvR[Ix] = 1+d = dimR[Ix]. Consider R[Ix] ⊆ Rs(I) ⊆ R[x]. Then dimvRs(I) ≤ +dimvR[Ix] = d+1 and dimvRs(I) ≥ dimvR[x] = d+1. This implies dimvRs(I) = d+1. +We know that dimRs(I) ≤ dimvRs(I). Hence dimRs(I) ≤ d + 1. +For the other inequality, note that Rs(I) = Rs(I)0 ⊕ Rs(I)+, where Rs(I)0 = R and +Rs(I)+ = Ix ⊕ I(2)x2 ⊕ · · · , and Rs(I)/Rs(I)+ ∼= R. Hence Rs(I)+ is a prime ideal of +Rs(I) and ht(Rs(I)+) > 0. Then we have +dim(Rs(I)/Rs(I)+) + ht(Rs(I)+) ≤ dimRs(I). +This implies that dimR + 1 ≤ dimRs(I). Therefore dimRs(I) = d + 1. +□ +Lemma 2.8. Let R be a Noetherian domain of dimension d. Then dimRs(I, x−1) = +d + 1. +Proof. Note that dimvR[x−1] = d + 1. Consider R[x−1] ⊆ Rs(I, x−1) ⊆ R[x, x−1]. By +[1, Lemma 1.15], dimvR[x−1] = dimvR[x, x−1]. Hence dimvRs(I, x−1) = 1 + d. Then we +have dimRs(I, x−1) ≤ dimvRs(I, x−1) = d + 1. +For the other inequality, from R[x−1] ⊆ Rs(I, x−1) ⊆ R[x, x−1], we observe that +Rs(I, x−1)x−1 = R[x, x−1]. Then we have +dimRs(I, x−1) ≥ dimRs(I, x−1)x−1 = dimR[x, x−1]. +Hence dimRs(I, x−1) ≥ d + 1. Therefore dimRs(I, x−1) = d + 1. +□ +If R is a Noetherian ring, then we know that the Rees algebra is a finitely generated +R-algebra. +Unlike Rees algebra, the symbolic Rees algebra is not necessarily finitely +generated R-algebra. For example, see [17]. +3. Existence of a unimodular element +Lemma 3.1. Let R be a Noetherian ring, I an ideal of R, and T be a multiplicatively +closed subset of R. Then T −1Rs(I) = T −1Rs(T −1I), where T −1Rs(T −1I) means the +symbolic Rees algebra of T −1R with respect to T −1I. +Proof. By definition Rs(I) = R ⊕ Ix ⊕ I(2)x2 ⊕ · · · , and since the localization commutes +with direct sums, we have +T −1Rs(I) = T −1R ⊕ T −1(I)x ⊕ T −1(I(2))x2 ⊕ · · · += T −1R ⊕ T −1(IR)x ⊕ T −1(I(2)R)x2 ⊕ · · · . + +EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE +5 +Now for n ≥ 1, +T −1(I(n)) = T −1( +� +Ass (R/I) +(InRp ∩ R)) += +� +Ass (T −1(R/I)) +T −1(InRp ∩ R) += +� +Ass (T −1R/T −1I)) +((T −1I)n(T −1R)p ∩ T −1R) += (T −1I)(n). +Since the localization commutes with direct sums, we also have +T −1Rs(T −1I) = T −1R ⊕ T −1I(T −1R)x ⊕ (T −1I(T −1R))(2)x2 ⊕ · · · . +But we already have (T −1I)(n) = T −1(I(n)), hence we conclude that T −1Rs(I) = +T −1Rs(T −1I). +□ +The following lemma is a particular case of the previous one. We use the following +version later. +Lemma 3.2. Let R be a commutative Noetherian domain and (0) ̸= I an ideal of R. +Then +(1) If T := set of all non-zero-divisors of R, T −1Rs(I) = T −1R[x]. +(2) If T = {1, a, a2, . . .} with a ∈ I is a non-zero-divisor of R, T −1Rs(I) = Ra[x]. +Proof. (1) Since R is a domain, T = R∗ = R\{0}. For n ≥ 1, I(n) is an ideal of R and +T ∩ I(n) ̸= ∅. This implies T −1(I(n)) = T −1R. Hence by 1st part of the proof of Lemma +3.1, we get T −1Rs(I) = T −1R[x]. +(2) By [15, Lemma 3.1], we have T −1R[Ix] = T −1R[(T −1I)x]. Since T ∩ I ̸= ∅ this +implies T −1R[Ix] = T −1R[x]. Consider +R[Ix] ⊆ Rs(I) ⊆ R[x]. +After localize at T, we have +T −1R[Ix] ⊆ T −1Rs(I) ⊆ T −1R[x]. +Therefore T −1Rs(I) = Ra[x]. +□ +Lemma 3.3. Let R be a commutative domain of dimension d and A = R1+aR, where a +is a non-zero non-unit element of R. Let P be a projective R-module of rank ≥ d. Let +Q := P1+aR. Then Q has a unimodular element. +Proof. Let ‘overbar’ denote the going modulo the ideal aR. Since (0) is the minimum +prime ideal in the domain R, we observe that dim(R) < d. By [6, Corollary 2.6], P has +a unimodular element. Therefore there exists p such that the order ideal OP (p) = R. +Let p ∈ P be a lift of p. Now observe that 1 + aR ⊂ of the order ideal OP (p). Hence +p1+aR is a unimodular element of Q. This finishes the proof. +□ +The following theorem is a result of Ravi A. Rao [14, Theorem 5.1(I)]. +Theorem 3.4. Let R be a commutative Noetherian ring of dimension d, and S be a +multiplicative closed set of non-zero-divisors of R[x]. Let A be a ring lying between R[x] +and S−1R[x]. Then for n ≥ d + 2, En(A) acts transitively on Um(An). +Theorem 3.5. Let R be a commutative Noetherian domain of dimension d and I an +ideal of R. Let A = Rs(I) or Rs(I, x−1) and P a finitely generated projective A-module +of rank ≥ d + 1. Then P has a unimodular element. + +6 +CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR +Proof. First, we assume that A = Rs(I). Let rank(P) = m > d. Since A is a subring of +an integral domain R[x], A is an integral domain. If I = (0), then I(n) = (0) this implies +A = R. In this case, the theorem follows from Serre [21]. If I = (1), then I(n) = (1) this +implies A = R[x]. In this case, the theorem follows from Plumstead [13, Corollary 4]. +So we can assume I ̸= (0) and I ̸= (1). By Lemma 2.7, dimRs(I) = d + 1. +Let T be the set of all non-zero-divisors of R. By Lemma 3.2(1), T −1A = T −1R[x], +where T −1R is a quotient field of R. Since T −1R[x] is a PID, T −1P is a free module +over T −1A. Since P is finitely generated, there exists t ∈ T such that Pt is a free Rs(I)t- +module. Let 0 ̸= a be a non-unit element of I. Then Pat is a free Rs(I)at-module. +Denote b = at. By Lemma 3.2(2), we have Rs(I)b ∼= Rb[x]. Now consider the following +Cartesian square of rings +Rs(I) +Rs(I)b ∼= Rb[x] +Rs(I)1+bRs(I) +Rb[x]1+bRs(I). +i1 +i2 +j1 +j2 +Since Pb is a free Ab-module of rank m, Pb ∼= Am +b . Hence Pb has a unimodular element say +p1. Since P1+bA is projective A1+bA-module, by Lemma 3.3(1), P1+bA has a unimodular +element say p2. Hence we have P1+bA ∼= p2A1+bA ⊕ Q, with projective A1+bA-module Q. +Now consider B = Ab(1+bA) = (1 + bA)−1Rs(I)b = (1 + bR)−1D−1Rb[x] = D−1R +′[x], +where D is a multiplicative closed subset of R +′[x] and R +′ = Rb(1+bR) is of dimension +d − 1. Since Pb(1+bA) is a free B-module of rank m, Pb(1+bA) ∼= Bm. Let ¯u and ¯v be the +image of p1 and p2 respectively in Pb(1+bA). Now consider the following cartesian square +of projective modules. +P +Pb +P1+bA +Pb(1+bA) ∼= Bm +i1 +i2 +j1 +j2 +Note that the ring B lying between R +′[x] and S−1R +′[x], where S is a set of all non-zero- +divisors in R +′[x]. Then by Theorem 3.4, there exists σ ∈ Em(B) such that σ(¯u) = ¯v. By +[18, Corollary 3.2] (see also [2, Proposition 3.2]), we can split σ as (σ1)b(σ2)1+bA, where +σ2 ∈ E(Pb) and σ1 ∈ E(P1+bA). Since (σ1)b(σ2)1+bA(¯u) = ¯v, for suitably changing p1 +and p2, we can assume that ¯u = ¯v. +Now consider the following fiber product diagram of projective modules. +P +Pb ∼= Am +b +A +Ab +P1+bA ∼= A1+bA ⊕ Q +Pb(1+bA) +A1+bA +Ab(1+bA) +ψ +ψ1 +ψ2 +Since ψ1 and ψ2 are surjective homomorphisms, from the above diagram, we get a sur- +jective homomorphism ψ : P ։ A. Hence there is p ∈ P such that ψ(p) = 1. Therefore +P has a unimodular element. + +EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE +7 +When A = Rs(I, x−1), the proof is a verbatim copy of the above. +However one +needs to use [11, Theorem 2.1] in the second corner. A diligent reader can work out the +details. +□ +Acknowledgement: H.P. Sarwar would like to thank S.E.R.B. Govt. of India for +the grant SRG/2020/000272. +References +[1] D.F. Anderson, A.Bouvier, D. Dobbs, M.Fontana and S.Kabbaj. On Jaffard domains, Expo. Math. +6 (1988), 145-175. +[2] S.M. Bhatwadekar, H. Lindel and R.A. Rao. 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On Serre dimension of monoid algebras and Segre extensions, +Journal of Pure and Applied Algebra 226 (2022), no. 9, Paper No. 107058, 16 pp. +[10] A. Krishna. Murthy’s conjecture on 0-cycles. Invent. Math. 217 (2019), no. 2, 549–602. +[11] S. Mandal. Basic elements and cancellation over Laurent polynomial rings, J. Algebra 79 (1982), +251-257. +[12] M.P. Murthy. Zero cycles and projective modules. The Annals of Mathematics, 140(2), 405, 1994. +[13] B. Plumstead. The conjectures of Eisenbud and Evans, Amer. J. Math. 105 (1983), 1417–1433. +[14] R.A. Rao. Stability theorems for overrings of polynomial rings. II. J. Algebra 78 (1982), no. 2, +437–444. +[15] R.A. Rao and H.P. Sarwar. Stability results for projective modules over Rees algebras, Journal of +Pure and Applied Algebra, 223 (2019), no 1, 1-9. +[16] D. Rees. On a problem of Zariski, Illinois Journal of Mathematics 2 (1958), no. 1, 145-149. +[17] P.C. Roberts. A prime ideal in a polynomial ring whose symbolic blow-up is not noetherian, Proc. +Amer. Math. Soc. 94 (1985), no. 4, 589-592. +[18] A. Roy. Application of patching diagrams to some questions about projective modules, J. Pure Appl. +Algebra 24 (1982), no. 3, 313-319. +[19] H.P. Sarwar, Some results about projective modules over monoid algebras. Comm. Algebra 44 +(2016), no. 5, 2256–2263. +[20] H.P. Sarwar. K0-stability over monoid algebras, Annals of K-theory, 6 (2021), no. 4, 629–649. +[21] J.-P. Serre, Modules projectifs et espaces fibr´es `a fibre vectorielle, 1958 S´eminaire P. Dubreil, M.-L. +Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Expos´e 23 18 pp. +[22] I. Swanson and C. Huneke. Integral closure of ideals, rings, and modules, London Mathematical +Society, Lecture Note Series 336, Cambridge, 2006. +(Chandan Bhaumik) Department of Mathematics, Indian Institute of Technology Kharag- +pur, Kharagpur 721302, West Bengal, India +Email address: cbhaumik11math@gmail.com +(H.P. Sarwar) Department of Mathematics, Indian Institute of Technology Kharagpur, +Kharagpur 721302, West Bengal, India +Email address: parvez@maths.iitkgp.ac.in +Email address: mathparvez@gmail.com + diff --git a/XNE5T4oBgHgl3EQfCQ64/content/tmp_files/load_file.txt b/XNE5T4oBgHgl3EQfCQ64/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b9b59a7197782782529a8cee9e6bd3273bc159a --- /dev/null +++ b/XNE5T4oBgHgl3EQfCQ64/content/tmp_files/load_file.txt @@ -0,0 +1,446 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf,len=445 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='05394v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='AC] 13 Jan 2023 EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE OVER SYMBOLIC REES ALGEBRAS CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let A be a symbolic (or an extended symbolic) Rees algebra (need not be Noetherian) of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let P be a finitely generated projective A-module of rank ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then P has a unimodular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' This improves the classical result of Serre for the mentioned class of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Introduction Let R be a commutative ring and P a finitely generated projective R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' An element p ∈ P is said to be a unimodular if there exists φ ∈ Hom(P, R) such that φ(p) = 1, in other words, P splits off a free summand of rank one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' P ∼= Q ⊕ R for some projective R-module Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' If R is a Noetherian commutative ring of dimension d, then a classical result of Serre [21] asserts that every finitely generated projective R- modules of rank > d has a unimodular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In fact, this is the best possible result in general as a common example which can be found easily in the literature is the tangent bundle over the real algebraic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Therefore if the rank(P) ≤ d, then the question that P has a unimodular element, is subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In the case rank(P) = dim(R), where R is an affine algebra over an algebraically closed field, there is a well developed obstruction theory which is due to Murthy [12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (For a proof of the hypotheses in [12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='7], see [10, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Other than Murthy’s works, there are various fascinating obstruction theory in the literature when rank(P) ≤ dim(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For this, we recommend the reader to look at MathSciNet citations of Murthy’s paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' We do not deal with such a fancy theory here as our paper is in the pursuit of discovering commutative rings where such an obstruction does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In this paper, we show that symbolic Ress algebra and extended symbolic Rees alge- bras (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='6) are examples of commutative rings where such an obstruction does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The importance of symbolic Rees algebra first comes from the coun- terexample of Hilbert’s 14th problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Rees [16] provided the first counterexample to Zariski’s version of Hilbert’s 14th problem by proving an ideal J of R whose symbolic Rees algebra Rs(J) is not finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' We prove the following result about an existence of a unimodular element in a projective module over a symbolic Rees algebra or over an extended symbolic Rees algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='5) Let R be a commutative noetherian domain of dimension d and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let A = Rs(I) or Rs(I, x−1) (symbolic or extended symbolic Rees algebra)(see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='6) and P a finitely generated projective A-module of rank ≥ d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then P has a unimodular element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' P ∼= Q⊕A for some projective A-module Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The similar results are obtained for polynomial rings by Plumstead [13], for Lau- rent polynomial rings by Mandal [11] and for overring of polynomial rings by Rao [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For Rees algebras, the analogous conjectures are considered in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The above results 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Primary 13C10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Secondary 19A13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' projective modules, unimodular element, symbolic Rees algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 1 2 CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR are generalized to polynomial and Laurent polynomial rings for several variables by Bhatwadekar–Roy [3] and Bhatwadekar–Lindel–Rao [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For monoid extension, exis- tence of unimodular problem is considered by Sarwar [19] [20], Keshari–Sarwar [8], and Keshari–Mathew [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative Noetherian domain of Krull dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then the Rees algebras over R are Noetherian but symbolic Rees algebras need not be Noetherian (for example, see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' However we can prove the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1 without using the assumption that symbolic Rees algebras are Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' This is possible because of Heitmann’s result [6] over non-Noetherian rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For an extension of Heitmann’s result, see the results of Gupta [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Our method is to use standard sheaf patching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' More precisely, we find two covers, then we prove the results separately on each cover, finally we patch/glue to get the results over the original space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' A similar technique is used in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' However there to prove the result in one of the two covers, they have used Plumstead’s generalized dimension function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Here we prove the similar result using Heitmann’s result as symbolic Rees algebras need not be Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In section 2, we have recalled some basic definitions and results which will be used throughout the paper, also we have studied the Krull dimension of (extended) symbolic Rees algebras via valuation dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='5 is proved in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Recollection of basic definitions and results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Throughout the paper, we assume that the rings are commutative with the unity and the modules are finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Also, we assume that projective module have the constant rank function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Now we recall some basic notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For a ring R, let dimR denote the Krull dimension of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let Ass (R) denote the set of associated prime ideals of R, and ht(I) denote the height of the ideal I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1 (Unimodular Element).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a ring R and M an R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For m ∈ M, we define an ideal OM(m) = {ϕ(m) : ϕ ∈ M∗ = HomR(M, R)} of R which is called the order ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The element m in M is called a unimodular element if OM(m) = R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' there exists a surjective R-linear map ϕ ∈ M∗ such that ϕ(m) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let Um (M) denote the set of all unimodular elements in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Valuation dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The notion of the valuation dimension was introduced by Jaffard [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let A be an integral domain, an overring B of A is a subring of the field of fractions K of A that contains A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' A ⊆ B ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Valuation overring of A means an overring of A which is a valuation ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2 (Valuation Dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be an integral domain, the valuation dimension of R is the supremum of dimensions of the valuation overring of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Valuation dimension of R is denoted by dimvR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' dimvR = Sup {dimV : V is a valuation overring of R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' [1, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1] Let R be an integral domain which is not a field, K is a field of fraction of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let F denote the algebraic extension of K, and d a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then we say, R have finite valuation dimension d (write dimvR = d) if the following equivalent conditions are satisfied: (1) Each valuation overring of R in F has dimension ≤ d and there exists a valuation overring of R in F of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (2) Each overring of R in F has dimension ≤ d and there exists a overring of R in F of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (3) dimR[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' , xd] = 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE 3 (4) dimR[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' , xn] = n + d if n ≥ d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' We say that dimvR = ∞ if there exists no d satisfying the conditions 1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For the sake of completeness, each field is assigned valuation dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Clearly dimR ≤ dimvR for every domain R and if B is an overring of A, then dimvB ≤ dimvA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Recollection of Rees algebra and Symbolic Rees Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative ring and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The Rees algebra (also known as blow-up algebra) of R with respect to I as a subring of R[x] define to be R[Ix] = { n � i=0 aixi : ai ∈ Ii} = � n≥0 Inxn ⊆ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The extended Rees algebra of R with respect to I as a subring of R[x, x−1] define to be R[Ix, x−1] = { n � i=−n aixi : ai ∈ Ii} = � n∈Z Inxn ⊆ R[x, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' where for n ≤ 0, In = R by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The following theorem is for the Krull dimension of Rees algebra and extended Rees algebra (see [22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a Noetherian ring and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' If dimR is finite, then (1) dimR[Ix] = dimR + 1, if I ⊈ p for some p ∈ Spec R with dim(R/p) = dimR and dimR[Ix] = dimR, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (2) dimR[Ix, x−1] = dimR + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a Noetherian domain of dimension d and (0) ̸= I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then dimvR[Ix] = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For further properties of Rees algebras R[Ix] and extended Rees algebras R[Ix, x−1], one can see [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' [Symbolic Power, Symbolic Rees Algebra] Let R be a commutative Noetherian ring and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The n-th symbolic power of I is an ideal of R defined by I(n) = � p∈Ass (R/I) (InRp ∩ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In general, given an ideal I of R the ordinary power In do not coincide with the symbolic power I(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In particular, if I = p ∈ Spec (R), then p(n) = pnRp ∩ R and if I is a maximal ideal m, then m(n) = mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Clearly from the definition I(1) = I, In ⊆ I(n) and I(n+1) ⊆ I(n) for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative noetherian ring and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' The symbolic Rees algebra of R with respect to I is a graded algebra defined as Rs(I) = { n � i=0 aixi : ai ∈ I(i)} = � n≥0 I(n)xn ⊆ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Sometimes it is also known as symbolic blow-up algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In particular, for all n ≥ 2 if I(n) = In, then the symbolic Rees algebra Rs(I) coincide with the Rees algebra R[Ix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' One can define the extended symbolic Rees algebra of an ideal I of R as an extension of Rs(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let us denote Rs(I, x−1) as extended symbolic Rees algebra define to be Rs(I, x−1) = { n � i=−n aixi : ai ∈ I(i)} = � n∈Z I(n)xn ⊆ R[x, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 4 CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR where for n ≤ 0, I(n) = R by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For further properties of Rs(I) and Rs(I, x−1), one can see [4], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative Noetherian domain of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then dimRs(I) = � d, if I = (0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' d + 1, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For I = (0), there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' So we can assume I ̸= (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='5, dimvR[Ix] = 1+d = dimR[Ix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Consider R[Ix] ⊆ Rs(I) ⊆ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then dimvRs(I) ≤ dimvR[Ix] = d+1 and dimvRs(I) ≥ dimvR[x] = d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' This implies dimvRs(I) = d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' We know that dimRs(I) ≤ dimvRs(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence dimRs(I) ≤ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For the other inequality, note that Rs(I) = Rs(I)0 ⊕ Rs(I)+, where Rs(I)0 = R and Rs(I)+ = Ix ⊕ I(2)x2 ⊕ · · · , and Rs(I)/Rs(I)+ ∼= R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence Rs(I)+ is a prime ideal of Rs(I) and ht(Rs(I)+) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then we have dim(Rs(I)/Rs(I)+) + ht(Rs(I)+) ≤ dimRs(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' This implies that dimR + 1 ≤ dimRs(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Therefore dimRs(I) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a Noetherian domain of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then dimRs(I, x−1) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Note that dimvR[x−1] = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Consider R[x−1] ⊆ Rs(I, x−1) ⊆ R[x, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By [1, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='15], dimvR[x−1] = dimvR[x, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence dimvRs(I, x−1) = 1 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then we have dimRs(I, x−1) ≤ dimvRs(I, x−1) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For the other inequality, from R[x−1] ⊆ Rs(I, x−1) ⊆ R[x, x−1], we observe that Rs(I, x−1)x−1 = R[x, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then we have dimRs(I, x−1) ≥ dimRs(I, x−1)x−1 = dimR[x, x−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence dimRs(I, x−1) ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Therefore dimRs(I, x−1) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' □ If R is a Noetherian ring, then we know that the Rees algebra is a finitely generated R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Unlike Rees algebra, the symbolic Rees algebra is not necessarily finitely generated R-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For example, see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Existence of a unimodular element Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a Noetherian ring, I an ideal of R, and T be a multiplicatively closed subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then T −1Rs(I) = T −1Rs(T −1I), where T −1Rs(T −1I) means the symbolic Rees algebra of T −1R with respect to T −1I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By definition Rs(I) = R ⊕ Ix ⊕ I(2)x2 ⊕ · · · , and since the localization commutes with direct sums, we have T −1Rs(I) = T −1R ⊕ T −1(I)x ⊕ T −1(I(2))x2 ⊕ · · · = T −1R ⊕ T −1(IR)x ⊕ T −1(I(2)R)x2 ⊕ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE 5 Now for n ≥ 1, T −1(I(n)) = T −1( � Ass (R/I) (InRp ∩ R)) = � Ass (T −1(R/I)) T −1(InRp ∩ R) = � Ass (T −1R/T −1I)) ((T −1I)n(T −1R)p ∩ T −1R) = (T −1I)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since the localization commutes with direct sums, we also have T −1Rs(T −1I) = T −1R ⊕ T −1I(T −1R)x ⊕ (T −1I(T −1R))(2)x2 ⊕ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' But we already have (T −1I)(n) = T −1(I(n)), hence we conclude that T −1Rs(I) = T −1Rs(T −1I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' □ The following lemma is a particular case of the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' We use the following version later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative Noetherian domain and (0) ̸= I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then (1) If T := set of all non-zero-divisors of R, T −1Rs(I) = T −1R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (2) If T = {1, a, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='} with a ∈ I is a non-zero-divisor of R, T −1Rs(I) = Ra[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (1) Since R is a domain, T = R∗ = R\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' For n ≥ 1, I(n) is an ideal of R and T ∩ I(n) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' This implies T −1(I(n)) = T −1R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence by 1st part of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1, we get T −1Rs(I) = T −1R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' (2) By [15, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1], we have T −1R[Ix] = T −1R[(T −1I)x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since T ∩ I ̸= ∅ this implies T −1R[Ix] = T −1R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Consider R[Ix] ⊆ Rs(I) ⊆ R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' After localize at T, we have T −1R[Ix] ⊆ T −1Rs(I) ⊆ T −1R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Therefore T −1Rs(I) = Ra[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative domain of dimension d and A = R1+aR, where a is a non-zero non-unit element of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let P be a projective R-module of rank ≥ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let Q := P1+aR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then Q has a unimodular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let ‘overbar’ denote the going modulo the ideal aR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since (0) is the minimum prime ideal in the domain R, we observe that dim(R) < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By [6, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='6], P has a unimodular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Therefore there exists p such that the order ideal OP (p) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let p ∈ P be a lift of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Now observe that 1 + aR ⊂ of the order ideal OP (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence p1+aR is a unimodular element of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' □ The following theorem is a result of Ravi A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Rao [14, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1(I)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative Noetherian ring of dimension d, and S be a multiplicative closed set of non-zero-divisors of R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let A be a ring lying between R[x] and S−1R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then for n ≥ d + 2, En(A) acts transitively on Um(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let R be a commutative Noetherian domain of dimension d and I an ideal of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let A = Rs(I) or Rs(I, x−1) and P a finitely generated projective A-module of rank ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then P has a unimodular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 6 CHANDAN BHAUMIK AND HUSNEY PARVEZ SARWAR Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' First, we assume that A = Rs(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let rank(P) = m > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since A is a subring of an integral domain R[x], A is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' If I = (0), then I(n) = (0) this implies A = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In this case, the theorem follows from Serre [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' If I = (1), then I(n) = (1) this implies A = R[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' In this case, the theorem follows from Plumstead [13, Corollary 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' So we can assume I ̸= (0) and I ̸= (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='7, dimRs(I) = d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let T be the set of all non-zero-divisors of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2(1), T −1A = T −1R[x], where T −1R is a quotient field of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since T −1R[x] is a PID, T −1P is a free module over T −1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since P is finitely generated, there exists t ∈ T such that Pt is a free Rs(I)t- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let 0 ̸= a be a non-unit element of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then Pat is a free Rs(I)at-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Denote b = at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2(2), we have Rs(I)b ∼= Rb[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Now consider the following Cartesian square of rings Rs(I) Rs(I)b ∼= Rb[x] Rs(I)1+bRs(I) Rb[x]1+bRs(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' i1 i2 j1 j2 Since Pb is a free Ab-module of rank m, Pb ∼= Am b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence Pb has a unimodular element say p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since P1+bA is projective A1+bA-module, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='3(1), P1+bA has a unimodular element say p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence we have P1+bA ∼= p2A1+bA ⊕ Q, with projective A1+bA-module Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Now consider B = Ab(1+bA) = (1 + bA)−1Rs(I)b = (1 + bR)−1D−1Rb[x] = D−1R ′[x], where D is a multiplicative closed subset of R ′[x] and R ′ = Rb(1+bR) is of dimension d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since Pb(1+bA) is a free B-module of rank m, Pb(1+bA) ∼= Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Let ¯u and ¯v be the image of p1 and p2 respectively in Pb(1+bA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Now consider the following cartesian square of projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' P Pb P1+bA Pb(1+bA) ∼= Bm i1 i2 j1 j2 Note that the ring B lying between R ′[x] and S−1R ′[x], where S is a set of all non-zero- divisors in R ′[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='4, there exists σ ∈ Em(B) such that σ(¯u) = ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' By [18, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2] (see also [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='2]), we can split σ as (σ1)b(σ2)1+bA, where σ2 ∈ E(Pb) and σ1 ∈ E(P1+bA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Since (σ1)b(σ2)1+bA(¯u) = ¯v, for suitably changing p1 and p2, we can assume that ¯u = ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Now consider the following fiber product diagram of projective modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' P Pb ∼= Am b A Ab P1+bA ∼= A1+bA ⊕ Q Pb(1+bA) A1+bA Ab(1+bA) ψ ψ1 ψ2 Since ψ1 and ψ2 are surjective homomorphisms, from the above diagram, we get a sur- jective homomorphism ψ : P ։ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Hence there is p ∈ P such that ψ(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Therefore P has a unimodular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' EXISTENCE OF UNIMODULAR ELEMENT IN A PROJECTIVE MODULE 7 When A = Rs(I, x−1), the proof is a verbatim copy of the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' However one needs to use [11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='1] in the second corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' A diligent reader can work out the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' □ Acknowledgement: H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Sarwar would like to thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' of India for the grant SRG/2020/000272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='iitkgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='in Email address: mathparvez@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNE5T4oBgHgl3EQfCQ64/content/2301.05394v1.pdf'} diff --git a/XdA0T4oBgHgl3EQfFP_h/content/2301.02031v1.pdf b/XdA0T4oBgHgl3EQfFP_h/content/2301.02031v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1bffb65b4dfd80ffcbab16f1434b396b744f5354 --- /dev/null +++ b/XdA0T4oBgHgl3EQfFP_h/content/2301.02031v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:552111f70b85baf601ff6896679e59b5352986121d0cd8fa54ff9319bb54eeb5 +size 19969852 diff --git a/Y9FIT4oBgHgl3EQfkCuy/content/tmp_files/2301.11299v1.pdf.txt b/Y9FIT4oBgHgl3EQfkCuy/content/tmp_files/2301.11299v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a91db84ff8874b30e3ddf8fe0925c924ef32a305 --- /dev/null +++ b/Y9FIT4oBgHgl3EQfkCuy/content/tmp_files/2301.11299v1.pdf.txt @@ -0,0 +1,1546 @@ +Draft version January 27, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +The frequency ratio and time delay of solar radio emissions with fundamental and harmonic +components +Xingyao Chen (陈星瑶 ) +,1 Eduard P. Kontar +,1 Daniel L. Clarkson +,1 and Nicolina Chrysaphi +2, 1 +1School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK +2LESIA, Observatoire de Paris, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, 5 place Jules Janssen, 92195 Meudon, +France +ABSTRACT +Solar radio bursts generated through the plasma emission mechanism produce radiation near the +local plasma frequency (fundamental emission) and double the plasma frequency (harmonic). While +the theoretical ratio of these two frequencies is close to 2, simultaneous observations give ratios ranging +from 1.6 to 2, suggesting either a ratio different from 2, a delay of the fundamental emission, or both. +To address this long-standing question, we conducted high frequency, high time resolution imaging +spectroscopy of type III and type J bursts with fine structures for both the fundamental and harmonic +components with LOFAR between 30 and 80 MHz. The short-lived and narrow frequency-band fine +structures observed simultaneously at fundamental and harmonic frequencies give a frequency ratio +of 1.66 and 1.73, similar to previous observations. However, frequency-time cross-correlations suggest +a frequency ratio of 1.99 and 1.95 with a time delay between the F and H emissions of 1.00 and +1.67 s, respectively for each event. Hence, simultaneous frequency ratio measurements different from +2 are caused by the delay of the fundamental emission. Among the processes causing fundamental +emission delays, anisotropic radio-wave scattering is dominant. Moreover, the levels of anisotropy and +density fluctuations reproducing the delay of fundamental emissions are consistent with those required +to simulate the source size and duration of fundamental emissions. Using these simulations we are able +to, for the first time, provide quantitative estimates of the delay time of the fundamental emissions +caused by radio-wave propagation effects at multiple frequencies, which can be used in future studies. +1. INTRODUCTION +Plasma emission is believed to be responsible for gen- +erating solar radio bursts at decimeter and longer wave- +lengths (e.g. Ginzburg & Zhelezniakov 1958; Dulk 1985; +Melrose 1987). Within the plasma emission theory, the +instability generates Langmuir waves at the local plasma +frequency fpe, where fpe = ωpe/2π = +� +e2n(r)/πme is +the electron plasma frequency, n(r) is the electron num- +ber density, and e and me are the electron charge and +mass. The coalescence of Langmuir waves and low fre- +quency ion-sound waves may produce electromagnetic +waves via nonlinear plasma processes, which is referred +to as fundamental emission (hereafter, F). The coales- +cence of counter-propagating Langmuir waves may pro- +duce the radio emission at 2fpe, which is referred to as +the second harmonic emission (hereafter, H). +Corresponding author: Xingyao Chen +Xingyao.Chen@glasgow.ac.uk +Several longstanding issues exist concerning the F and +H emissions—one problem in particular is how to dis- +tinguish between them. When a single burst is observed +(and not an F-H pair), it is difficult to identify whether +this burst is the result of F or H emissions, with the +identification relying on the degree of their polarisation. +The polarisation degree of the F radio waves is predicted +to be higher than the H (Dulk 1985). Suzuki & Dulk +(1985) reported that the polarisation degree of the F +components are all larger than the H components from +714 F-H pairs of type III bursts, that being 0.35 (F) and +0.12 (H) on average. However, the polarisation degree +varies even within one burst (Dulk & Suzuki 1980) or +even within one fundamental component (Kontar et al. +2017). When there are two components observed, some +additional arguments can suggest F-H pairs: the F typ- +ically has higher intensities, shorter rise times, higher +polarisations, and near half drift rates of the higher fre- +quency component. +There are also multiple issues in debate—for exam- +ple, F and H emission may be generated over different +arXiv:2301.11299v1 [astro-ph.SR] 26 Jan 2023 + +ID2 +timescales because they are produced through related, +but different, processes. Moreover, it is unclear whether +F and H emissions are produced at the same spatial +location. In this paper, we consider a remarkable prob- +lem related to their frequency ratio RH/F, i.e. the ratio +of the H to F components observed at the same time. +While the theory gives a harmonic frequency ratio very +close to 2, the observations suggest a range from 1.6-2, +averaging at 1.8 (Wild et al. 1954; Stewart 1974). +Multiple studies have previously indicated time delays +of the F with respect to the H, in a range from 1 s up +to 7 s, and H/F frequency ratios in a range of 1.74-1.94 +(Hughes & Harkness 1963; Stewart 1974; Robinson & +Cairns 1998; Dorovskyy et al. 2015; Koval et al. 2016; +Melnik et al. 2018), yet there is no agreement on the +mechanisms involved. For example, an early explana- +tion by Wild et al. (1954) for the observed difference +from the theoretical prediction being that the observed +spectrum consists of the H with only the higher frequen- +cies of the F band; Stewart (1974) suggested that the F +emission had reduced escape efficiency; the model by +Robinson & Cairns (1998) stated that the F was re- +absorbed above the local plasma frequency; Dorovskyy +et al. (2015) suggested lower group velocities of the F +inside a modeled magnetic loop. +There are few studies aimed at the analysis of the H/F +frequency ratios, because firstly, the F emissions are not +always observed along with their corresponding H emis- +sions and are difficult to be clearly defined; secondly, +the H/F frequency ratios are usually used as evidence +for the H components, while they are close to 2 then the +two emission branches may be regarded as F-H pairs; +and thirdly, it is hard to quantitatively compare a F fre- +quency with the corresponding H component in order to +calculate the H/F frequency ratios. +From previous studies, the cause of the time delay +of the F components has been mostly explained by their +different electromagnetic wave group velocities and radio +wave propagation effects in an inhomogeneous corona +(Itkina et al. 1993; Melnik et al. 2018). However, there +was no quantitative estimation of the H/F frequency +ratios and delay times from radio wave propagation ef- +fects. There seems to be no obvious dependencies of H/F +frequency ratios on frequencies from previous observa- +tions, which suggests that the propagation effects may +be dominant due to various characteristic parameters +of the background density fluctuations in a turbulent +corona. +In this paper, we study the H/F frequency ratios and +delay times between the F and H emissions from type +III and type J bursts with striae observed by the LOw +Frequency ARray (LOFAR; van Haarlem et al. 2013) +with high temporal, spectral and spatial resolutions in +a frequency range of 30-80 MHz. J-type and/or U-type +solar radio bursts is a variant of type III bursts and is +believed to be generated from electron beams traveling +along closed magnetic loops (Maxwell & Swarup 1958; +Labrum & Stewart 1970; Hillaris et al. 1990; Aschwan- +den et al. 1992; Aurass & Klein 1997; Dorovskyy et al. +2010; Fernandes et al. 2012; Reid & Kontar 2017). From +a featureless radio burst with F and H components, the +frequency ratio can be calculated from the frequencies +at the maximum intensities of F and H components at +each time. The time delay can be derived from the times +at the peak intensities at f and 2f. However, it does not +seem possible to determine the H/F frequency ratio and +delay time simultaneously. We implement the correla- +tion of type III and type J bursts that present striae in +both F and H components, allowing clear determination +of the frequency ratio. The fine structures are crucial +to this method because striae in the F components have +their counterparts in the H components and can be well +correlated to determine the frequency ratio and delay +time simultaneously. The variants of type III bursts in +a form of ‘J’ or ‘U’ in the dynamic spectrum can also +give the frequency ratio and delay time simultaneously +from their similar shapes but it is worth noting that they +are different if measured at the turning point and start- +ing frequency of type U or J bursts. The isolines with +different intensity levels of the U or J structure would +also affect the estimations of frequency ratio and delay +time. +From the dynamic spectra of type III bursts with +striae, the observed H/F frequency ratios are less than +the theoretical ratio, suggesting that the observed F +branch is delayed. We suggest that the time delay be- +tween F and H components can be explained by a combi- +nation of different group velocities and scattering effects, +with the latter forming a larger contribution. As the F +component is emitted at the local plasma frequency, it +undergoes a stronger scattering effect than the H com- +ponent, which lengthens the propagation path of the +F emission and leads to a delayed F band. +For the +first time, we quantitatively estimate the delay times +between the F and H components from ray-tracing sim- +ulations of radio wave propagation (Kontar et al. 2019). +Based on the diagnosis by Chen et al. (2020) in which +they estimated the turbulent coronal parameters by +combined analysis of scattering simulations and imaging +observations from one type IIIb radio burst, our estima- +tions of delay times from both different group velocities +and scattering propagation effects can be well matched +with that from observations. + +3 +Freq +Time +𝑡delay +𝑓Ftrue +𝑓Fobs +𝑓H +Freq +Time +harmonic +Fobs +Ftrue +𝑡delay +Figure 1. A physical scenario of two dynamic spectra showing the time delay between the observed and intrinsic F components +of the normal type III burst (left panel) and the type III burst with striae fine structures (right panel). +In Section 2, we present the theoretical H/F frequency +ratio. A schematic illustration of the delayed F compo- +nent in the type III burst with striae is presented in Sec- +tion 3. Section 4 describes the characteristic parameters +from type III and type J burst observations, including +their observed H/F frequency ratios and delay times. +Section 5 shows the results from ray-tracing simulations +of radio wave propagation. Section 6 is our summary. +2. EMISSION NEAR FUNDAMENTAL AND +HARMONIC FREQUENCIES +The plasma emission mechanism is widely accepted +for the generation of solar radio bursts at decimeter and +longer wavelengths (e.g. Ginzburg & Zhelezniakov 1958; +Dulk 1985; McLean & Labrum 1985; Zheleznyakov 1996; +Ratcliffe et al. 2014). The F emission can be generated +through scattering of Langmuir waves by ions and/or +their interaction with ion-sound waves. The H emission +is generated by the coalescence of the Langmuir waves +with back-scattered Langmuir waves, so emits at the +summed frequencies of two Langmuir waves. +Conservation of energy and momentum (frequency +and wavenumber) leads to the fundamental emission be- +ing close to the Langmuir wave frequency +ωL ≃ ωpe +� +1 + 3 +2 +v2 +Te +v2 +� +where vTe is the electron thermal speed, and v is the +phase speed of the waves. For the typical parameters of +type III solar radio bursts, v/vTe is about 10 (e.g. Rat- +cliffe et al. 2014; Reid & Kontar 2021), so the deviation +from plasma frequency ∆ω is as small as +∆ω/ωpe ≃ 0.015 +Similarly, the harmonic frequency is close to the twice +of the Langmuir wave frequency. Hence the ratio of har- +monic to fundamental should be very close to 2, within +1 − 2% for the typical parameters in type III solar radio +bursts. +3. SCHEMATIC ILLUSTRATION +The schematic illustration outlined in Figure 1 clearly +explains a delayed F component leads to the derived +H/F frequency ratio to be less than 2 from observations +of F-H pairs. The observed F component is delayed by a +time tdelay compared to the intrinsic F component. The +frequency of the observed F branch is larger than the +intrinsic F branch at each time, so the H/F frequency +ratio normally calculated between the observed H and F +components will be less than the theoretical frequency +ratio. The intrinsic H/F frequency ratio should be be- +tween the observed H component and the intrinsic F +component instead of the observed F component. Fur- +thermore, the radio source imaging of the H component +should be coincident with an earlier F branch instead of +the F emission at the same time. +We also show a scenario for a type III burst with striae +in the right panel from Figure 1. With the striae, we +can determine the frequency ratio and delay time si- +multaneously, which is not feasible for normal type III +bursts without fine structures. Ideally, striae in the F +components are expected to have counterparts in the H +components, yet the striae in the H are normally not as +apparent, which may be the result of the weak intensity +of the H components and the limited dynamic range of +antennas. Nonetheless, the striae in our analysis are well +observed and correlated with both F and H components +from LOFAR observations. +4. OBSERVATIONS + +4 +11:56:50 +11:56:55 +11:57:00 +Start Time (16-Apr-15 11:56:48) +30 +40 +50 +60 +70 +80 +Frequency [MHz] +11:56:50 +11:56:55 +11:57:00 +Start Time (16-Apr-15 11:56:48) +30 +40 +50 +60 +70 +80 +Frequency [MHz] +1 +14 +200 +Flux [sfu] +(a) Type III +12:19:55 +12:20:00 +12:20:05 +Start Time (07-May-15 12:19:53) +30 +40 +50 +60 +70 +80 +Frequency [MHz] +12:19:55 +12:20:00 +12:20:05 +Start Time (07-May-15 12:19:53) +30 +40 +50 +60 +70 +80 +Frequency [MHz] +0.1 +4.5 +200.0 +Flux [sfu] +(b) Type J +Figure 2. The dynamic spectra of type III (a) and J (b) bursts with both F and H components observed by LOFAR. The same +time range for both F and H components, but twice the frequency range for the H component, is selected to derive their drift +rates, indicated by the solid rectangular boxes. The green lines are the best fit through all the positions of the fitted Gaussian +peaks using a linear fitting function. +The solar type III radio burst (left panel in Figure 2) +at around 11:57:00 UT on 16-April-2015 and the type +J burst (right panel in Figure 2) at 12:20:00 UT on +07-May-2015 are observed by the LOw Frequency AR- +ray (LOFAR; van Haarlem et al. 2013). LOFAR is a +large interferometric radio telescope with high spectro- +scopic and imaging capabilities, located primarily in the +Netherlands with a number of international stations in +other European countries. +It was completed in 2012 +by the Netherlands Institute for Radio Astronomy (AS- +TRON) and can observe with the Low Band Antenna +(LBA) and the High Band Antenna (HBA), optimized +for 30 - 80 MHz and 120 - 240 MHz, respectively. The +type III and J bursts here are observed by the tied-array +beam forming mode simultaneously with a maximum +frequency resolution of ∼ 12.2 kHz and time resolution +of ∼ 10 ms (Kontar et al. 2017). +We show our analysis of well observed type III and J +bursts with both F and H components and striae, de- +rive their frequency ratios and delay times, and compare +with those estimated from radio wave propagation sim- +ulations. +4.1. Overview of the type III burst +The type III burst presented in Figure 2 is composed +of two branches—the F branch between 30-65 MHz and +H branch between 30-72 MHz. The background is sub- +tracted by using quiet periods prior to the bursts, and +only the burst properties are analyzed. Their frequency +and time resolutions are 12.2 kHz and 52.4 ms, respec- +tively. +The type III-IIIb burst with F-H pairs is well ana- +lyzed in several papers. +Kontar et al. (2017) demon- +strated that radio wave propagation effects dominated +the observed spatial characterization of radio burst +images. +Using a model developed by Kontar et al. +(2019) to quantitatively study radio-wave propagation +in anisotropic density fluctuations, Chen et al. (2020) +found that anisotropic scattering simulations can repro- +duce the observed time profiles, centroid locations, and +source sizes of the type IIIb radio burst. +From anal- +ysis of the dynamic spectrum, Sharykin et al. (2018) +provided statistically significant properties of individual +striae, Chen et al. (2018) explained that the striae fine +structures were caused by the background density fluc- +tuations, and Kolotkov et al. (2018) demonstrated that +the striae frequency drift can be modulated by a prop- +agating fast wave train. In this study, we focus on the +delay time and frequency ratio between the F and H +components. +The F emission started at around 11:56:54 UT and +ended at around 11:56:59 UT. Each distinct stria be- +tween 30-40 MHz contributes to a mean striae lifetime +of about 1 s, with longer duration times at lower fre- +quencies. The same time range and twice the frequency + +5 +11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 +Start Time (16-Apr-15 11:56:53) +30 +31 +32 +33 +34 +35 +Frequency [MHz] +11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 +Start Time (16-Apr-15 11:56:53) +30 +31 +32 +33 +34 +35 +Frequency [MHz] +1 +17 +300 +Flux [sfu] +(a) Spectrum of type III burst +11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 +Start Time (16-Apr-15 11:56:53) +45 +50 +55 +60 +65 +70 +Frequency [MHz] +(a) Spectrum of type III burst +11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 +Start Time (16-Apr-15 11:56:53) +45 +50 +55 +60 +65 +70 +Frequency [MHz] +1.0 +4.5 +20.0 +Flux [sfu] +Type III (H) +Type III (F) +RH/F: 1.99 td: -1.00 +12:19:56 +12:19:58 +12:20:00 +12:20:02 +12:20:04 +Start Time (07-May-15 12:19:55) +33 +34 +35 +36 +37 +38 +Frequency [MHz] +12:19:56 +12:19:58 +12:20:00 +12:20:02 +12:20:04 +Start Time (07-May-15 12:19:55) +33 +34 +35 +36 +37 +38 +Frequency [MHz] +1 +10 +100 +Flux [sfu] +(c) Spectrum of type J burst +12:19:56 +12:19:58 +12:20:00 +12:20:02 +12:20:04 +Start Time (07-May-15 12:19:55) +55 +60 +65 +70 +75 +Frequency [MHz] +(c) Spectrum of type J burst +12:19:56 +12:19:58 +12:20:00 +12:20:02 +12:20:04 +Start Time (07-May-15 12:19:55) +55 +60 +65 +70 +75 +Frequency [MHz] +1 +4 +15 +Flux [sfu] +Type J (H) +Type J (F) +RH/F: 1.95 td: -1.66 +(b) Correlation results +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Delay time [s] +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +RH/F +(b) Correlation results +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +Delay time [s] +1.5 +1.6 +1.7 +1.8 +1.9 +2.0 +RH/F +-0.496 +0.208 +0.911 +Type III +(d) Correlation results +-3 +-2 +-1 +0 +Delay time [s] +1.70 +1.75 +1.80 +1.85 +1.90 +1.95 +2.00 +RH/F +(d) Correlation results +-3 +-2 +-1 +0 +Delay time [s] +1.70 +1.75 +1.80 +1.85 +1.90 +1.95 +2.00 +RH/F +-0.41 +0.18 +0.77 +Type J +Figure 3. Correlation results for type III and type J bursts: (a), (c) Enlarged dynamic spectra of the F (below) and H (upper) +components. The solid rectangular box in the H spectrum will slip with given time and frequency lags. The selected F spectrum +in the solid rectangular box is correlated with each slipped H spectrum. (b), (d) Two-dimensional cross correlation results +between the selected F and each slipped H (spectral) boxes. The peak correlation coefficients at each frequency ratio are shown +by green dots. The maximal correlation coefficient is marked by the plus symbol. If there is no time delay, the frequency ratio +at the peak correlation coefficient is marked by the cross symbol. + +6 +range for the H branch (60-70 MHz) with respect to the +F branch (30-35 MHz) are selected for analysis. +The +H/F frequency ratios are measured in two ways in our +study: from their drift rates and the cross correlations +between F and H spectra. The delay times are derived +from the peak time intervals of the F and H flux profiles +and cross correlations between F and H branches. +4.2. Frequency drift rates +The frequency drift rate df +dt for the F emission at the +local plasma frequency is +dfF +dt += +dfpe +dt . +Since fpe = +� +e2n(r)/πme is a function of background electron den- +sity, then it can be written as +dfF +dt += +dfpe +dn(r) +dn(r) +dr +dr +dt = +fpe +2n +dn(r) +dr +dr +dt . For the H emission at double the plasma fre- +quency, the frequency drift rate is dfH +dt = d2fpe +dn(r) +dn(r) +dr +dr +dt = +2fpe +2n +dn(r) +dr +dr +dt . Therefore, the theoretical frequency drift +rate ratio between H and F emissions is DH/F += +dfH +dt / dfF +dt = 2, which can be used for evidence of the har- +monic branches. +From the dynamic spectrum, the time profiles at each +frequency are fitted with a 1D Gaussian function. The +peak times at the maximum fitted flux at each frequency +are marked. They follow a linear function over small +frequency ranges. They are then fitted using a linear +function of f = df +dtt + C, seen from the two green lines +in the dynamic spectra (Figure 2) for both the F and +H components. The drift rates with 1-sigma errors are +df 1 +F +dt = −5.68 ± 0.16 MHz s−1 and df 1 +H +dt = −10.30 ± 0.12 +MHz s−1 for the F and H components respectively. The +drift rate of the H component is around twice that of the +F component, which can be further deduced as evidence +of a F-H pair. +Defining the drift rates for the F and H components as +DF = dfF +dt and DH = dfH +dt , then considering the drift rate +functions fF = DFt + CF and fH = DHt + CH, one can +derive a function of their H/F frequency ratio, Rdr +H/F = +fH +fF = +DH +DF + (CH − DH +DF CF)f −1 +F . +Their H/F frequency +ratios are then calculated between the two green lines +from Figure 2. The frequency ratios range from 1.65 to +1.67 in the F frequency range of 30-35 MHz. +4.3. Cross correlations between the F and H branches +The H component is distinguished from the F, seen +from Figure 3(a). We select part of the F component in +a frequency range of 30-35 MHz by considering that the +cut-off frequency of the H emission is around 70 MHz. +In order to derive a cross correlation map, the fre- +quency range and time range need to be selected for the +H component. The 2D cross correlation is then calcu- +lated between the selected F component (the white box +in the lower panel from Figure 3(a)) and the slipped +H component (the white box, as an example at one in- +stance, in the upper panel from Figure 3(a)). +The frequency ratio is set up to range from 1.5 to +2.0 with a ratio lag of 0.01. A frequency ratio of 1.5 +corresponds to a frequency range of 45.0-52.5 MHz for +the H, and a frequency ratio of 2.0 will determine a H +frequency range of 60-70 MHz. The time delay of the +H is set up to range from -1.84 s to 1.26 s, which is the +time difference between the start time of the H and the +start time of the F component at 11:56:55.8 UT. The +time step is set to be 0.05 s. We keep the same time +interval between the start and ending times for both F +and H components. The time range for the F is fixed +from 0 s (11:56:55.8 UT) to 3.10 s (11:56:58.9 UT) and +the time range for the H is slipped and changed with +each delay. For example, while the time delay for the H +is -1 s, the H time range is from -1 s (11:56:54.8 UT) to +2.10 s (11:56:57.9 UT). +In order to search for any time-delays and frequency +ratios between the F and H spectra, we create two- +dimensional cross-correlation functions (CCFs) as fol- +lows: +CCF = +� +ij +(Xij − X) × � +ij +(Yij − Y ) +� +(� +ij +Xij − X)2 × (� +ij +Yij − Y )2 +Here Xij and Yij are two sets of spectra of the F and H +components. We loop over all delay times and frequency +ratios and compute the overlap and correlation for each +shift. The effective correlation coefficients range from 0 +to 1, meaning no correlation and maximum correlation, +respectively. The cross correlation map can be seen in +Figure 3(b). +Uncertainties of the time and frequency ratio lags are +determined using intensity randomization subset sam- +pling by taking an observed dynamic spectrum and cre- +ating 50 variations where the observed intensity is varied +by I ±δI. The background flux level before the burst at +each frequency is taken as the uncertainty on the flux, +around 1 sfu, similar to Kontar et al. (2017). δI is ran- +domly taken from a normal distribution with a mean of +zero, and a standard deviation of one. The average fre- +quency ratio and delay time are obtained and the errors +are taken from the sum of the standard deviations, delay +time, frequency ratio resolutions and also the time and +frequency resolutions of LOFAR. +From the cross correlation map, the peak correla- +tion coefficients at each frequency ratio give the best- +matched delay times (dot symbols). The plus symbol +shows the maximal correlation coefficient at a delay time +of 1.00± 0.05 s and a frequency ratio of 1.99± 0.01 (seen +from Figure 3(b)), which means that the H emission in + +7 +56:56.0 56:56.5 56:57.0 56:57.5 56:58.0 56:58.5 +Start Time (16-Apr-15 11:56:55) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Normalized Flux +F 30.04 MHz +H 60.06 MHz +Delay 1.05 s +Delay(fit) 0.94 s +Shifted F(cross) 0.94 s +(a) +30 +31 +32 +33 +34 +35 +Frequency of F [MHz] +0.5 +1.0 +1.5 +2.0 +Delay time [s] +Delay +Delay(fit) +Delay(cross) +(b) +Figure 4. Statistical delay times for the type III burst: (a) Normalized time profiles of the F and H emissions at 30 and 60 +MHz, respectively. Their Gaussian fit profiles are shown by dashed lines. The peak times of observed and fitted time profiles +are marked by the vertical solid and dashed lines, respectively. The time profile of the F component was shifted by a time lag +of 0.94 s from the cross correlation between the F and H emissions (solid red line). (b) Statistical delay times from observations +(black), fits (blue) and cross correlations (red) between F and H emissions between 30 MHz to 35 MHz. +the 60-70 MHz range is best matched with the F emis- +sions in 30-35 MHz with a delay time of 1.00 s. In other +words, the F emission is delayed by 1 s with respect to +the H emission. If there is no delay time between the F +and H components, the frequency ratio is 1.66± 0.01, as +seen from the cross symbol in Figure 3(b). +4.4. Statistical delay times +The normalized time profiles at frequencies of 30 MHz +(F, black solid line) and 60 MHz (H, blue solid line) and +their Gaussian fitted profiles (dashed lines) are shown as +an example in Figure 4(a). The vertical lines mark the +peak times of the flux curves. The delay times derived +from the intervals between the peak times of the original +and fitted flux profiles between 30 MHz (F) and 60 MHz +(H) are 1.05 s and 0.94 s, respectively. The time profiles +of the F and H components are also cross correlated, +which gives a delay time of 0.94 s for the maximum cross +correlation coefficient. After correcting for a time lag of +0.94 s, the shifted F component is shown by the red solid +line in Figure 4(a). These three methods (estimations +using the original, the fitted, and the cross correlation +profiles) give similar delay times of around 1 second at +30 MHz. +We also statistically obtain the delay times from the +flux profiles at each frequency between 30-35 MHz (F) +and twice the frequency for the H components, shown +in Figure 4(b). It shows the averaged time delays for +the original (black), fitted (blue), and the cross corre- +lated (red) time profiles are 1.19 s, 1.14 s and 1.08 s, +respectively. +Type III Burst +Type J Burst +t +F +Dur [s] +1.27 +2.19 +t +H +Dur [s] +1.19 +2.10 +fmid [MHz] +32.5 +36.0 +dfF +dt [MHz/s] +−5.68 ± 0.16 +−2.38 ± 0.11 +dfH +dt [MHz/s] +−10.30 ± 0.12 +−4.58 ± 0.14 +Rdr +H/F +1.65-1.67 +1.70-1.73 +t +pk +d +[s] +1.19 +2.04 +t +fit +d [s] +1.14 +1.97 +t +corr +d +[s] +1.08 +1.91 +Rcorr(td=0) +H/F +1.66 +1.73 +Rcorr(max) +H/F +1.99 +1.95 +tcorr(max) +d +[s] +1.00 +1.67 +Table 1. Characteristic parameters of the type III and type +J bursts, including t +F +Dur (the averaged duration given by the +FWHM from Gaussian fits for the F), t +H +Dur (averaged FWHM +duration for the H), fmid (middle frequency of the F com- +ponent), dfF +dt (drift rate of the F), dfH +dt (drift rate of the H), +Rdr +H/F (frequency ratio calculated between the two drifting +rate lines of the F and H), t +pk +d +(averaged delay times calcu- +lated from the peak intervals of the flux curves of the F and +H), t +fit +d (averaged delay times derived from the peak intervals +of the fitted flux curves), t +corr +d +(averaged delay times from the +cross correlations between the F and H at each frequency), +Rcorr(td=0) +H/F +(frequency ratio at the peak correlation coefficient +for the case of no time delay), Rcorr(max) +H/F +(frequency ratio at +the maximal correlation coefficient from the 2D cross corre- +lation between the F and H spectra), tcorr(max) +d +(delay time +at the maximal correlation coefficient from the 2D cross cor- +relation). + +8 +4.5. Type J burst +The F component of the type J burst was observed +between 30-80 MHz, with the H component between 52- +80 MHz. +The F emission started at around 12:19:56 +UT and ended with a long tail in the LOFAR observing +frequency range. +The same method to derive the frequency ratios and +delay times are implemented for type J as for type III +burst. All characteristic parameters are listed in Table +1. +The frequency drift rate of the F branch is about +df 2 +F +dt = −2.38±0.11 MHz s−1 while that for the H branch +is roughly twice that of the F branch at df 2 +H +dt = −4.58 ± +0.14 MHzs−1. Considering the lower drift rates than the +normal type III burst and near ”J” shape, the burst is +identified as a variant of the type III burst called a type +J burst (Reid & Kontar 2017). +The frequency ratios +calculated from the drift rates range from 1.70 to 1.73 +in the F frequency range of 33-39 MHz. +The results from cross correlations between the F and +H components are shown in Figure 3(d). It’s worth not- +ing that the lower cut-off frequency for the H is around +55 MHz, so the frequency ratio is set up to be from 1.67 +(55 MHz/33 MHz) to 2.00. The delay times are set up +from -3.15 s to 0.75 s. The frequency ratio and delay +time are 1.95± 0.01 s and 1.67± 0.03 s at the maximal +correlation coefficient from the two dimensional cross +correlation map. The frequency ratio with no time de- +lay is 1.73± 0.01 at the peak correlation coefficient. +The delay times derived from peak time intervals of +the original flux profiles, the fitted flux profiles, and the +cross correlation between the F and H flux curves are +2.04 s, 1.97 s and 1.91 s, respectively, averaged in a +frequency range of 33-39 MHz (F) and twice those fre- +quencies for the H components. +5. SIMULATIONS OF RADIO WAVE +PROPAGATION +The observed properties of radio waves, including time +profiles, source positions, sizes, and emission directivity +can be strongly affected by the inhomogeneous density +fluctuations as they propagate through the turbulent +corona. +There are very few studies on the time pro- +files that quantitatively investigate the delay times be- +tween F and H emissions resulting from their propaga- +tion through the turbulent coronal medium. In order to +quantitatively study the effects of radio wave propaga- +tion on the H/F frequency ratio and delay times between +the F and H components, we use ray tracing simulations +of radio wave propagation developed by Kontar et al. +(2019). +5.1. Brief introduction of the simulation set-up +28 +30 +32 +34 +36 +38 +Frequency [MHz] +0.36 +0.38 +0.40 +0.42 +Delay time (rays) [s] +Cq= 0 +Figure 5. Delay time as a function of frequency calculated +using simulations without scattering: Cq = 0 R−1 +⊙ , frequency +ratio RH/F=1.82, and heliocentric angle θ = 5◦. +In the simulations, the radio waves are treated as a +number of rays (105 in our case) with positions r and +wavenumbers k. Initially, they are seen as a point source +located at a given position which is related to the emit- +ting frequency. Then the radio waves propagate in the +turbulent corona and undergo refraction effects mainly +caused by the large-scale density gradient, as well as +scattering effects caused by small-scale density pertur- +bations. +Their positions and wave vectors are calcu- +lated from the numerical solutions of the Fokker-Planck +equation and Hamilton’s equations in an unmagnetized +plasma (seen in Kontar et al. (2019)). All rays arrive at a +sphere where the scattering is assumed to be negligible. +Their arrival times, final positions and wave vectors are +recorded to produce the time profiles and source images. +Importantly, the diffusion coefficient (equation 14 +in Kontar et al. 2019) is derived to describe the +anisotropic scattering effects, which is related to the +emitting frequency, the levels of the density fluctua- +tion, the scale height of the turbulence, and the density +anisotropy. The simulated properties of the radio waves +are mainly determined by the following: the frequency +ratio over the local plasma frequency, the spectrum- +weighted mean wavenumber of density fluctuations Cq +defined as 4πl−1/3 +i +l−2/3 +o +ϵ2 = Cqr−0.88 (where r is related +to the emitting frequency in units of solar radii R⊙, +ϵ2 = ⟨δn2⟩/n2 is the variance of density fluctuations, +and li and lo give the inner and outer scales of the den- +sity turbulence), the anisotropic parameter α, and the +heliocentric angle θ between the line of sight and source +position in the ecliptic plane. +5.2. Cases without scattering + +9 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Time of arrival - free propagation time [s] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Normalized Number of Rays +F 30.01× 1.10 MHz +H 30.01× 2.00 MHz +Delay (rays) 1.00 s +(a) +28 +30 +32 +34 +36 +38 +Frequency [MHz] +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Delay time (rays) [s] +Cq= 2300 +(b) +Figure 6. Ray tracing simulation results: (a) Time profiles of both the F and H emissions at a local plasma frequency of 30 +MHz, with a mean wavenumber of density fluctuations of Cq = 2300 R−1 +⊙ , anisotropy α=0.25, and heliocentric angle θ = 5◦. +(b) Simulated delay times taken from time intervals between the peak intensities of the F and H emission in a frequency range +from 30-36 MHz. The error bars represent the time bin width used for the histogram of the photon arrival times. +Firstly, we investigate the delay times between F and +H emissions without scattering effects (from small-scale +density fluctuations) by assuming the Cq number to +be 0. From the dispersion relation for electromagnetic +waves in an unmagnetized plasma, the group velocity +vgr = c2k/ω and ω2 = ω2 +pe + k2c2 will be different for +the F emission at ωpe and H emission at 2ωpe. In this +case, the time delay is caused by different group veloc- +ities of the F and H components and refraction effects +from the large scale density perturbation. +The F components emit at a frequency close to the lo- +cal plasma frequency, which will be more strongly scat- +tered than the H components that emit at a higher fre- +quency. Thus, the peak time of the F will arrive later +than that for the H. The simulated delay times are de- +fined from the time interval between the peaks of the +histograms showing the photon arrival times of the F +and H emissions. +In the case of no scattering on small-scale density fluc- +tuations, the time bins for the F and H are 0.002 s so an +error of +� +(δtF/2)2 + (δtH/2)2 ∼ 0.001 s is directly con- +sidered. The statistical errors are from the finite number +of photons in the simulations, such that a larger number +of photons would give a smaller error in the simulated +delay time. The delay times vary from 0.41 ± 0.001 s to +0.37 ± 0.001 s at frequencies from 30 to 36 MHz, shown +in Figure 5. It can be seen that the delay times caused +without scattering (on small-scale density fluctuations) +are apparently too small compared to the delay times +from type III and type J burst observations. In the fol- +lowing, the delay times are investigated using multiple +simulation parameters for anisotropic scattering which +are necessary in order to explain solar radio emissions +at meter to kilometer wavelengths, as shown by previ- +ous studies (Kontar et al. 2019; Kuznetsov et al. 2020; +Chen et al. 2020; Musset et al. 2021; Zhang et al. 2021; +Clarkson et al. 2021). +5.3. Delay times for multiple simulation parameters +We take the same simulation parameters from Chen +et al. (2020), in which they found that the observed +time profiles, source sizes, and motion of the type III- +IIIb burst at 32 MHz (the same type III burst in Fig- +ure 2) can be simultaneously explained with anisotropic +radio-wave scattering due to turbulence with parame- +ters Cq = 2300 R−1 +⊙ , α=0.25, at a heliocentric angle of +θ = 5◦. +The simulated time profiles of both F and H compo- +nents at a local plasma frequency of 30 MHz are pre- +sented by the histogram of the photon arrival times, +shown in Figure 6(a). +We consider the F frequency +1.1fpe and the H frequency 2fpe (giving a frequency ra- +tio of RH/F=1.82), the same as Chen et al. (2020). The +F components undergo a stronger scattering effect and +arrive later than the H components. As a result, the de- +lay time is 1.00 ± 0.04 s, measured as the time between +the peak times of the flux curves, which is close to the +averaged delay time from the observation of the type III +burst. +The simulated delay times in a frequency range from +30 to 36 MHz are presented in Figure 6(b). +The de- + +10 +28 +30 +32 +34 +36 +38 +Frequency [MHz] +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Delay time (rays) [s] +RH/F= 1.90 +RH/F= 1.82 +RH/F= 1.74 +RH/F= 1.67 +RH/F= 1.60 +(a) +28 +30 +32 +34 +36 +38 +Frequency [MHz] +0.6 +0.8 +1.0 +1.2 +1.4 +Delay time (rays) [s] +θ= 0° +θ= 10° +θ= 20° +θ= 30° +θ= 40° +θ= 50° +(b) +28 +30 +32 +34 +36 +38 +Frequency [MHz] +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Delay time (rays) [s] +Cq= 80 +Cq= 1200 +Cq= 2300 +Cq= 4300 +(c) +28 +30 +32 +34 +36 +38 +Frequency [MHz] +0 +1 +2 +3 +4 +5 +Delay time (rays) [s] +α= 0.10 +α= 0.25 +α= 0.40 +α= 0.55 +α= 0.70 +(d) +Figure 7. Delay times from ray tracing simulation results for (a) multiple H/F frequency ratios (RH/F=1.60-1.90), (b) he- +liocentric angles (θ = 0◦ − 50◦), (c) multiple mean wavenumbers of the density fluctuations (Cq=80, 1200, 2300 and 4300 +R−1 +⊙ ), and (d) anisotropy parameters (α=0.10, 0.25, 0.40, 0.55 and 0.70). For the additional parameters assumed, (a) θ = 5◦, +Cq = 2300 R−1 +⊙ , α=0.25, (b) RH/F=1.82, Cq = 2300 R−1 +⊙ , α=0.25, (c) RH/F=1.82, θ = 5◦, α=0.25, (d) RH/F=1.82, θ = 5◦, +Cq = 2300 R−1 +⊙ , respectively. The error bars represent the time bin width for the simulated time profiles. +lay times and frequencies follow a linear relation from +simulations, where the longer delay times correspond +to emissions at lower frequencies. The delay times are +roughly 1.00 ± 0.04 s at 30 MHz and 0.90 ± 0.03 s at +36 MHz. +The frequency ratios of the F component are simulated +as being 1.05, 1.10, 1.15, 1.20 and 1.25 times the local +plasma frequency fpe while the frequency ratio of the H +component is set to be 2fpe, which give H/F frequency +ratios of 1.90, 1.82, 1.74, 1.67 and 1.60 respectively. The +simulated delay times for multiple H/F frequency ratios +are shown in Figure 7(a). They have a linear relationship +with frequency for all assumed H/F frequency ratios. +The delay times are about 1.20 ± 0.05 s and 0.68 ± 0.04 +s respectively for the H/F frequency ratios of 1.90 and +1.60 at fpe = 30 MHz. The delay time is longer for a +higher H/F frequency ratio, which is reasonable because +radio emission emitting closer to the plasma frequency +undergoes stronger scattering and thus incurs a longer +delay. +The delay times for multiple heliocentric angles θ of +the source varying from 0 to 50 degrees are represented +by the different colors in Figure 7(b). The delay times +change from 1.01 ± 0.04 s (0◦) to 1.16 ± 0.04 s (50◦) + +11 +0.0 +0.4 +0.8 +1.2 +Delay time (rays) [s] +1.6 +1.7 +1.8 +1.9 +2.0 +RH/F +Cq=80 RΟ + • +-1 +α= 0.10 +α= 0.25 +α= 0.40 +α= 0.55 +0 +1 +2 +3 +Delay time (rays) [s] +1.6 +1.7 +1.8 +1.9 +2.0 + +Cq=1200 RΟ + • +-1 +0 +1 +2 +3 +4 +Delay time (rays) [s] +1.6 +1.7 +1.8 +1.9 +2.0 + +Cq=2300 RΟ + • +-1 +Figure 8. Ray tracing simulation results at 30 MHz: frequency ratio versus delay time. The different colors in three panels +represent a mixture of Cq=80, 1200, 2300 R−1 +⊙ +and α=0.10, 0.25, 0.40 and 0.55. +at fpe = 30 MHz for a H/F frequency ratio of 1.82. It +seems that the heliocentric angle has a weak influence +on the delay times which become only slightly extended +for larger heliocentric angles. +We also investigate the delay times for multiple lev- +els of density fluctuations by simulating for multiple +spectrum-weighted mean wavenumbers of density fluc- +tuations (the Cq parameter), which are combinations of +the density fluctuation level and inner and outer scales of +the density fluctuations. Delay times for Cq=80, 1200, +2300, 4300 R−1 +⊙ +with an anisotropy parameter of 0.25 +are shown in Figure 7(c). The delay times are 0.64 ± +0.01 s to 1.09 ± 0.05 s at fpe = 30 MHz, for Cq= 80 +and 4300 R−1 +⊙ , respectively. As expected, stronger den- +sity fluctuations for a larger Cq number will result in +stronger scattering and thus longer delay times. +Anisotropic density fluctuations that are predomi- +nantly in the perpendicular direction to the magnetic +field are required to describe the observed solar radio +bursts (Kontar et al. 2017, 2019; Kuznetsov et al. 2020; +Chen et al. 2020; Musset et al. 2021). When α < 1, +radio wave propagation aligns (to a larger degree) with +the radial direction, leading to a narrower time profile. +When α = 1, it represents the case of isotropic density +fluctuations. +Anisotropy parameters of α=0.10, 0.25, +0.40, 0.55, 0.70 are considered, as shown in Figure 7(d). +Anisotropic scattering has a significant effect on the time +profiles. Strong anisotropy can highly reduce the dura- +tion of the radio emissions and delay times between the +F and H. Weak anisotropic scattering with α = 0.70 +produces a delay of 4.19 ± 0.12 s, whereas the delay is +reduced to 0.87 ± 0.01 s for strong anisotropic scattering +with α = 0.10 at fpe = 30 MHz. +5.4. Frequency ratio and delay time +While inputting the density fluctuation parameters Cq +and their anisotropy α in our numerical models, we can +determine the frequency ratio vs. +delay time at each +frequency. Figure 8 shows our predictions of the rela- +tion between frequency ratios and delay times for Cq = +[80, 1200, 2300] and α=[0.10, 0.25, 0.40, 0.55] at 30 +MHz. +Consequently, we make it feasible to quantita- +tively predict the relation between the frequency ratio +and delay time between the F and H emissions by deduc- +ing the density fluctuation properties from the spectral +and imaging radio observations and then applying those +parameters in ray tracing simulations. +6. SUMMARY +We presented the H/F frequency ratios and delay +times of the F component from observations of type +III and type J bursts with striae and compared them +to those resulting from radio wave propagation simula- +tions. +The striae in F and H components are expected to +correlate with each other. Cross correlations between +the spectra of F and H with striae are carried out to +find their best matched counterparts and can give the +frequency ratio and delay times simultaneously. +The +statistical delay times averaged at 1.1 s and 2.0 s with +frequency ratios of 1.99 and 1.95 for the type III and +type J burst, respectively. Without correcting for the +delay times, the frequency ratios at the same time are +1.66 and 1.73—which are significantly different from the +theoretical prediction of 2. +For the plasma emission mechanism, both the F and +H emissions are generated at the same coronal location +but normally the H emissions arrive earlier than the F +emissions. The earlier arrival of the H emissions may +be caused by the combination of two effects: the faster +group velocity and weaker scattering effects than those +on the F emissions. We estimate the delay times caused +by the different group velocities and propagation effects +through ray-tracing simulations (Kontar et al. 2019). + +12 +From our simulations, we quantitatively show the de- +lay times between the F and H emissions for multiple +H/F frequency ratios, heliocentric angles, density fluc- +tuation levels, and anisotropy parameters. When there +is no scattering from small-scale density fluctuations, +the delay time is caused by the different group velocities +and the refraction effects from large scale density fluctu- +ations. Such delay time is estimated to be around 0.41 s +at a local plasma frequency of 30 MHz, which is not suf- +ficient to explain the observed delay time. If we adopt +the same characteristic parameters as in the ray tracing +simulations that successfully reproduced the observed +properties of the same type IIIb burst analysed in Chen +et al. (2020), the simulated delay time (∼ 1.00±0.10 s +at 30 MHz) between the F and H components is very +close to the observed delay times derived from the orig- +inal (∼1.05 s at 30 MHz), fitted (∼0.94 s at 30 MHz) +and cross correlated (∼0.94 s at 30 MHz) time profiles +of the type IIIb burst, implying that propagation effects +have a main contribution to the delay times between F +and H emissions. +It may be deduced from the simulations in Figure +7, that a stronger scattering environment and weaker +anisotropy give a longer delay time. From the context of +observations, stronger scattering and weaker anisotropy +would cause a longer burst duration, which itself may +then imply longer delay times, as seen in Table 1. The +delay times vary from one event to another, likely due +to radio-wave propagation effects which vary with the +coronal properties from event to event. The observed +F component is delayed with respect to the observed H +component at each time point, which may be one of the +reasons that the source positions of F and H components +do not coincide at the same time. The delay of the F +component with respect to the H component could con- +tribute to the fact that the F and H sources do not co- +incide when imaged, alongside the effects of radio-wave +propagation which cause a larger shift of the F sources +away from their true position compared to the H sources +(Chrysaphi et al. 2018; Kontar et al. 2019; Chen et al. +2020). +Radio-wave propagation effects lead to a delay of the +F with respect to their H counterparts, producing an ob- +served frequency ratio lower than the theoretical ratio of +∼ 2. Radio burst F-H pair observations with fine struc- +tures can be used to derive this delay time and retrieve +the theoretical frequency ratio between striae counter- +parts. The delay time is dependent on the anisotropic +turbulent conditions that vary between events. We show +that the same parameters that reproduce the decay +times and source sizes can also predict the delay times +in radio-wave scattering simulations, offering a quan- +titative solution to the long-standing question of why +different frequency ratios are often observed. +XC and EPK are supported by STFC consolidated grant +ST/T000422/1. +DLC and EPK are thankful to Dstl +for the funding through the UK-France PhD Scheme +(contract DSTLX-1000106007). NC thanks CNES for +its financial support. +We gratefully acknowledge the +UK-France collaboration grant provided by the British +Council Hubert Curien Alliance Programme that con- +tributed to the completion of this work. The authors ac- +knowledge the support by the international team grant +(http://www.issibern.ch/teams/lofar/) from ISSI Bern, +Switzerland. This paper is based (in part) on data ob- +tained from facilities of the International LOFAR Tele- +scope (ILT) under project code LC8 027. LOFAR (van +Haarlem et al. 2013) is the Low-Frequency Array de- +signed and constructed by ASTRON. It has observing, +data processing, and data storage facilities in several +countries, that are owned by various parties (each with +their own funding sources), and that are collectively op- +erated by the ILT Foundation under a joint scientific pol- +icy. The ILT resources have benefited from the following +recent major funding sources: CNRS-INSU, Observa- +toire de Paris and Universit´e d’Orl´eans, France; BMBF, +MIWF-NRW, MPG, Germany; Science Foundation Ire- +land (SFI), Department of Business, Enterprise and In- +novation (DBEI), Ireland; NWO, The Netherlands; The +Science and Technology Facilities Council, UK; Ministry +of Science and Higher Education, Poland. XC thanks +NSFC Grants 12003048. +REFERENCES +Aschwanden, M. J., Bastian, T. S., Benz, A. O., & Brosius, +J. W. 1992, ApJ, 391, 380, doi: 10.1086/171353 +Aurass, H., & Klein, K. L. 1997, A&AS, 123, 279, +doi: 10.1051/aas:1997161 +Chen, X., Kontar, E. P., Chrysaphi, N., et al. 2020, ApJ, +905, 43, doi: 10.3847/1538-4357/abc24e +Chen, X., Kontar, E. P., Yu, S., et al. 2018, ApJ, 856, 73, +doi: 10.3847/1538-4357/aaa9bf +Chrysaphi, N., Kontar, E. P., Holman, G. D., & Temmer, +M. 2018, ApJ, 868, 79, doi: 10.3847/1538-4357/aae9e5 + +13 +Clarkson, D. L., Kontar, E. 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V. 1996, Astrophysics and Space Science +Library, Vol. 204, Radiation in Astrophysical Plasmas +(Dordrecht, Netherlands: Springer), +doi: 10.1007/978-94-009-0201-5 + diff --git a/Y9FIT4oBgHgl3EQfkCuy/content/tmp_files/load_file.txt b/Y9FIT4oBgHgl3EQfkCuy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..93e7b1fa4d04eb8df5e47f7f8e3f93ac35903fcf --- /dev/null +++ b/Y9FIT4oBgHgl3EQfkCuy/content/tmp_files/load_file.txt @@ -0,0 +1,1101 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf,len=1100 +page_content='Draft version January 27, 2023 Typeset using LATEX twocolumn style in AASTeX631 The frequency ratio and time delay of solar radio emissions with fundamental and harmonic components Xingyao Chen (陈星瑶 ) ,1 Eduard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Kontar ,1 Daniel L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Clarkson ,1 and Nicolina Chrysaphi 2, 1 1School of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 2LESIA, Observatoire de Paris, Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, 5 place Jules Janssen, 92195 Meudon, France ABSTRACT Solar radio bursts generated through the plasma emission mechanism produce radiation near the local plasma frequency (fundamental emission) and double the plasma frequency (harmonic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' While the theoretical ratio of these two frequencies is close to 2, simultaneous observations give ratios ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 to 2, suggesting either a ratio different from 2, a delay of the fundamental emission, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' To address this long-standing question, we conducted high frequency, high time resolution imaging spectroscopy of type III and type J bursts with fine structures for both the fundamental and harmonic components with LOFAR between 30 and 80 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The short-lived and narrow frequency-band fine structures observed simultaneously at fundamental and harmonic frequencies give a frequency ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='66 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='73, similar to previous observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' However, frequency-time cross-correlations suggest a frequency ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='99 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95 with a time delay between the F and H emissions of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 s, respectively for each event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Hence, simultaneous frequency ratio measurements different from 2 are caused by the delay of the fundamental emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Among the processes causing fundamental emission delays, anisotropic radio-wave scattering is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Moreover, the levels of anisotropy and density fluctuations reproducing the delay of fundamental emissions are consistent with those required to simulate the source size and duration of fundamental emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Using these simulations we are able to, for the first time, provide quantitative estimates of the delay time of the fundamental emissions caused by radio-wave propagation effects at multiple frequencies, which can be used in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' INTRODUCTION Plasma emission is believed to be responsible for gen- erating solar radio bursts at decimeter and longer wave- lengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ginzburg & Zhelezniakov 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Dulk 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Melrose 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Within the plasma emission theory, the instability generates Langmuir waves at the local plasma frequency fpe, where fpe = ωpe/2π = � e2n(r)/πme is the electron plasma frequency, n(r) is the electron num- ber density, and e and me are the electron charge and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The coalescence of Langmuir waves and low fre- quency ion-sound waves may produce electromagnetic waves via nonlinear plasma processes, which is referred to as fundamental emission (hereafter, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The coales- cence of counter-propagating Langmuir waves may pro- duce the radio emission at 2fpe, which is referred to as the second harmonic emission (hereafter, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Corresponding author: Xingyao Chen Xingyao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Chen@glasgow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='uk Several longstanding issues exist concerning the F and H emissions—one problem in particular is how to dis- tinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' When a single burst is observed (and not an F-H pair), it is difficult to identify whether this burst is the result of F or H emissions, with the identification relying on the degree of their polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The polarisation degree of the F radio waves is predicted to be higher than the H (Dulk 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Suzuki & Dulk (1985) reported that the polarisation degree of the F components are all larger than the H components from 714 F-H pairs of type III bursts, that being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='35 (F) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12 (H) on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' However, the polarisation degree varies even within one burst (Dulk & Suzuki 1980) or even within one fundamental component (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' When there are two components observed, some additional arguments can suggest F-H pairs: the F typ- ically has higher intensities, shorter rise times, higher polarisations, and near half drift rates of the higher fre- quency component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' There are also multiple issues in debate—for exam- ple, F and H emission may be generated over different arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11299v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='SR] 26 Jan 2023 ID2 timescales because they are produced through related, but different, processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Moreover, it is unclear whether F and H emissions are produced at the same spatial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In this paper, we consider a remarkable prob- lem related to their frequency ratio RH/F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' the ratio of the H to F components observed at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' While the theory gives a harmonic frequency ratio very close to 2, the observations suggest a range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6-2, averaging at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 (Wild et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 1954;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Stewart 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Multiple studies have previously indicated time delays of the F with respect to the H, in a range from 1 s up to 7 s, and H/F frequency ratios in a range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='74-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 (Hughes & Harkness 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Stewart 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Robinson & Cairns 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Dorovskyy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Koval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Melnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2018), yet there is no agreement on the mechanisms involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For example, an early explana- tion by Wild et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (1954) for the observed difference from the theoretical prediction being that the observed spectrum consists of the H with only the higher frequen- cies of the F band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Stewart (1974) suggested that the F emission had reduced escape efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' the model by Robinson & Cairns (1998) stated that the F was re- absorbed above the local plasma frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Dorovskyy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2015) suggested lower group velocities of the F inside a modeled magnetic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' There are few studies aimed at the analysis of the H/F frequency ratios, because firstly, the F emissions are not always observed along with their corresponding H emis- sions and are difficult to be clearly defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' secondly, the H/F frequency ratios are usually used as evidence for the H components, while they are close to 2 then the two emission branches may be regarded as F-H pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' and thirdly, it is hard to quantitatively compare a F fre- quency with the corresponding H component in order to calculate the H/F frequency ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From previous studies, the cause of the time delay of the F components has been mostly explained by their different electromagnetic wave group velocities and radio wave propagation effects in an inhomogeneous corona (Itkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Melnik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' However, there was no quantitative estimation of the H/F frequency ratios and delay times from radio wave propagation ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' There seems to be no obvious dependencies of H/F frequency ratios on frequencies from previous observa- tions, which suggests that the propagation effects may be dominant due to various characteristic parameters of the background density fluctuations in a turbulent corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In this paper, we study the H/F frequency ratios and delay times between the F and H emissions from type III and type J bursts with striae observed by the LOw Frequency ARray (LOFAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2013) with high temporal, spectral and spatial resolutions in a frequency range of 30-80 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' J-type and/or U-type solar radio bursts is a variant of type III bursts and is believed to be generated from electron beams traveling along closed magnetic loops (Maxwell & Swarup 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Labrum & Stewart 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Hillaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Aschwan- den et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Aurass & Klein 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Dorovskyy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Reid & Kontar 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From a featureless radio burst with F and H components, the frequency ratio can be calculated from the frequencies at the maximum intensities of F and H components at each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The time delay can be derived from the times at the peak intensities at f and 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' However, it does not seem possible to determine the H/F frequency ratio and delay time simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We implement the correla- tion of type III and type J bursts that present striae in both F and H components, allowing clear determination of the frequency ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The fine structures are crucial to this method because striae in the F components have their counterparts in the H components and can be well correlated to determine the frequency ratio and delay time simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The variants of type III bursts in a form of ‘J’ or ‘U’ in the dynamic spectrum can also give the frequency ratio and delay time simultaneously from their similar shapes but it is worth noting that they are different if measured at the turning point and start- ing frequency of type U or J bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The isolines with different intensity levels of the U or J structure would also affect the estimations of frequency ratio and delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From the dynamic spectra of type III bursts with striae, the observed H/F frequency ratios are less than the theoretical ratio, suggesting that the observed F branch is delayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We suggest that the time delay be- tween F and H components can be explained by a combi- nation of different group velocities and scattering effects, with the latter forming a larger contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' As the F component is emitted at the local plasma frequency, it undergoes a stronger scattering effect than the H com- ponent, which lengthens the propagation path of the F emission and leads to a delayed F band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For the first time, we quantitatively estimate the delay times between the F and H components from ray-tracing sim- ulations of radio wave propagation (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Based on the diagnosis by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2020) in which they estimated the turbulent coronal parameters by combined analysis of scattering simulations and imaging observations from one type IIIb radio burst, our estima- tions of delay times from both different group velocities and scattering propagation effects can be well matched with that from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 3 Freq Time 𝑡delay 𝑓Ftrue 𝑓Fobs 𝑓H Freq Time harmonic Fobs Ftrue 𝑡delay Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' A physical scenario of two dynamic spectra showing the time delay between the observed and intrinsic F components of the normal type III burst (left panel) and the type III burst with striae fine structures (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In Section 2, we present the theoretical H/F frequency ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' A schematic illustration of the delayed F compo- nent in the type III burst with striae is presented in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Section 4 describes the characteristic parameters from type III and type J burst observations, including their observed H/F frequency ratios and delay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Section 5 shows the results from ray-tracing simulations of radio wave propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Section 6 is our summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' EMISSION NEAR FUNDAMENTAL AND HARMONIC FREQUENCIES The plasma emission mechanism is widely accepted for the generation of solar radio bursts at decimeter and longer wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ginzburg & Zhelezniakov 1958;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Dulk 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' McLean & Labrum 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Zheleznyakov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ratcliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The F emission can be generated through scattering of Langmuir waves by ions and/or their interaction with ion-sound waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The H emission is generated by the coalescence of the Langmuir waves with back-scattered Langmuir waves, so emits at the summed frequencies of two Langmuir waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Conservation of energy and momentum (frequency and wavenumber) leads to the fundamental emission be- ing close to the Langmuir wave frequency ωL ≃ ωpe � 1 + 3 2 v2 Te v2 � where vTe is the electron thermal speed, and v is the phase speed of the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For the typical parameters of type III solar radio bursts, v/vTe is about 10 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Rat- cliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Reid & Kontar 2021), so the deviation from plasma frequency ∆ω is as small as ∆ω/ωpe ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='015 Similarly, the harmonic frequency is close to the twice of the Langmuir wave frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Hence the ratio of har- monic to fundamental should be very close to 2, within 1 − 2% for the typical parameters in type III solar radio bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' SCHEMATIC ILLUSTRATION The schematic illustration outlined in Figure 1 clearly explains a delayed F component leads to the derived H/F frequency ratio to be less than 2 from observations of F-H pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The observed F component is delayed by a time tdelay compared to the intrinsic F component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency of the observed F branch is larger than the intrinsic F branch at each time, so the H/F frequency ratio normally calculated between the observed H and F components will be less than the theoretical frequency ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The intrinsic H/F frequency ratio should be be- tween the observed H component and the intrinsic F component instead of the observed F component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Fur- thermore, the radio source imaging of the H component should be coincident with an earlier F branch instead of the F emission at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We also show a scenario for a type III burst with striae in the right panel from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' With the striae, we can determine the frequency ratio and delay time si- multaneously, which is not feasible for normal type III bursts without fine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ideally, striae in the F components are expected to have counterparts in the H components, yet the striae in the H are normally not as apparent, which may be the result of the weak intensity of the H components and the limited dynamic range of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Nonetheless, the striae in our analysis are well observed and correlated with both F and H components from LOFAR observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' OBSERVATIONS 4 11:56:50 11:56:55 11:57:00 Start Time (16-Apr-15 11:56:48) 30 40 50 60 70 80 Frequency [MHz] 11:56:50 11:56:55 11:57:00 Start Time (16-Apr-15 11:56:48) 30 40 50 60 70 80 Frequency [MHz] 1 14 200 Flux [sfu] (a) Type III 12:19:55 12:20:00 12:20:05 Start Time (07-May-15 12:19:53) 30 40 50 60 70 80 Frequency [MHz] 12:19:55 12:20:00 12:20:05 Start Time (07-May-15 12:19:53) 30 40 50 60 70 80 Frequency [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 Flux [sfu] (b) Type J Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The dynamic spectra of type III (a) and J (b) bursts with both F and H components observed by LOFAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The same time range for both F and H components, but twice the frequency range for the H component, is selected to derive their drift rates, indicated by the solid rectangular boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The green lines are the best fit through all the positions of the fitted Gaussian peaks using a linear fitting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The solar type III radio burst (left panel in Figure 2) at around 11:57:00 UT on 16-April-2015 and the type J burst (right panel in Figure 2) at 12:20:00 UT on 07-May-2015 are observed by the LOw Frequency AR- ray (LOFAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' LOFAR is a large interferometric radio telescope with high spectro- scopic and imaging capabilities, located primarily in the Netherlands with a number of international stations in other European countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It was completed in 2012 by the Netherlands Institute for Radio Astronomy (AS- TRON) and can observe with the Low Band Antenna (LBA) and the High Band Antenna (HBA), optimized for 30 - 80 MHz and 120 - 240 MHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The type III and J bursts here are observed by the tied-array beam forming mode simultaneously with a maximum frequency resolution of ∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 kHz and time resolution of ∼ 10 ms (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We show our analysis of well observed type III and J bursts with both F and H components and striae, de- rive their frequency ratios and delay times, and compare with those estimated from radio wave propagation sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Overview of the type III burst The type III burst presented in Figure 2 is composed of two branches—the F branch between 30-65 MHz and H branch between 30-72 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The background is sub- tracted by using quiet periods prior to the bursts, and only the burst properties are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Their frequency and time resolutions are 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 kHz and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 ms, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The type III-IIIb burst with F-H pairs is well ana- lyzed in several papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2017) demon- strated that radio wave propagation effects dominated the observed spatial characterization of radio burst images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Using a model developed by Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2019) to quantitatively study radio-wave propagation in anisotropic density fluctuations, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2020) found that anisotropic scattering simulations can repro- duce the observed time profiles, centroid locations, and source sizes of the type IIIb radio burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From anal- ysis of the dynamic spectrum, Sharykin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2018) provided statistically significant properties of individual striae, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2018) explained that the striae fine structures were caused by the background density fluc- tuations, and Kolotkov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2018) demonstrated that the striae frequency drift can be modulated by a prop- agating fast wave train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In this study, we focus on the delay time and frequency ratio between the F and H components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The F emission started at around 11:56:54 UT and ended at around 11:56:59 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Each distinct stria be- tween 30-40 MHz contributes to a mean striae lifetime of about 1 s, with longer duration times at lower fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The same time range and twice the frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (16-Apr-15 11:56:53) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (16-Apr-15 11:56:53) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Flux [sfu] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='(a) Spectrum of type III burst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (16-Apr-15 11:56:53) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='(a) Spectrum of type III burst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11:56:54 11:56:55 11:56:56 11:56:57 11:56:58 11:56:59 11:57:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (16-Apr-15 11:56:53) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:19:56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:19:58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (07-May-15 12:19:55) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='37 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Flux [sfu] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='(c) Spectrum of type J burst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:19:56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:19:58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (07-May-15 12:19:55) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='(c) Spectrum of type J burst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:19:56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:19:58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12:20:04 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Start Time (07-May-15 12:19:55) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Frequency [MHz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Flux [sfu] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Type J (H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='Type J (F) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='RH/F: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95 td: -1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 RH/F (b) Correlation results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 Delay time [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 RH/F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='911 Type III (d) Correlation results 3 2 1 0 Delay time [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 RH/F (d) Correlation results 3 2 1 0 Delay time [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 RH/F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='77 Type J Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Correlation results for type III and type J bursts: (a), (c) Enlarged dynamic spectra of the F (below) and H (upper) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The solid rectangular box in the H spectrum will slip with given time and frequency lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The selected F spectrum in the solid rectangular box is correlated with each slipped H spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (b), (d) Two-dimensional cross correlation results between the selected F and each slipped H (spectral) boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The peak correlation coefficients at each frequency ratio are shown by green dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The maximal correlation coefficient is marked by the plus symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' If there is no time delay, the frequency ratio at the peak correlation coefficient is marked by the cross symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 6 range for the H branch (60-70 MHz) with respect to the F branch (30-35 MHz) are selected for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The H/F frequency ratios are measured in two ways in our study: from their drift rates and the cross correlations between F and H spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times are derived from the peak time intervals of the F and H flux profiles and cross correlations between F and H branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Frequency drift rates The frequency drift rate df dt for the F emission at the local plasma frequency is dfF dt = dfpe dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Since fpe = � e2n(r)/πme is a function of background electron den- sity, then it can be written as dfF dt = dfpe dn(r) dn(r) dr dr dt = fpe 2n dn(r) dr dr dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For the H emission at double the plasma fre- quency, the frequency drift rate is dfH dt = d2fpe dn(r) dn(r) dr dr dt = 2fpe 2n dn(r) dr dr dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Therefore, the theoretical frequency drift rate ratio between H and F emissions is DH/F = dfH dt / dfF dt = 2, which can be used for evidence of the har- monic branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From the dynamic spectrum, the time profiles at each frequency are fitted with a 1D Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The peak times at the maximum fitted flux at each frequency are marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' They follow a linear function over small frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' They are then fitted using a linear function of f = df dtt + C, seen from the two green lines in the dynamic spectra (Figure 2) for both the F and H components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The drift rates with 1-sigma errors are df 1 F dt = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='16 MHz s−1 and df 1 H dt = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12 MHz s−1 for the F and H components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The drift rate of the H component is around twice that of the F component, which can be further deduced as evidence of a F-H pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Defining the drift rates for the F and H components as DF = dfF dt and DH = dfH dt , then considering the drift rate functions fF = DFt + CF and fH = DHt + CH, one can derive a function of their H/F frequency ratio, Rdr H/F = fH fF = DH DF + (CH − DH DF CF)f −1 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Their H/F frequency ratios are then calculated between the two green lines from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency ratios range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='65 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 in the F frequency range of 30-35 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Cross correlations between the F and H branches The H component is distinguished from the F, seen from Figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We select part of the F component in a frequency range of 30-35 MHz by considering that the cut-off frequency of the H emission is around 70 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In order to derive a cross correlation map, the fre- quency range and time range need to be selected for the H component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The 2D cross correlation is then calcu- lated between the selected F component (the white box in the lower panel from Figure 3(a)) and the slipped H component (the white box, as an example at one in- stance, in the upper panel from Figure 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency ratio is set up to range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 with a ratio lag of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' A frequency ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 corresponds to a frequency range of 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0-52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 MHz for the H, and a frequency ratio of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 will determine a H frequency range of 60-70 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The time delay of the H is set up to range from -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='84 s to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='26 s, which is the time difference between the start time of the H and the start time of the F component at 11:56:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The time step is set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We keep the same time interval between the start and ending times for both F and H components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The time range for the F is fixed from 0 s (11:56:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 UT) to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 s (11:56:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 UT) and the time range for the H is slipped and changed with each delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For example, while the time delay for the H is -1 s, the H time range is from -1 s (11:56:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 UT) to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 s (11:56:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In order to search for any time-delays and frequency ratios between the F and H spectra, we create two- dimensional cross-correlation functions (CCFs) as fol- lows: CCF = � ij (Xij − X) × � ij (Yij − Y ) � (� ij Xij − X)2 × (� ij Yij − Y )2 Here Xij and Yij are two sets of spectra of the F and H components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We loop over all delay times and frequency ratios and compute the overlap and correlation for each shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The effective correlation coefficients range from 0 to 1, meaning no correlation and maximum correlation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The cross correlation map can be seen in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Uncertainties of the time and frequency ratio lags are determined using intensity randomization subset sam- pling by taking an observed dynamic spectrum and cre- ating 50 variations where the observed intensity is varied by I ±δI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The background flux level before the burst at each frequency is taken as the uncertainty on the flux, around 1 sfu, similar to Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' δI is ran- domly taken from a normal distribution with a mean of zero, and a standard deviation of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The average fre- quency ratio and delay time are obtained and the errors are taken from the sum of the standard deviations, delay time, frequency ratio resolutions and also the time and frequency resolutions of LOFAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From the cross correlation map, the peak correla- tion coefficients at each frequency ratio give the best- matched delay times (dot symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The plus symbol shows the maximal correlation coefficient at a delay time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s and a frequency ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='99± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01 (seen from Figure 3(b)), which means that the H emission in 7 56:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 56:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 56:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 56:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 56:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 56:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 Start Time (16-Apr-15 11:56:55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 Normalized Flux F 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 MHz H 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='06 MHz Delay 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s Delay(fit) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s Shifted F(cross) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s (a) 30 31 32 33 34 35 Frequency of F [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 Delay time [s] Delay Delay(fit) Delay(cross) (b) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Statistical delay times for the type III burst: (a) Normalized time profiles of the F and H emissions at 30 and 60 MHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Their Gaussian fit profiles are shown by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The peak times of observed and fitted time profiles are marked by the vertical solid and dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The time profile of the F component was shifted by a time lag of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s from the cross correlation between the F and H emissions (solid red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (b) Statistical delay times from observations (black), fits (blue) and cross correlations (red) between F and H emissions between 30 MHz to 35 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' the 60-70 MHz range is best matched with the F emis- sions in 30-35 MHz with a delay time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In other words, the F emission is delayed by 1 s with respect to the H emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' If there is no delay time between the F and H components, the frequency ratio is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='66± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01, as seen from the cross symbol in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Statistical delay times The normalized time profiles at frequencies of 30 MHz (F, black solid line) and 60 MHz (H, blue solid line) and their Gaussian fitted profiles (dashed lines) are shown as an example in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The vertical lines mark the peak times of the flux curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times derived from the intervals between the peak times of the original and fitted flux profiles between 30 MHz (F) and 60 MHz (H) are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The time profiles of the F and H components are also cross correlated, which gives a delay time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s for the maximum cross correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' After correcting for a time lag of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s, the shifted F component is shown by the red solid line in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' These three methods (estimations using the original, the fitted, and the cross correlation profiles) give similar delay times of around 1 second at 30 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We also statistically obtain the delay times from the flux profiles at each frequency between 30-35 MHz (F) and twice the frequency for the H components, shown in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It shows the averaged time delays for the original (black), fitted (blue), and the cross corre- lated (red) time profiles are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='19 s, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='14 s and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='08 s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Type III Burst Type J Burst t F Dur [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='19 t H Dur [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 fmid [MHz] 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 dfF dt [MHz/s] −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='16 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11 dfH dt [MHz/s] −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='14 Rdr H/F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='65-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='73 t pk d [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 t fit d [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='97 t corr d [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='91 Rcorr(td=0) H/F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='73 Rcorr(max) H/F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95 tcorr(max) d [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Characteristic parameters of the type III and type J bursts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' including t F Dur (the averaged duration given by the FWHM from Gaussian fits for the F),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' t H Dur (averaged FWHM duration for the H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' fmid (middle frequency of the F com- ponent),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' dfF dt (drift rate of the F),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' dfH dt (drift rate of the H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Rdr H/F (frequency ratio calculated between the two drifting rate lines of the F and H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' t pk d (averaged delay times calcu- lated from the peak intervals of the flux curves of the F and H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' t fit d (averaged delay times derived from the peak intervals of the fitted flux curves),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' t corr d (averaged delay times from the cross correlations between the F and H at each frequency),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Rcorr(td=0) H/F (frequency ratio at the peak correlation coefficient for the case of no time delay),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Rcorr(max) H/F (frequency ratio at the maximal correlation coefficient from the 2D cross corre- lation between the F and H spectra),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' tcorr(max) d (delay time at the maximal correlation coefficient from the 2D cross cor- relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Type J burst The F component of the type J burst was observed between 30-80 MHz, with the H component between 52- 80 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The F emission started at around 12:19:56 UT and ended with a long tail in the LOFAR observing frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The same method to derive the frequency ratios and delay times are implemented for type J as for type III burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' All characteristic parameters are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency drift rate of the F branch is about df 2 F dt = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='11 MHz s−1 while that for the H branch is roughly twice that of the F branch at df 2 H dt = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='14 MHzs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Considering the lower drift rates than the normal type III burst and near ”J” shape, the burst is identified as a variant of the type III burst called a type J burst (Reid & Kontar 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency ratios calculated from the drift rates range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='73 in the F frequency range of 33-39 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The results from cross correlations between the F and H components are shown in Figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It’s worth not- ing that the lower cut-off frequency for the H is around 55 MHz, so the frequency ratio is set up to be from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 (55 MHz/33 MHz) to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times are set up from -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='15 s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='75 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency ratio and delay time are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01 s and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='03 s at the maximal correlation coefficient from the two dimensional cross correlation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency ratio with no time de- lay is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='73± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01 at the peak correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times derived from peak time intervals of the original flux profiles, the fitted flux profiles, and the cross correlation between the F and H flux curves are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 s, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='97 s and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='91 s, respectively, averaged in a frequency range of 33-39 MHz (F) and twice those fre- quencies for the H components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' SIMULATIONS OF RADIO WAVE PROPAGATION The observed properties of radio waves, including time profiles, source positions, sizes, and emission directivity can be strongly affected by the inhomogeneous density fluctuations as they propagate through the turbulent corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' There are very few studies on the time pro- files that quantitatively investigate the delay times be- tween F and H emissions resulting from their propaga- tion through the turbulent coronal medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In order to quantitatively study the effects of radio wave propaga- tion on the H/F frequency ratio and delay times between the F and H components, we use ray tracing simulations of radio wave propagation developed by Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Brief introduction of the simulation set-up 28 30 32 34 36 38 Frequency [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='42 Delay time (rays) [s] Cq= 0 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Delay time as a function of frequency calculated using simulations without scattering: Cq = 0 R−1 ⊙ , frequency ratio RH/F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82, and heliocentric angle θ = 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In the simulations, the radio waves are treated as a number of rays (105 in our case) with positions r and wavenumbers k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Initially, they are seen as a point source located at a given position which is related to the emit- ting frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Then the radio waves propagate in the turbulent corona and undergo refraction effects mainly caused by the large-scale density gradient, as well as scattering effects caused by small-scale density pertur- bations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Their positions and wave vectors are calcu- lated from the numerical solutions of the Fokker-Planck equation and Hamilton’s equations in an unmagnetized plasma (seen in Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' All rays arrive at a sphere where the scattering is assumed to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Their arrival times, final positions and wave vectors are recorded to produce the time profiles and source images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Importantly, the diffusion coefficient (equation 14 in Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2019) is derived to describe the anisotropic scattering effects, which is related to the emitting frequency, the levels of the density fluctua- tion, the scale height of the turbulence, and the density anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The simulated properties of the radio waves are mainly determined by the following: the frequency ratio over the local plasma frequency, the spectrum- weighted mean wavenumber of density fluctuations Cq defined as 4πl−1/3 i l−2/3 o ϵ2 = Cqr−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='88 (where r is related to the emitting frequency in units of solar radii R⊙, ϵ2 = ⟨δn2⟩/n2 is the variance of density fluctuations, and li and lo give the inner and outer scales of the den- sity turbulence), the anisotropic parameter α, and the heliocentric angle θ between the line of sight and source position in the ecliptic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Cases without scattering 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 Time of arrival - free propagation time [s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 Normalized Number of Rays F 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01× 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 MHz H 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01× 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 MHz Delay (rays) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 s (a) 28 30 32 34 36 38 Frequency [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 Delay time (rays) [s] Cq= 2300 (b) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ray tracing simulation results: (a) Time profiles of both the F and H emissions at a local plasma frequency of 30 MHz, with a mean wavenumber of density fluctuations of Cq = 2300 R−1 ⊙ , anisotropy α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, and heliocentric angle θ = 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (b) Simulated delay times taken from time intervals between the peak intensities of the F and H emission in a frequency range from 30-36 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The error bars represent the time bin width used for the histogram of the photon arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Firstly, we investigate the delay times between F and H emissions without scattering effects (from small-scale density fluctuations) by assuming the Cq number to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From the dispersion relation for electromagnetic waves in an unmagnetized plasma, the group velocity vgr = c2k/ω and ω2 = ω2 pe + k2c2 will be different for the F emission at ωpe and H emission at 2ωpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In this case, the time delay is caused by different group veloc- ities of the F and H components and refraction effects from the large scale density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The F components emit at a frequency close to the lo- cal plasma frequency, which will be more strongly scat- tered than the H components that emit at a higher fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Thus, the peak time of the F will arrive later than that for the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The simulated delay times are de- fined from the time interval between the peaks of the histograms showing the photon arrival times of the F and H emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In the case of no scattering on small-scale density fluc- tuations, the time bins for the F and H are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='002 s so an error of � (δtF/2)2 + (δtH/2)2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='001 s is directly con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The statistical errors are from the finite number of photons in the simulations, such that a larger number of photons would give a smaller error in the simulated delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times vary from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='001 s to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='001 s at frequencies from 30 to 36 MHz, shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It can be seen that the delay times caused without scattering (on small-scale density fluctuations) are apparently too small compared to the delay times from type III and type J burst observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' In the fol- lowing, the delay times are investigated using multiple simulation parameters for anisotropic scattering which are necessary in order to explain solar radio emissions at meter to kilometer wavelengths, as shown by previ- ous studies (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Kuznetsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Musset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Clarkson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Delay times for multiple simulation parameters We take the same simulation parameters from Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2020), in which they found that the observed time profiles, source sizes, and motion of the type III- IIIb burst at 32 MHz (the same type III burst in Fig- ure 2) can be simultaneously explained with anisotropic radio-wave scattering due to turbulence with parame- ters Cq = 2300 R−1 ⊙ , α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, at a heliocentric angle of θ = 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The simulated time profiles of both F and H compo- nents at a local plasma frequency of 30 MHz are pre- sented by the histogram of the photon arrival times, shown in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We consider the F frequency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1fpe and the H frequency 2fpe (giving a frequency ra- tio of RH/F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82), the same as Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The F components undergo a stronger scattering effect and arrive later than the H components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' As a result, the de- lay time is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 s, measured as the time between the peak times of the flux curves, which is close to the averaged delay time from the observation of the type III burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The simulated delay times in a frequency range from 30 to 36 MHz are presented in Figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The de- 10 28 30 32 34 36 38 Frequency [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 Delay time (rays) [s] RH/F= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90 RH/F= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82 RH/F= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='74 RH/F= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 RH/F= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 (a) 28 30 32 34 36 38 Frequency [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 Delay time (rays) [s] θ= 0° θ= 10° θ= 20° θ= 30° θ= 40° θ= 50° (b) 28 30 32 34 36 38 Frequency [MHz] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 Delay time (rays) [s] Cq= 80 Cq= 1200 Cq= 2300 Cq= 4300 (c) 28 30 32 34 36 38 Frequency [MHz] 0 1 2 3 4 5 Delay time (rays) [s] α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 (d) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Delay times from ray tracing simulation results for (a) multiple H/F frequency ratios (RH/F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90), (b) he- liocentric angles (θ = 0◦ − 50◦), (c) multiple mean wavenumbers of the density fluctuations (Cq=80, 1200, 2300 and 4300 R−1 ⊙ ), and (d) anisotropy parameters (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For the additional parameters assumed, (a) θ = 5◦, Cq = 2300 R−1 ⊙ , α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, (b) RH/F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82, Cq = 2300 R−1 ⊙ , α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, (c) RH/F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82, θ = 5◦, α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, (d) RH/F=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82, θ = 5◦, Cq = 2300 R−1 ⊙ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The error bars represent the time bin width for the simulated time profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' lay times and frequencies follow a linear relation from simulations, where the longer delay times correspond to emissions at lower frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times are roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 s at 30 MHz and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='03 s at 36 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The frequency ratios of the F component are simulated as being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='15, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='20 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25 times the local plasma frequency fpe while the frequency ratio of the H component is set to be 2fpe, which give H/F frequency ratios of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='74, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='67 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The simulated delay times for multiple H/F frequency ratios are shown in Figure 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' They have a linear relationship with frequency for all assumed H/F frequency ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times are about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 s respectively for the H/F frequency ratios of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='90 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='60 at fpe = 30 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay time is longer for a higher H/F frequency ratio, which is reasonable because radio emission emitting closer to the plasma frequency undergoes stronger scattering and thus incurs a longer delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times for multiple heliocentric angles θ of the source varying from 0 to 50 degrees are represented by the different colors in Figure 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times change from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 s (0◦) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='04 s (50◦) 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='2 Delay time (rays) [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 RH/F Cq=80 RΟ 1 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40 α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55 0 1 2 3 Delay time (rays) [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 Cq=1200 RΟ 1 0 1 2 3 4 Delay time (rays) [s] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 Cq=2300 RΟ 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ray tracing simulation results at 30 MHz: frequency ratio versus delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The different colors in three panels represent a mixture of Cq=80, 1200, 2300 R−1 ⊙ and α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' at fpe = 30 MHz for a H/F frequency ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It seems that the heliocentric angle has a weak influence on the delay times which become only slightly extended for larger heliocentric angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We also investigate the delay times for multiple lev- els of density fluctuations by simulating for multiple spectrum-weighted mean wavenumbers of density fluc- tuations (the Cq parameter), which are combinations of the density fluctuation level and inner and outer scales of the density fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Delay times for Cq=80, 1200, 2300, 4300 R−1 ⊙ with an anisotropy parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25 are shown in Figure 7(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01 s to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s at fpe = 30 MHz, for Cq= 80 and 4300 R−1 ⊙ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' As expected, stronger den- sity fluctuations for a larger Cq number will result in stronger scattering and thus longer delay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Anisotropic density fluctuations that are predomi- nantly in the perpendicular direction to the magnetic field are required to describe the observed solar radio bursts (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2017, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Kuznetsov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Musset et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' When α < 1, radio wave propagation aligns (to a larger degree) with the radial direction, leading to a narrower time profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' When α = 1, it represents the case of isotropic density fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Anisotropy parameters of α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 are considered, as shown in Figure 7(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Anisotropic scattering has a significant effect on the time profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Strong anisotropy can highly reduce the dura- tion of the radio emissions and delay times between the F and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Weak anisotropic scattering with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='70 produces a delay of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='12 s, whereas the delay is reduced to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='01 s for strong anisotropic scattering with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 at fpe = 30 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Frequency ratio and delay time While inputting the density fluctuation parameters Cq and their anisotropy α in our numerical models, we can determine the frequency ratio vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' delay time at each frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Figure 8 shows our predictions of the rela- tion between frequency ratios and delay times for Cq = [80, 1200, 2300] and α=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='55] at 30 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Consequently, we make it feasible to quantita- tively predict the relation between the frequency ratio and delay time between the F and H emissions by deduc- ing the density fluctuation properties from the spectral and imaging radio observations and then applying those parameters in ray tracing simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' SUMMARY We presented the H/F frequency ratios and delay times of the F component from observations of type III and type J bursts with striae and compared them to those resulting from radio wave propagation simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The striae in F and H components are expected to correlate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Cross correlations between the spectra of F and H with striae are carried out to find their best matched counterparts and can give the frequency ratio and delay times simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The statistical delay times averaged at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='1 s and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='0 s with frequency ratios of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='99 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='95 for the type III and type J burst, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Without correcting for the delay times, the frequency ratios at the same time are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='66 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='73—which are significantly different from the theoretical prediction of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' For the plasma emission mechanism, both the F and H emissions are generated at the same coronal location but normally the H emissions arrive earlier than the F emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The earlier arrival of the H emissions may be caused by the combination of two effects: the faster group velocity and weaker scattering effects than those on the F emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We estimate the delay times caused by the different group velocities and propagation effects through ray-tracing simulations (Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 12 From our simulations, we quantitatively show the de- lay times between the F and H emissions for multiple H/F frequency ratios, heliocentric angles, density fluc- tuation levels, and anisotropy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' When there is no scattering from small-scale density fluctuations, the delay time is caused by the different group velocities and the refraction effects from large scale density fluctu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Such delay time is estimated to be around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='41 s at a local plasma frequency of 30 MHz, which is not suf- ficient to explain the observed delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' If we adopt the same characteristic parameters as in the ray tracing simulations that successfully reproduced the observed properties of the same type IIIb burst analysed in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' (2020), the simulated delay time (∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='00±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='10 s at 30 MHz) between the F and H components is very close to the observed delay times derived from the orig- inal (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='05 s at 30 MHz), fitted (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s at 30 MHz) and cross correlated (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='94 s at 30 MHz) time profiles of the type IIIb burst, implying that propagation effects have a main contribution to the delay times between F and H emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It may be deduced from the simulations in Figure 7, that a stronger scattering environment and weaker anisotropy give a longer delay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' From the context of observations, stronger scattering and weaker anisotropy would cause a longer burst duration, which itself may then imply longer delay times, as seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay times vary from one event to another, likely due to radio-wave propagation effects which vary with the coronal properties from event to event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The observed F component is delayed with respect to the observed H component at each time point, which may be one of the reasons that the source positions of F and H components do not coincide at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay of the F component with respect to the H component could con- tribute to the fact that the F and H sources do not co- incide when imaged, alongside the effects of radio-wave propagation which cause a larger shift of the F sources away from their true position compared to the H sources (Chrysaphi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Kontar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Radio-wave propagation effects lead to a delay of the F with respect to their H counterparts, producing an ob- served frequency ratio lower than the theoretical ratio of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Radio burst F-H pair observations with fine struc- tures can be used to derive this delay time and retrieve the theoretical frequency ratio between striae counter- parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The delay time is dependent on the anisotropic turbulent conditions that vary between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We show that the same parameters that reproduce the decay times and source sizes can also predict the delay times in radio-wave scattering simulations, offering a quan- titative solution to the long-standing question of why different frequency ratios are often observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' XC and EPK are supported by STFC consolidated grant ST/T000422/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' DLC and EPK are thankful to Dstl for the funding through the UK-France PhD Scheme (contract DSTLX-1000106007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' NC thanks CNES for its financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' We gratefully acknowledge the UK-France collaboration grant provided by the British Council Hubert Curien Alliance Programme that con- tributed to the completion of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The authors ac- knowledge the support by the international team grant (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='issibern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content='ch/teams/lofar/) from ISSI Bern, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' This paper is based (in part) on data ob- tained from facilities of the International LOFAR Tele- scope (ILT) under project code LC8 027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' LOFAR (van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' 2013) is the Low-Frequency Array de- signed and constructed by ASTRON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively op- erated by the ILT Foundation under a joint scientific pol- icy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The ILT resources have benefited from the following recent major funding sources: CNRS-INSU, Observa- toire de Paris and Universit´e d’Orl´eans, France;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' BMBF, MIWF-NRW, MPG, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Science Foundation Ire- land (SFI), Department of Business, Enterprise and In- novation (DBEI), Ireland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' NWO, The Netherlands;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' The Science and Technology Facilities Council, UK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' Ministry of Science and Higher Education, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9FIT4oBgHgl3EQfkCuy/content/2301.11299v1.pdf'} +page_content=' XC thanks NSFC Grants 12003048.' metadata={'source': 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Nijmegen, the Netherlands +{marck.vandervegt, nils.jansen, sebastian.junges}@ru.nl +Abstract. Multiple-environment MDPs (MEMDPs) capture finite sets +of MDPs that share the states but differ in the transition dynamics. +These models form a proper subclass of partially observable MDPs (POMDPs). +We consider the synthesis of policies that robustly satisfy an almost-sure +reachability property in MEMDPs, that is, one policy that satisfies a +property for all environments. For POMDPs, deciding the existence of +robust policies is an EXPTIME-complete problem. In this paper, we show +that this problem is PSPACE-complete for MEMDPs, while the policies +in general require exponential memory. We exploit the theoretical results +to develop and implement an algorithm that shows promising results in +synthesizing robust policies for various benchmarks. +1 +Introduction +Markov decision processes (MDPs) are the standard formalism to model sequen- +tial decision making under uncertainty. A typical goal is to find a policy that +satisfies a temporal logic specification [5]. Probabilistic model checkers such as +Storm [22] and Prism [30] efficiently compute such policies. A concern, how- +ever, is the robustness against potential perturbations in the environment. MDPs +cannot capture such uncertainty about the shape of the environment. +Multi-environment MDPs (MEMDPs) [36,14] contain a set of MDPs, called +environments, over the same state space. The goal in MEMDPs is to find a sin- +gle policy that satisfies a given specification in all environments. MEMDPs are, +for instance, a natural model for MDPs with unknown system dynamics, where +several domain experts provide their interpretation of the dynamics [11]. These +different MDPs together form a MEMDP. MEMDPs also arise in other domains: +In security applications, a natural example is a (static) password that must be +guessed. In robotics, a MEMDP captures the fact that we do not know the posi- +tion of some static obstacle. MEMDPs can be interpreted as a (disjoint) union +of MDPs in which an agent only has partial observation, i.e., every MEMDP can +be cast into a linearly larger partially observable MDP (POMDP) [27]. Indeed, +some famous examples for POMDPs are in fact MEMDPs, such as RockSam- +ple [39] and Hallway [31]. Solving POMDPs is notoriously hard [32], and thus, +it is worthwhile to investigate natural subclasses. +We consider almost-sure specifications where the probability needs to be one +to reach a set of target states. In MDPs, it suffices to consider memoryless + +2 +M. van der Vegt, N. Jansen, S. Junges +policies. Constructing such policies can be efficiently implemented by means of +a graph-search [5]. For MEMDPs, that consist of a finite set of environments, +each described by a MDP, we consider the following problem: +Compute one policy that almost-surely reaches the target in all environments. +Such a policy robustly satisfies an almost-sure specification for a set of MDPs. +Our approach. Inspired by work on POMDPs, we construct a belief-observation +MDP[16] that tracks the states of the MDPs and the (support of the) belief +over potential environments. We show that a policy satisfying the almost-sure +property in this MDP also robustly satisfies the property in the MEMDP. +Although the belief-observation MDP is exponentially larger than the MEMDP, +we exploit its particular structure to create a PSPACE algorithm to decide +whether such a robust policy exists. The essence of the algorithm is a recur- +sive construction of a fragment of the belief-observation MDP, restricted to a +setting in which the belief-support is fixed. Such an approach is possible, as +the belief in a MEMDP behaves monotonically: Once we know that we are not +in a particular environment, we never lose this knowledge. This behavior is in +contrast to POMDPs, where there is no monotonic behavior in belief-supports. +The difference is essential: Deciding almost-sure reachability in POMDPs is +EXPTIME-complete [37,19]. In contrast, the problem of deciding whether a pol- +icy for almost-sure reachability in a MEMDP exists is indeed PSPACE-complete. +We show the hardness using a reduction from the true quantified Boolean formula +problem. Finally, we cannot hope to extract a policy with such an algorithm, as +the smallest policy for MEMDPs may require exponential memory in the number +of environments. +The PSPACE algorithm itself recomputes many results. For practical pur- +poses, we create an algorithm that iteratively explores parts of the belief-observation +MDP. The algorithm additionally uses the MEMDP structure to generalize the +set of states from which a winning policy exists and deduce efficient heuristics +for guiding the exploration. The combination of these ingredients leads to an +efficient and competitive prototype on top of the model checker Storm. +Related work. We categorize related work in three areas. +MEMDPs. Almost-sure reachability for MEMDPs for exactly two environments +have been studied by [36]. We extend the results to arbitrarily many environ- +ments. This is nontrivial: For two environments, the decision problem has a poly- +nomial time routine [36], whereas we show that the problem is PSPACE-complete +for an arbitrary number of environments. MEMDPs and closely related models +such as hidden-model MDPs, hidden-parameter MDPs, multi-model MDPs, and +concurrent MDPs [11,2,40,10] have been considered for quantitative properties1. +The typical approach is to consider approximative algorithms for the undecid- +able problem in POMDPs [14] or adapt reinforcement learning algorithms [3,28]. +These approximations are not applicable to almost-sure properties. +1 Hidden-parameter MDPs are different than MEMDPs in that they assume a prior +over different MDPs. However, for almost-sure properties, this difference is irrelevant. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +3 +POMDPs. One can build an underlying potentially infinite belief-MDP [27] that +corresponds to the POMDP – using model checkers [35,7,8] to verify this MDP +can answer the question for MEMDPs. For POMDPs, almost-sure reachability +is decidable in exponential time [37,19] via a construction similar to ours. Most +qualitative properties beyond almost-sure reachability are undecidable [4,15]. +Two dedicated algorithms that limit the search to policies with small mem- +ory requirements and employ a SAT-based approach [12,26] to this NP-hard +problem [19] are implemented in Storm. We use them as baselines. +Robust models. The high-level representation of MEMDPs is structurally simi- +lar to featured MDPs [18,1] that represent sets of MDPs. The proposed tech- +niques are called family-based model checking and compute policies for every +MDP in the family, whereas we aim to find one policy for all MDPs. Interval +MDPs [25,43,23] and SGs [38] do not allow for dependencies between states +and thus cannot model features such as various obstacle positions. Parametric +MDPs [2,44,24] assume controllable uncertainty and do not consider robustness +of policies. +Contributions. We establish PSPACE-completeness for deciding almost-sure +reachability in MEMDPs and show that the corresponding policies may be expo- +nentially large. Our iterative algorithm builds fragments of the belief-observation +MDP. The prototype is the first algorithm specific to almost-sure reachability +in MEMDPs. A empirical evaluation shows that the iterative algorithm outper- +forms approaches dedicated to POMDPs. +2 +Problem Statement +In this section, we provide some background and formalize the problem state- +ment. +For a set X, Dist(X) denotes the set of probability distributions over X. For +a given distribution d ∈ Dist(X), we denote its support as Supp(d). For a finite +set X, let unif(X) denote the uniform distribution. For x ∈ X, we use dirac(x) +to denote the Dirac distribution on x. We use short-hand notation for functions +and distributions, f = [x �→ a, y �→ b] means that f(x) = a and f(y) = b. We +write P (X) for the powerset of X. For n ∈ N we write [n] = {i ∈ N | 1 ≤ i ≤ n}. +Definition 1 (MDP). A Markov Decision Process is a tuple M = ⟨S, A, ιinit, p⟩ +where S is the finite set of states, A is the finite set of actions, ιinit ∈ Dist(S) is +the initial state distribution, and p: S × A → Dist(S) is the transition function. +The transition function is total, that is, for notational convenience MDPs are +input-enabled. This requirement does not affect the generality of our results. A +path of an MDP is a sequence π = (s0, a0)(s1, a1) . . . sn such that ιinit(s0) > 0 +and p(si, ai, si+1) > 0 for all 0 ≤ i < n. The last state of π is last(π) = sn. We +denote the set of all finite paths by Path = (S ×A)∗ ×S, and the paths starting +in a state from S′ ⊆ S by Path(S′) = {(s0, a0) . . . sn ∈ Path | s0 ∈ S′}. The +set of reachable states from S′ is Reachable(S′) = {last(π) | π ∈ Path(S′)}. If + +4 +M. van der Vegt, N. Jansen, S. Junges +N1 : +s0 +s1 +q1, q2 +q1, q2 +a1 +a2, a3 +N2 : +s0 +s1 +q2 +q2 +q1 +q1 +a2 +a1, a3 +N3 : +s0 +q1, q2 +a3 +a1, a2 +Fig. 1: Example MEMDP +S′ = Supp(ιinit) we just call them the reachable states. The MDP restricted to a +set of reachable states from a distribution d ∈ Dist(S) is ReachFragment(M, d), +where d is the new initial distribution. A state s of the MDP is absorbing if +Reachable({s}) = {s}. An MDP is acyclic, if each state is either absorbing or +not reachable from its successor states. +Action choices are resolved by a policy σ: Path → Dist(A) that maps paths +to distributions over actions. A policy of the form σ: S → Dist(A) is called mem- +oryless, deterministic if we have σ: Path → A; and, memoryless deterministic +for σ: S → A. For an MDP M, we denote the probability of a policy σ reaching +some target set T ⊆ S starting in state s as PrM(s → T | σ). More precisely, +PrM(s → T | σ) denotes the probability of all paths from s reaching T under σ. +We simplify to PrM(T | σ) if s is distributed according to ιinit. +Definition 2 (MEMDP). A Multiple Environment MDP is a tuple N = +⟨S, A, ιinit, {pi}i∈I⟩ with S, A, ιinit as for MDPs, and {pi}i∈I is a set of tran- +sition functions, where I is a finite set of environment indices. +Intuitively, MEMDPs form sets of MDPs (environments) that share states and +actions, but differ in the transition probabilities. For MEMDP N with index set I +and a set I′ ⊆ I, we define the restriction of environments as the MEMDP N↓I′ = +⟨S, A, ιinit, {pi}i∈I′⟩. Given an environment i ∈ I, we denote its corresponding +MDP as Ni = ⟨S, A, ιinit, pi⟩. A MEMDP with only one environment is an MDP. +Paths and policies are defined on the states and actions of MEMDPs and do not +differ from MDP policies. A MEMDP is acyclic, if each MDP is acyclic. +Example 1. Figure 1 shows an example of a MEMDP with three environments. +An agent can perform two questions, q1 and q2. The response is either to ‘switch’ +the state (from s1 to s2 or vice versa), or to ‘stay’, i.e., take a self-loop). For +example, in environment N1, the response to both questions is to switch. In N2, +the response to q1 is to stay, while the response to q2 is to switch. The agent can +also guess the environment using a1, a2, a3. The idea is that guessing ai leads to +the target set { } only in environment i. An agent must therefore deduce the +environment via q1, q2 to be able to make a guaranteed correct guess. +■ +Definition 3 (Almost-Sure Reachability Property). An almost-sure reach- +ability property is defined by a set of target states T ⊆ S. A policy σ satisfies the +almost-sure reachability property T for MEMDP N = ⟨S, A, ιinit, {pi}i∈I⟩ iff: +∀i ∈ I : PrNi(T | σ) = 1 + +Robust Almost-Sure Reachability in Multi-Environment MDPs +5 +In other words, a policy σ satisfies an almost-sure reachability property T , called +winning, if and only if the probability of reaching T within each MDP is one. By +extension, a state s ∈ S is winning if there exists a winning policy when starting +in state s. Policies and states that are not winning are losing. +We will now define both the decision and policy problem: +Given a MEMDP N and an almost-sure reachability property T . +The Decision Problem asks to decide if a policy exists that satisfies T . +The Policy Problem asks to compute such a policy, if it exists. +In Section 4 we discuss the computational complexity of the decision problem. +Following up, in Section 5 we present our algorithm for solving the policy prob- +lem. Details on its implementation and evaluation will be presented in Section 6. +3 +A Reduction To Belief-Observation MDPs +In this section, we reduce the policy problem, and thus also the decision prob- +lem, to finding a policy in an exponentially larger belief-observation MDP. This +reduction is an elementary building block for the construction of our PSPACE +algorithm and the practical implementation, both presented in this paper. Ad- +ditional information such as proofs for statements in this and later sections are +given in the technical report [41]. +3.1 +Partially observable MDPs +As a first step, we formalise the interpretation of MEMDPs as a POMDP. +Definition 4 (POMDP). A partially observable MDP (POMDP) is a tuple +⟨M, Z, O⟩ where M = ⟨S, A, ιinit, p⟩ is an MDP, Z is a set of observations, and +O: S → Z the observation function. +A partially observable MDP is an MDP where states are labelled with obser- +vations. We lift O to paths and use O(π) = O(s1)a1O(s2) . . . O(sn). We are +interested in observation-based policies σ, i.e., policies s.t. for π, π′ ∈ Path, +O(π) = O(π′) implies σ(π) = σ(π′). +A MEMDP can be cast into a POMDP +that is made up as the disjoint union: +Definition 5 (Union-POMDP). Given an MEMDP N = ⟨S, A, ιinit, {pi}i∈I⟩ +we define its union-POMDP N⊔ = ⟨⟨S′, A, ι′ +init, p′⟩, Z, O⟩, with states S′ = S×I, +initial distribution ι′ +init(⟨s, i⟩) = ιinit(s) · |I|−1, transitions p′(⟨s, i⟩, a)(⟨s′, i⟩) = +pi(s, a)(s′), observations Z = S, and observation function O(⟨s, i⟩) = s. +A policy may observe the state s but not in which MDP we are. This forces any +observation-based policy to take the same choice in all MEMDPs. +Lemma 1. Given MEMDP N, there exists a winning policy iff there exists an +observation-based policy σ such that PrN⊔(T | σ) = 1. +The statement follows as, first, any observation-based policy of the POMDP can +be applied to the MEMDP, second, vice versa, any MEMDP policy is observation- +based, and third, the induced MCs under these policies are isomorphic. + +6 +M. van der Vegt, N. Jansen, S. Junges +3.2 +Belief-observation MDPs +For POMDPs, memoryless policies are not sufficient, which makes computing +policies intricate. We therefore add the information that the history — i.e., +the path until some point — contains. In MEMDPs, this information is the +(environment-)belief (support) J ⊆ I, as the set of environments that are consis- +tent with a path in the MEMDP. Given a belief J ⊆ I and a state-action-state +pair s +a−→ s′, then we define Up(J, s, a, s′) = {i ∈ J | pi(s, a, s′) > 0}, i.e., the +subset of environments in which the transition exists. For a path π ∈ Path, we +define its corresponding belief B(π) ⊆ I recursively as: +B(s0) = I +and +B(π · sas′)) = Up(B(π · s), s, a, s′) +The belief in a MEMDP monotonically decreases along a path, i.e., if we know +that we are not in a particular environment, this remains true indefinitely. +We aim to use a model where memoryless policies suffice. To that end, we +cast MEMDPs into the exponentially larger belief-observation MDPs [16]2. +Definition 6 (BOMDP). For a MEMDP N = ⟨S, A, ιinit, {pi}i∈I⟩, we define +its belief-observation MDP (BOMDP) as a POMDP GN = ⟨⟨S′, A, ι′ +init, p′⟩, Z, O⟩ +with states S′ = S × I × P (I), initial distribution ι′ +init(⟨s, j, I⟩) = ιinit(s) · |I|−1, +transition relation p′(⟨s, j, J⟩, a)(⟨s′, j, J′⟩) = pj(s, a, s′) with J′ = Up(J, s, a, s′), +observations Z = S × P (I), and observation function O(⟨s, j, J⟩) = ⟨s, J⟩. +Compared to the union-POMDP, BOMDPs also track the belief by updating it +accordingly. We clarify the correspondence between paths of the BOMDP and +the MEMDP. For a path π through the MEMDP, we can mimic this path exactly +in the MDPs Nj for j ∈ Bπ. As we track Bπ in the state, we can deduce from +the BOMDP state in which environments we can be. +Lemma 2. For MEMDP N and the path ⟨s1, j, J1⟩a1⟨s2, j, J2⟩ . . . ⟨sn, j, Jn⟩ of +the BOMDP GN , let j ∈ J1. Then: Jn ̸= ∅ and the path (s1, v1) . . . sn exists in +MDP Ni iff i ∈ J1 ∩ Jn. +Consequently, the belief of a path can be uniquely determined by the observation +of the last state reached, hence the name belief-observation MDPs. +Lemma 3. For every pair of paths π, π′ in a BOMDP, we have: +B(π) = B(π′) +implies +O(last(π)) = O(last(π′)). +For notation, we define SJ = {⟨s, j, J⟩ | j ∈ J, s ∈ S}, and analogously write +ZJ = {⟨s, J⟩ | s ∈ S}. We lift the target states T to states in the BOMDP: +TGN = {⟨s, j, J⟩ | s ∈ T, J ⊆ I, j ∈ J} and define target observations TZ = +O(TGN ). +Definition 7 (Winning in a BOMDP). Let GN be a BOMDP with target +observations TZ. An observation-based policy σ is winning from some observation +z ∈ Z, if for all s ∈ O−1(z) it holds that PrGN (s → O−1(TZ) | σ) = 1. +2 This translation is notationally simpler than going via the union-POMDP. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +7 +Furthermore, a policy σ is winning if it is winning for the initial distribution ιinit. +An observation z is winning if there exists a winning policy for z. The winning +region WinT +GN is the set of all winning observations. +Almost-sure winning in the BOMDP corresponds to winning in the MEMDP. +Theorem 1. There exists a winning policy for a MEMDP N with target states +T iff there exists a winning policy in the BOMDP GN with target states TGN . +Intuitively, the important aspect is that for almost-sure reachability, observation- +based memoryless policies are sufficient [13]. For any such policy, the induced +Markov chains on the union-POMDP and the BOMDP are bisimilar [16]. +BOMDPs make policy search conceptually easier. First, as memoryless poli- +cies suffice for almost-sure reachability, winning regions are independent of fixed +policies: For policies σ and σ′ that are winning in observation z and z′, re- +spectively, there must exist a policy ˆσ that is winning for both z and z′. Sec- +ond, winning regions can be determined in polynomial time in the size of the +BOMDP [16]. +3.3 +Fragments of BOMDPs +To avoid storing the exponentially sized BOMDP, we only build fragments: We +may select any set of observations as frontier observations and make the states +with those observations absorbing. We later discuss the selection of frontiers. +Definition 8 (Sliced BOMDP). For a BOMDP GN = ⟨⟨S, A, ιinit, p⟩, Z, O⟩ +and a set of frontier observations F ⊆ Z, we define a BOMDP GN |F = +⟨⟨S, A, ιinit, p′⟩, Z, O⟩ with: +∀s ∈ S, a ∈ A: p′(s, a) = +� +dirac(s) +if O(s) ∈ F, +p(s, a) +otherwise. +We exploit this sliced BOMDP to derive constraints on the set of winning states. +Lemma 4. For every BOMDP GN with states S and targets T and for all fron- +tier observations F ⊆ Z it holds that: WinT +GN |F ⊆ WinT +GN ⊆ WinT ∪F +GN |F . +Making (non-target) observations absorbing extends the set of losing observa- +tions, while adding target states extends the set of winning observations. +4 +Computational Complexity +The BOMDP above yields an exponential time and space algorithm via Theo- +rem 1. We can avoid the exponential memory requirement. This section shows +the PSPACE-completeness of deciding whether a winning policy exists. +Theorem 2. The almost-sure reachability decision problem is PSPACE-complete. +The result follows from Lemmas 11 and 10 below. In Section 4.3, we show that +representing the winning policy itself may however require exponential space. + +8 +M. van der Vegt, N. Jansen, S. Junges +{1, 2, 3} +{2, 3} +{1, 2} +{1} +{2} +{3} +{1, 3} +Fig. 2: The environment graph for our running example. +4.1 +Deciding Almost-Sure Winning for MEMDPs in PSPACE +We develop an algorithm with a polynomial memory footprint. The algorithm +exploits locality of cyclic behavior in the BOMDP, as formalized by an acyclic +environment graph and local BOMDPs that match the nodes in the environment +graph. The algorithm recurses on the environment graph while memorizing re- +sults from polynomially many local BOMDPs. +The graph-structure of BOMDPs. First, along a path of the MEMDP, we +will only gain information and are thus able to rule out certain environments [14]. +Due to the monotonicity of the update operator, we have for any BOMDP that +⟨s, j, J⟩ ∈ Reachable(⟨s′, j, J′⟩) implies J ⊆ J′. We define a graph over environ- +ment sets that describes how the belief-support can update over a run. +Definition 9 (Environment graph). +Let N be a MEMDP and p the tran- +sition function of GN . The environment graph GE N = (VN , EN ) for N is a +directed graph with vertices VN = P (I) and edges +EN = {⟨J, J′⟩ | ∃s, s′ ∈ S, a ∈ A, j ∈ I.p(⟨s, j, J⟩, a, ⟨s′, j, J′⟩) > 0 and J ̸= J′}. +Example 2. Figure 2 shows the environment graph for the MEMDP in Ex. 1. It +consists of the different belief-supports. For example, the transition from {1, 2, 3} +to {2, 3} and to {1} is due to the action q1 in state s0, as shown in Fig. 1. +■ +Paths in the environment graph abstract paths in the BOMDP. Path fragments +where the belief-support remains unchanged are summarized into one step, as +we do not create edges of the form ⟨J, J⟩. We formalize this idea: Let π = +⟨s1, j, J1⟩a1⟨s2, j, J2⟩ . . . ⟨sn, j, Jn⟩ be a path in the BOMDP. For any J ⊆ I, we +call π a J-local path, if Ji = J for all i ∈ [n]. +Lemma 5. For a MEMDP N with environment graph GE N , there is a path +J1 . . . Jn iff there is a path π = π1 . . . πn in GN s.t. every πi is Ji-local. +The shape of the environment graph is crucial for the algorithm we develop. +Lemma 6. Let GE N = (VN , EN ) be an environment graph for MEMDP N. +First, EN (J, J′) implies J′ ⊊ J. Thus, G is acyclic and has maximal path length +|I|. The maximal outdegree of the graph is |S|2|A|. +The monotonicity regarding J, J′ follows from definition of the belief update. +The bound on the outdegreee is a consequence from Lemma 9 below. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +9 +Local belief-support BOMDPs. Before we continue, we remark that the +(future) dynamics in a BOMDP only depend on the current state and set of +environments. More formally, we capture this intuition as follows. +Lemma 7. Let GN be a BOMDP with states S′. For any state ⟨s, j, J⟩ ∈ S′, let +N ′ = ReachFragment(N↓J, dirac(s)) and Y = � +i∈J⟨s, i, J⟩. Then: +ReachFragment(GN , unif(Y )) = GN ′. +The key insight is that restricting does not change the transition functions for +the environments j ∈ J. Furthermore, using monotonicity of the update, we only +reach BOMDP-states whose behavior is determined by the environments in J. +This intuition allows us to analyze the BOMDP locally and lift the results +to the complete BOMDP. We define a local BOMDP as the part of a BOMDP +starting in any state in SJ. All states not in SJ are made absorbing. +Definition 10 (Local BOMDP). Given a MEMDP N with BOMDP GN and +a set of environments J. The local BOMDP for environments J is the fragment +LocG(J) = ReachFragment(GN↓J |F, unif(SJ)) +where +F = Z \ ZJ . +This definition of a local BOMDP coincides with a fragment of the complete +BOMDP. We then mark exactly the winning observations restricted to the envi- +ronment sets J′ ⊊ J as winning in the local BOMDP and compute all winning +observations in the local BOMDP. These observations are winning in the com- +plete BOMDP. The following concretization of Lemma 4 formalizes this. +Lemma 8. Consider a MEMDP N and a subset of environments J. +Win +T ′ +GN +LocG(J) ∩ ZJ = Win +TGN +GN +∩ ZJ +with +T ′ +GN = TGN ∪ (Win +TGN +GN +\ ZJ). +Furthermore, local BOMDPs are polynomially bounded in the size of the MEMDP. +Lemma 9. Let N be a MEMDP with states S and actions A. LocG(J) has at +most O(|S|2 · |A| · |J|) states and O(|S|2 · |A| · |J|2) transitions3. +A PSPACE algorithm. We present Algorithm 1 for the MEMDP decision +problem, which recurses depth-first over the paths in the environment graph4. +We first state the correctness and the space complexity of this algorithm. +Lemma 10. ASWinning in Alg. 1 solves the decision problem in PSPACE. +To prove correctness, we first note that Search(N, J, T ) computes Win +TGN +GN ∩ZJ. +We show this by induction over the structure of the environment graph. For all +J without outgoing edges, the local BOMDP coincides with a BOMDP just for +3 The number of transitions is the number of nonzero entries in p +4 In contrast to depth-first-search, we do not memorize nodes we visited earlier. + +10 +M. van der Vegt, N. Jansen, S. Junges +Algorithm 1 Search algorithm +1: function Search(MEMDP N = ⟨S, A, {pi}i∈I, ιinit⟩, J ⊆ I, T ⊆ S) +2: +T ′ ← {⟨s, j, J⟩ | j ∈ J, s ∈ T } +3: +for J′ s.t. EN (J, J′) do +⊲ Consider the edges in the env. graph (Def. 9) +4: +WJ′ ← Search(N, J′, T ) +⊲ Recursion! +5: +T ′ ← T ′ ∪ {⟨s, j, J′⟩ | j ∈ J, ⟨s, J′⟩ ∈ WJ′} +6: +return WinT ′ +LocG(J) ∩ ZJ ⊲ Construct BOMDP as in Def. 10, then model check +7: +8: function ASWinning(MEMDP N = ⟨S, A, {pi}i∈I, ιinit⟩, T ⊆ S) +9: +return O(Supp(ιinit)) ⊆ Search(N, I, T ) +x +M1 : +x⊥ +x⊤ +α⊗ +1 +2 +1 +2 +y +W +F +α⊗ +α⊗ +⊤ +⊥ +x +M2 : +x⊤ +x⊥ +α⊗ +1 +2 +1 +2 +y +W +F +α⊗ +α⊗ +⊤ +⊥ +Fig. 3: Constructed MEMDP for the QBF formula ∀x∃y +� +(x ∨ y) ∧ (¬x ∨ ¬y) +� +. +environments J (Lemma 7). Otherwise, observe that T ′ in l. 5 coincides with its +definition in Lemma 8 and thus, by the same lemma, we return Win +TGN +GN ∩ZJ. To +finalize the proof, a winning policy exists in the MEMDP if the observation of +the initial states of the BOMDP are winning (Theorem 1). The algorithm must +terminate as it recurses over all paths of a finite acyclic graph, see Lemma 6. +Following Lemma 9, the number of frontier states is then bounded by |S|2 · |A|. +The main body of the algorithm therefore requires polynomial space, and the +maximal recursion depth (stack height) is |I| (Lemma 6). Together, this yields +a space complexity in O(|S|2 · |A| · |I|2). +4.2 +Deciding Almost-Sure Winning for MEMDPs Is PSPACE-hard +It is not possible to improve the algorithm beyond PSPACE. +Lemma 11. The MEMDP decision problem is PSPACE-hard. +Hardness holds even for acyclic MEMDPs and uses the following fact. +Lemma 12. On winning acyclic MEMDPs deterministic winning policies exist. +In particular, almost-sure reachability coincides with avoiding the sink states. +This is a safety property. For safety, deterministic policies are sufficient, as ran- +domization visits only additional states, which is not beneficial for safety. +Regarding Lemma 11, we sketch a polynomial-time reduction from the PSPACE- +complete TQBF problem [20] problem to the MEMDP decision problem. Let Ψ +be a QBF formula, Ψ = ∃x1∀y1∃x2∀y2 . . . ∃xn∀yn +� +Φ +� +with Φ a Boolean formula +in conjunctive normal form. The problem is to decide whether Ψ is true. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +11 +Example 3. Consider the QBF formula Ψ = ∀x∃y +� +(x ∨ y) ∧ (¬x ∨ ¬y) +� +. We +construct a MEMDP with an environment for every clause, see Figure 35. The +state space consists of three states for each variable v ∈ V : the state v and +the states v⊤ and v⊥ that encode their assignment. Additionallly, we have a +dedicated target W and sink state F. We consider three actions: The actions true +(⊤) and false (⊥) semantically describe the assignment to existentially quantified +variables. The action any α⊗ is used for all other states. Every environment +reaches the target state iff one literal in the clause is assigned true. +In the example, intuitively, a policy should assign the negation of x to y. +Formally, the policy σ, characterized by σ(π · y) = ⊤ iff x⊥ ∈ π, is winning. +■ +As a consequence of this construction, we may also deduce the following theorem. +Theorem 3. Deciding whether a memoryless winning policy exists is NP-complete. +The proof of NP hardness uses a similar construction for the propositional SAT +fragment of QBF, without universal quantifiers. Additionally, the problem for +memoryless policies is in NP, because one can nondeterministically guess a (poly- +nomially sized) memoryless policy and verify in each environment independently. +4.3 +Policy Problem +Policies, mapping histories to actions, are generally infinite objects. However, we +may extract winning policies from the BOMDP, which is (only) exponential in +the MEMDP. Finite state controllers [34] are a suitable and widespread represen- +tation of policies that require only a finite amount of memory, formally defined in +??. Intuitively, the number of memory states reflects the number of equivalence +classes of histories that a policy can distinguish. In general, we cannot hope to +find smaller policies than those obtained via a BOMDP. +Theorem 4. There is a family of MEMDPs {N n}n≥1 where for each n, N n +has 2n environments and O(n) states and where every winning policy for N n +requires at least 2n memory states. +We illustrate the witness. Consider a family of MEMDPs {N n}n, where N n has +2n MDPs, 4n states partitioned into two parts, and at most 2n outgoing actions +per state. We outline the MEMDP N in Figure 4. In the first part, there is only +one action per state. The notation is as follows: in state s0 and MDP N n +1 , we +transition with probability one to state a0, whereas in N n +2 we transition with +probability one to state b0. In every other MDP, we transition with probability +one half to either state. In state s1, we do the analogous construction for envi- +ronments 3, 4, and all others. A path s0b0 . . . is thus consistent with every MDP +except N n +1 . The first part ends in state sn. By construction, there are 2n paths +ending in sn. Each of them is (in)consistent with a unique set of n environments. +In the second part, a policy may guess n times an environment by selecting an +action αi for every i ∈ [2n]. Only in MDP N n +i , action αi leads to a target state. +5 We depict a slightly simplified MEMDP for conciseness. + +12 +M. van der Vegt, N. Jansen, S. Junges +s0 +a1 +b1 +i = 1 +i = 2 +i ̸= 1 +i ̸= 2 +1 +2 +1 +2 +s1 +a2 +b2 +i = 3 +i = 4 +i ̸= 3 +i ̸= 4 +1 +2 +1 +2 +. . . +sn +an +bn +i = n − 1 +i = n +i ̸= n − 1 +i ̸= n +1 +2 +1 +2 +g1 +W +g2 +gn +αi +αi +αi +gn+1 +αj +αj +αj +Mi : i ∈ [2n] +j ̸= i +1 +2 +Fig. 4: Witness for exponential memory requirement for winning policies. +In all other MDPs, the transition leads from state gj to gj+1. The state gn+1 is +absorbing in all MDPs. Importantly, after taking an action αi and arriving in +gj+1, there is (at most) one more MDP inconsistent with the path. +Every MEMDP N n in this family has a winning policy which takes σ(π·gi) = +α2i−1 if ai−1 ∈ π and σ(π · gi) = α2i otherwise. Furthermore, when arriving in +state sn, the state of a finite memory controller must reflect the precise set of +environments consistent with the history. These are 2n many. The proof shows +that if we store less information, two paths will lead to the same memory state, +but with different sets of environments being consistent with these paths. As we +can rule out only n environments using the n actions in the second part of the +MEMDP, we cannot ensure winning in every environment. +5 +A Partial Game Exploration Algorithm +In this section, we present an algorithm for the policy problem. We tune the +algorithm towards runtime instead of memory complexity, but aim to avoid +running out of memory. We use several key ingredients to create a pragmatic +variation of Alg. 1, with support for extracting the winning policy. +First, we use an abstraction from BOMDPs to a belief stochastic game (BSG) +similar to [45] that reduces the number of states and simplifies the iterative +construction6. Second, we tailor and generalize ideas from bounded model check- +ing [6] to build and model check only a fragment of the BSG, using explicit +partial exploration approaches as in, e.g., [33,9,42,29]. Third, our exploration +does not continuously extend the fragment, but can also prune this fragment by +using the model checking results obtained so far. The structure of the BSG as +captured by the environment graph makes the approach promising and yields +some natural heuristics. Fourth, the structure of the winning region allows to +generalize results to unseen states. We thereby operationalize an idea from [26] in +a partial exploration context. Finally, we analyze individual MDPs as an efficient +and significant preprocessing step. In the following we discuss these ingredients. +6 At the time of writing, we were unaware of a polytime algorithm for BOMDPs. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +13 +Algorithm 2 Policy finding algorithm +1: function FindPolicy(MEMDP N = ⟨S, A, {pi}i∈I, ιinit⟩, targets T ⊆ S) +2: +W ← {⟨s, J⟩ | s ∈ T, J ⊆ I}; L ← ∅; i ← 1; Sinit ← Supp(ιinit) × {I} +3: +while Sinit ̸⊆ (W ∪ L) do +4: +⟨B, F⟩ ← GenerateGameSlice(N, W, L, i) +5: +W ← W ∪ WinW +B +6: +L ← L ∪ S \ WinW ∪F +B +7: +i ← i + 1 +8: +if Sinit ⊆ W then return ExtractPolicy(W ) else return ⊥ +Abstraction to Belief Support Games. We briefly recap stochastic games +(SG). See [38,17] for more details. +Definition 11 (SG). A stochastic game is a tuple B = ⟨M, S1, S2⟩, where +M = ⟨S, A, ιinit, p⟩ is an MDP and (S1, S2) is a partition of S. +S1 are Player 1 states, and S2 are Player 2 states. As common, we also ‘partition’ +(memoryless deterministic) policies into two functions σ1 : S1 → A and σ1 : S2 → +A. A Player 1 policy σ1 is winning for state s if Pr(T | σ1, σ2) for all σ2. We +(re)use WinT +BN to denote the set of states with a winning policy. +We apply a game-based abstraction to group states that have the same ob- +servation. Player 1 states capture the observation in the BOMDP, i.e., tuples +⟨s, J⟩ of MEMDP states s and subsets J of the environments. Player 1 selects +the action a, the result is Player 2 state ⟨⟨s, J⟩, a⟩. Then Player 2 chooses an +environment j ∈ J, and the game mimicks the outgoing transition from ⟨s, j, J⟩, +i.e., it mimicks the transition from s in Nj. Formally: +Definition 12 (BSG). Let GN be a BOMDP with GN = ⟨⟨S, A, ιinit, p⟩, Z, O⟩. +A belief support game BN for GN is an SG BN = ⟨⟨S′, A′, ι′ +init, p⟩, S1, S2⟩ with +S′ = S1 ∪ S2 as usual, Player 1 states S1 = Z, Player 2 states S2 = Z × A, +actions A′ = A ∪ I, initial distribution ι′ +init(⟨s, I⟩) = � +i∈I ιinit(⟨s, i, I⟩), and the +(partial) transition function p defined separately for Player 1 and 2: +p′(z, a) = dirac(⟨z, a⟩) +(Player 1) +p′(⟨z, a⟩, j, z′) = p(⟨s, j, J⟩, a)(⟨s′, j, J′⟩) with z = ⟨s, J⟩, z′ = ⟨s′, J′⟩ +(Player 2) +Lemma 13. An (acylic) MEMDP N with target states T is winning if(f) there +exists a winning policy in the BSG BN with target states TZ. +Thus, on acyclic MEMDPs, a BSG-based algorithm is sound and complete, how- +ever, on cyclic MDPs, it may not find the winning policy. The remainder of the +algorithm is formulated on the BSG, we use sliced BSGs as the BSG of a sliced +BOMDP, or equivalently, as a BSG with some states made absorbing. +Main algorithm. +We outline Algorithm 2 for the policy problem. We track +the sets of almost-sure observations and losing observations (states in the BSG). + +14 +M. van der Vegt, N. Jansen, S. Junges +Algorithm 3 Game generation algorithm +1: function GenerateGameSlice(MEMDP N, W , L, i) +2: +Q ← {sι}; E = {sι} +3: +while s ∈ Q and |E| ≤ Bound[i] exists do +4: +E ← E ∪ {s} +⊲ Mark s as explored +5: +B ← BN |(S \ E) +⊲ Extend game, cut-off everything not explored +6: +Q ← Reachable(B) \ (E ∪ W ∪ L) +⊲ Add newly reached states +7: +return B, Q +Initially, target states are winning. Furthermore, via a simple preprocessing, we +determine some winning and losing states on the individual MDPs. +We iterate until the initial state is winning or losing. Our algorithm constructs +a sliced BSG and decides on-the-fly whether a state should be a frontier state, +returning the sliced BSG and the used frontier states. We discuss the implemen- +tation below. For the sliced BSG, we compute the winning region twice: Once +assuming that the frontier states are winning, once assuming they are loosing. +This yields an approximation of the winning and losing states, see Lemma 4. +Soundness. The algorithm is sound, assuming that the BN is indeed a sliced +BSG with frontier F. In particular, then the following invariant holds: +W ⊆ WinT +BN +and +L ∩ WinT +BN = ∅. +This invariant exploits that from a sliced BSG we can (implicitly) slice the +complete BSG while preserving the winning status of every state, formalized +below. In future iterations we only explore the implicitly sliced BSG. +Lemma 14. Given W ⊆ Win +TBN +BN +and L ⊆ S \ Win +TBN +BN : Win +TBN +BN += Win +TBN ∪W +BN |W∪L +Termination depends on the sliced game generation. It suffices to ensure that in +the long run, either W or L grow as there are only finitely many states. If W +and L remain the same longer than some number of iterations,W ∪ L will be +used as frontier. Then, the new game will suffice to determine if s ∈ W in one +shot. +Generating the sliced BSG. Algorithm 3 outlines the generation of the sliced +BSG. In particular, we explore the implicit BSG from the initial state but make +every state that we do not explicitly explore absorbing. In every iteration, we first +check if there are states in Q left to explore and if the number of explored states +in E is below a threshold Bound[i]. Then, we take a state from the priority queue +and add it to E. We find new reachable states7 and add them to the queue Q. +Generalizing the winning and losing states. +We can determine that a +state in the game BN is winning without ever exploring it. Therefore, observe +the following natural property of MEMDPs. +7 In l. 5 we do not rebuild the game B from scratch but incrementally construct the +data structures. Likewise, reachable states are a direct byproduct of this construc- +tion. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +15 +Lemma 15. A winning policy in MEMDP N is winning in N↓J for any J. +A direct consequence is the following statement for two environments J1 ⊆ J2: +⟨s, J2⟩ ∈ WinT +BN +implies +⟨s, J1⟩ ∈ WinT +BN . +Consequently, we can store W (and symmetrically, L) as follows. For every +MEMDP state s ∈ S, Ws = {J | ⟨s, J⟩ ∈ W} is downward closed on the +partial order P = (I, ⊂). This allows for efficient storage: We only have to store +the set of pairwise maximal elements, i.e., the antichain, +W max +s += {J ∈ Ws | ∀J′ ∈ Ws with J ̸⊆ J′}. +To determine whether ⟨s, J⟩ is winning, we check whether J ⊆ J′ for some +J′ ∈ W max +s +. Adding J to W max +s +requires removing all J′ ⊆ J and then adding J. +Note, however, that |W max +s +| is still exponential in |I| in the worst case. +Selection of heuristics. +The algorithm allows some degrees of freedom. We +evaluate the following aspects empirically. (1) The maximal size bound[i] of a +sliced BSG at iteration i is critical. If it is too small, the sets W and L will grow +slowly in every iteration. The trade-off is further complicated by the fact that +the sets W and L may generalize to unseen states. (2) For a fixed bound[i], it +is unclear how to prioritize the exploration of states. The PSPACE algorithm +suggests that going deep is good, whereas the potential for generalization to +unseen states is largest when going broad. (3) Finally, there is overhead in com- +puting both W and L. If there is a winning policy, we only need to compute W. +However, computing L may ensure that we can prune parts of the state space. +A similar observation holds for computing W on unsatisfiable instances. +Remark 1. Algorithm 2 can be mildly tweaked to meet the PSPACE algorithm in +Algorithm 1. The priority queue must ensure to always include complete (reach- +able) local BSGs and to explore states ⟨s, J⟩ with small J first. Furthermore, W +and L require regular pruning, and we cannot extract a policy if we prune W to +a polynomial size bound. Practically, we may write pruned parts of W to disk. +Extracting policies. +A (randomized) winning policy can be extracted from +the final set W. Any policy σ with Supp(σ(π)) = W for any path π is winning. +For a deterministic winning policy, we additionally store the policies after model +checking a fragment of the game, and apply the translation from Theorem 13. +6 +Experiments +We highlight two aspects: (1) A comparison of our prototype to existing baselines +for POMDPs, and (2) an examination of the exploration heuristics. The appendix +contains details on the implementation, the benchmarks, and more results. +Implementation. We provide a novel PArtial Game Exploration (PaGE) proto- +type, based on Algorithm 2, on top of the probabilistic model checker Storm [22]. + +16 +M. van der Vegt, N. Jansen, S. Junges +1 +9 +90 +900 +1 +9 +90 +900 +TO +MO +TO +MO +PaGE (default) +POMDP-bel +1 +9 +90 +900 +1 +9 +90 +900 +TO +MO +TO +MO +PaGE (default) +POMDP-SAT +1 +9 +90 +900 +1 +9 +90 +900 +TO +MO +TO +MO +PaGE (default) +PaGE (pos entrpy) +Fig. 5: Performance of baselines and novel PaGE algorithm +We represent MEMDPs using the Prism language with integer constants. Every +assignment to these constants induces an explicit MDP. SGs are constructed and +solved using existing data structures and graph algorithms. +Setup. We create a set of benchmarks inspired by the POMDP and MEMDP +literature [26,12,21]. We consider a combination of satisfiable and unsatisfiable +benchmarks. In the latter case, a winning policy does not exist. We construct +POMDPs from MEMDPs as in Definition 5. As baselines, we use the follow- +ing two existing POMDP algorithms. For almost-sure properties, a belief-MDP +construction [7] acts similar to an efficiently engineered variant of our game- +construction, but tailored towards more general quantitative properties. A SAT- +based approach [26] aims to find increasingly larger policies. We evaluate all +benchmarks on a system with a 3GHz Intel Core i9-10980XE processor. We use +a time limit of 30 minutes and a memory limit of 32 GB. +Results. Figure 5 shows the (logscale) performance comparisons between differ- +ent configurations8. Green circles reflect satisfiable and red crosses unsatisfiable +benchmarks. On the x-axis is PaGE in its default configuration. The first plot +compares to the belief-MDP construction. The tailored heuristics and represen- +tation of the belief-support give a significant edge in almost all cases. The few +points below the line are due to a higher exploration rate when building the +state space. The second plot compares to the SAT-based approach, which is +only suitable for finding policies, not for disproving their existence. This ap- +proach implicitly searches for a particular class of policies, whose structure is +not appropriate for some MEMDPs. The third plot compares PaGE in the de- +fault configuration – with negative entropy as priority function – with PaGE +using positive entropy. As expected, different priorities have a significant impact +on the performance. +Table 1 shows an overview of satisfiable and unsatisfiable benchmarks. Each +table shows the number of environments, states, and actions-per-state in the +MEMDP. For PaGE, we include both the default configuration (negative en- +tropy) and variation (positive entropy). For both configurations, we provide +columns with the time and the maximum size of the BSG constructed. We also +8 Every point ⟨x, y⟩ in the graph reflects a benchmarks which was solved by the con- +figuration on the x-axis in x time and by the configuration on the y-axis in y time. +Points above the diagonal are thus faster for the configuration on the x-axis. + +Robust Almost-Sure Reachability in Multi-Environment MDPs +17 +Table 1: Satisfiable and unsatisfiable benchmark results +PaGE(posentr) PaGE(negentr) Belief SAT +|I| +|S| |A| +t +n +t +n +t +t +Grid +19 +132 +4 +0.2 +3002 +0.2 +3002 +0.6 +3.7 +39 +152 +4 +0.4 +9007 +1.6 +41029 +12.6 121.3 +199 +474 +4 +6.4 +337177 +MO +MO +TO +Catch +256 +625 +4 +6.6 +93614 +5.9 +41094 +3.8 +TO +256 +6561 +4 +40.1 +749295 +32.6 337899 +9.1 +TO +256 14641 +4 +82.5 1826922 +65.3 338079 +16.2 +TO +Exp +8 +19 +9 +0.1 +349 +0.1 +349 +0.1 +75.9 +20 +43 +21 +131.4 +192163 +197.6 448443 217.6 +TO +24 +51 +25 +TO +MO +MO +TO +Frogger +10 +1200 +4 +0.2 +1200 +0.2 +1200 +22.7 +1.4 +20 +1200 +4 +0.4 +1200 +0.5 +1200 +MO +3.9 +80 +4000 +4 +4.4 +4000 +4.4 +4000 +TO 597.3 +99 +4000 +4 +5.9 +8001 +6.1 +8001 +TO +TO +PaGE(posentr) PaGE(negentr) Belief +|I| |S| |A| +t +n +t +n +t +MMind +16 21 16 +0.1 +1003 +0.2 +1445 +0.3 +27 17 27 +0.5 +5167 +0.5 +7579 +2.0 +32 25 32 +0.6 +7799 +0.9 +11809 +4.2 +81 21 81 +41.1 170291 +38.6 296407 +MO +Exp +20 42 21 +0.8 +9005 +173.8 388127 +576.1 +24 50 25 +8.3 +41022 +MO +MO +32 66 33 +347.7 337177 +MO +MO +include the time for the two baselines. Unsurprisingly, the number of states to be +explored is a good predictor for the performance and the relative performance +is as in Fig. 5. +7 +Conclusion +This paper considers multi-environment MDPs with an arbitrary number of en- +vironments and an almost-sure reachability objective. We show novel and tight +complexity bounds and use these insights to derive a new algorithm. This algo- +rithm outperforms approaches for POMDPs on a broad set of benchmarks. For +future work, we will apply an algorithm directly on the BOMDP [16]. + +18 +M. van der Vegt, N. Jansen, S. Junges +Data-Availability Statement +Supplementary material related to this paper is openly available on Zenodo at: +https://doi.org/10.5281/zenodo.7560675 +References +1. Roman Andriushchenko, Milan Ceska, Sebastian Junges, Joost-Pieter Katoen, and +Simon Stupinsk´y. PAYNT: A tool for inductive synthesis of probabilistic programs. +In CAV, volume 12759 of LNCS, pages 856–869. Springer, 2021. +2. 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Control., 66(3):1040– +1054, 2021. + diff --git a/YNFIT4oBgHgl3EQfjSvo/content/tmp_files/load_file.txt b/YNFIT4oBgHgl3EQfjSvo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00b3f499226f13df9c96a99e765c732f6e35e6f2 --- /dev/null +++ b/YNFIT4oBgHgl3EQfjSvo/content/tmp_files/load_file.txt @@ -0,0 +1,897 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf,len=896 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='11296v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='LO] 26 Jan 2023 Robust Almost-Sure Reachability in Multi-Environment MDPs Marck van der Vegt , Nils Jansen , and Sebastian Junges Radboud University, Nijmegen, the Netherlands {marck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='vandervegt, nils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='jansen, sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='junges}@ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='nl Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Multiple-environment MDPs (MEMDPs) capture finite sets of MDPs that share the states but differ in the transition dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' These models form a proper subclass of partially observable MDPs (POMDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We consider the synthesis of policies that robustly satisfy an almost-sure reachability property in MEMDPs, that is, one policy that satisfies a property for all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For POMDPs, deciding the existence of robust policies is an EXPTIME-complete problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In this paper, we show that this problem is PSPACE-complete for MEMDPs, while the policies in general require exponential memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We exploit the theoretical results to develop and implement an algorithm that shows promising results in synthesizing robust policies for various benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1 Introduction Markov decision processes (MDPs) are the standard formalism to model sequen- tial decision making under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A typical goal is to find a policy that satisfies a temporal logic specification [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Probabilistic model checkers such as Storm [22] and Prism [30] efficiently compute such policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A concern, how- ever, is the robustness against potential perturbations in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' MDPs cannot capture such uncertainty about the shape of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Multi-environment MDPs (MEMDPs) [36,14] contain a set of MDPs, called environments, over the same state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The goal in MEMDPs is to find a sin- gle policy that satisfies a given specification in all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' MEMDPs are, for instance, a natural model for MDPs with unknown system dynamics, where several domain experts provide their interpretation of the dynamics [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' These different MDPs together form a MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' MEMDPs also arise in other domains: In security applications, a natural example is a (static) password that must be guessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In robotics, a MEMDP captures the fact that we do not know the posi- tion of some static obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' MEMDPs can be interpreted as a (disjoint) union of MDPs in which an agent only has partial observation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', every MEMDP can be cast into a linearly larger partially observable MDP (POMDP) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Indeed, some famous examples for POMDPs are in fact MEMDPs, such as RockSam- ple [39] and Hallway [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Solving POMDPs is notoriously hard [32], and thus, it is worthwhile to investigate natural subclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We consider almost-sure specifications where the probability needs to be one to reach a set of target states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In MDPs, it suffices to consider memoryless 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Constructing such policies can be efficiently implemented by means of a graph-search [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For MEMDPs, that consist of a finite set of environments, each described by a MDP, we consider the following problem: Compute one policy that almost-surely reaches the target in all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Such a policy robustly satisfies an almost-sure specification for a set of MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Inspired by work on POMDPs, we construct a belief-observation MDP[16] that tracks the states of the MDPs and the (support of the) belief over potential environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We show that a policy satisfying the almost-sure property in this MDP also robustly satisfies the property in the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Although the belief-observation MDP is exponentially larger than the MEMDP, we exploit its particular structure to create a PSPACE algorithm to decide whether such a robust policy exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The essence of the algorithm is a recur- sive construction of a fragment of the belief-observation MDP, restricted to a setting in which the belief-support is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Such an approach is possible, as the belief in a MEMDP behaves monotonically: Once we know that we are not in a particular environment, we never lose this knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This behavior is in contrast to POMDPs, where there is no monotonic behavior in belief-supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The difference is essential: Deciding almost-sure reachability in POMDPs is EXPTIME-complete [37,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In contrast, the problem of deciding whether a pol- icy for almost-sure reachability in a MEMDP exists is indeed PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We show the hardness using a reduction from the true quantified Boolean formula problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Finally, we cannot hope to extract a policy with such an algorithm, as the smallest policy for MEMDPs may require exponential memory in the number of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The PSPACE algorithm itself recomputes many results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For practical pur- poses, we create an algorithm that iteratively explores parts of the belief-observation MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The algorithm additionally uses the MEMDP structure to generalize the set of states from which a winning policy exists and deduce efficient heuristics for guiding the exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The combination of these ingredients leads to an efficient and competitive prototype on top of the model checker Storm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We categorize related work in three areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' MEMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Almost-sure reachability for MEMDPs for exactly two environments have been studied by [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We extend the results to arbitrarily many environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This is nontrivial: For two environments, the decision problem has a poly- nomial time routine [36], whereas we show that the problem is PSPACE-complete for an arbitrary number of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' MEMDPs and closely related models such as hidden-model MDPs, hidden-parameter MDPs, multi-model MDPs, and concurrent MDPs [11,2,40,10] have been considered for quantitative properties1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The typical approach is to consider approximative algorithms for the undecid- able problem in POMDPs [14] or adapt reinforcement learning algorithms [3,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' These approximations are not applicable to almost-sure properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1 Hidden-parameter MDPs are different than MEMDPs in that they assume a prior over different MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' However, for almost-sure properties, this difference is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 3 POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' One can build an underlying potentially infinite belief-MDP [27] that corresponds to the POMDP – using model checkers [35,7,8] to verify this MDP can answer the question for MEMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For POMDPs, almost-sure reachability is decidable in exponential time [37,19] via a construction similar to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Most qualitative properties beyond almost-sure reachability are undecidable [4,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Two dedicated algorithms that limit the search to policies with small mem- ory requirements and employ a SAT-based approach [12,26] to this NP-hard problem [19] are implemented in Storm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We use them as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The high-level representation of MEMDPs is structurally simi- lar to featured MDPs [18,1] that represent sets of MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The proposed tech- niques are called family-based model checking and compute policies for every MDP in the family, whereas we aim to find one policy for all MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Interval MDPs [25,43,23] and SGs [38] do not allow for dependencies between states and thus cannot model features such as various obstacle positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Parametric MDPs [2,44,24] assume controllable uncertainty and do not consider robustness of policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We establish PSPACE-completeness for deciding almost-sure reachability in MEMDPs and show that the corresponding policies may be expo- nentially large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Our iterative algorithm builds fragments of the belief-observation MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The prototype is the first algorithm specific to almost-sure reachability in MEMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A empirical evaluation shows that the iterative algorithm outper- forms approaches dedicated to POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 2 Problem Statement In this section, we provide some background and formalize the problem state- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a set X, Dist(X) denotes the set of probability distributions over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a given distribution d ∈ Dist(X), we denote its support as Supp(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a finite set X, let unif(X) denote the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For x ∈ X, we use dirac(x) to denote the Dirac distribution on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We use short-hand notation for functions and distributions, f = [x �→ a, y �→ b] means that f(x) = a and f(y) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We write P (X) for the powerset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For n ∈ N we write [n] = {i ∈ N | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 1 (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A Markov Decision Process is a tuple M = ⟨S, A, ιinit, p⟩ where S is the finite set of states, A is the finite set of actions, ιinit ∈ Dist(S) is the initial state distribution, and p: S × A → Dist(S) is the transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The transition function is total, that is, for notational convenience MDPs are input-enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This requirement does not affect the generality of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A path of an MDP is a sequence π = (s0, a0)(s1, a1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' sn such that ιinit(s0) > 0 and p(si, ai, si+1) > 0 for all 0 ≤ i < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The last state of π is last(π) = sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We denote the set of all finite paths by Path = (S ×A)∗ ×S, and the paths starting in a state from S′ ⊆ S by Path(S′) = {(s0, a0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' sn ∈ Path | s0 ∈ S′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The set of reachable states from S′ is Reachable(S′) = {last(π) | π ∈ Path(S′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' If 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges N1 : s0 s1 q1, q2 q1, q2 a1 a2, a3 N2 : s0 s1 q2 q2 q1 q1 a2 a1, a3 N3 : s0 q1, q2 a3 a1, a2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1: Example MEMDP S′ = Supp(ιinit) we just call them the reachable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The MDP restricted to a set of reachable states from a distribution d ∈ Dist(S) is ReachFragment(M, d), where d is the new initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A state s of the MDP is absorbing if Reachable({s}) = {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An MDP is acyclic, if each state is either absorbing or not reachable from its successor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Action choices are resolved by a policy σ: Path → Dist(A) that maps paths to distributions over actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A policy of the form σ: S → Dist(A) is called mem- oryless, deterministic if we have σ: Path → A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' and, memoryless deterministic for σ: S → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For an MDP M, we denote the probability of a policy σ reaching some target set T ⊆ S starting in state s as PrM(s → T | σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' More precisely, PrM(s → T | σ) denotes the probability of all paths from s reaching T under σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We simplify to PrM(T | σ) if s is distributed according to ιinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 2 (MEMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A Multiple Environment MDP is a tuple N = ⟨S, A, ιinit, {pi}i∈I⟩ with S, A, ιinit as for MDPs, and {pi}i∈I is a set of tran- sition functions, where I is a finite set of environment indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Intuitively, MEMDPs form sets of MDPs (environments) that share states and actions, but differ in the transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For MEMDP N with index set I and a set I′ ⊆ I, we define the restriction of environments as the MEMDP N↓I′ = ⟨S, A, ιinit, {pi}i∈I′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Given an environment i ∈ I, we denote its corresponding MDP as Ni = ⟨S, A, ιinit, pi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A MEMDP with only one environment is an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Paths and policies are defined on the states and actions of MEMDPs and do not differ from MDP policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A MEMDP is acyclic, if each MDP is acyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Figure 1 shows an example of a MEMDP with three environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An agent can perform two questions, q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The response is either to ‘switch’ the state (from s1 to s2 or vice versa), or to ‘stay’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', take a self-loop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For example, in environment N1, the response to both questions is to switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In N2, the response to q1 is to stay, while the response to q2 is to switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The agent can also guess the environment using a1, a2, a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The idea is that guessing ai leads to the target set { } only in environment i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An agent must therefore deduce the environment via q1, q2 to be able to make a guaranteed correct guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ■ Definition 3 (Almost-Sure Reachability Property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An almost-sure reach- ability property is defined by a set of target states T ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A policy σ satisfies the almost-sure reachability property T for MEMDP N = ⟨S, A, ιinit, {pi}i∈I⟩ iff: ∀i ∈ I : PrNi(T | σ) = 1 Robust Almost-Sure Reachability in Multi-Environment MDPs 5 In other words, a policy σ satisfies an almost-sure reachability property T , called winning, if and only if the probability of reaching T within each MDP is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' By extension, a state s ∈ S is winning if there exists a winning policy when starting in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Policies and states that are not winning are losing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We will now define both the decision and policy problem: Given a MEMDP N and an almost-sure reachability property T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The Decision Problem asks to decide if a policy exists that satisfies T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The Policy Problem asks to compute such a policy, if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In Section 4 we discuss the computational complexity of the decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Following up, in Section 5 we present our algorithm for solving the policy prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Details on its implementation and evaluation will be presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 3 A Reduction To Belief-Observation MDPs In this section, we reduce the policy problem, and thus also the decision prob- lem, to finding a policy in an exponentially larger belief-observation MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This reduction is an elementary building block for the construction of our PSPACE algorithm and the practical implementation, both presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Ad- ditional information such as proofs for statements in this and later sections are given in the technical report [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='1 Partially observable MDPs As a first step, we formalise the interpretation of MEMDPs as a POMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 4 (POMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A partially observable MDP (POMDP) is a tuple ⟨M, Z, O⟩ where M = ⟨S, A, ιinit, p⟩ is an MDP, Z is a set of observations, and O: S → Z the observation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A partially observable MDP is an MDP where states are labelled with obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We lift O to paths and use O(π) = O(s1)a1O(s2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' O(sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We are interested in observation-based policies σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', policies s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' for π, π′ ∈ Path, O(π) = O(π′) implies σ(π) = σ(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A MEMDP can be cast into a POMDP that is made up as the disjoint union: Definition 5 (Union-POMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Given an MEMDP N = ⟨S, A, ιinit, {pi}i∈I⟩ we define its union-POMDP N⊔ = ⟨⟨S′, A, ι′ init, p′⟩, Z, O⟩, with states S′ = S×I, initial distribution ι′ init(⟨s, i⟩) = ιinit(s) · |I|−1, transitions p′(⟨s, i⟩, a)(⟨s′, i⟩) = pi(s, a)(s′), observations Z = S, and observation function O(⟨s, i⟩) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A policy may observe the state s but not in which MDP we are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This forces any observation-based policy to take the same choice in all MEMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Given MEMDP N, there exists a winning policy iff there exists an observation-based policy σ such that PrN⊔(T | σ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The statement follows as, first, any observation-based policy of the POMDP can be applied to the MEMDP, second, vice versa, any MEMDP policy is observation- based, and third, the induced MCs under these policies are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 Belief-observation MDPs For POMDPs, memoryless policies are not sufficient, which makes computing policies intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We therefore add the information that the history — i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', the path until some point — contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In MEMDPs, this information is the (environment-)belief (support) J ⊆ I, as the set of environments that are consis- tent with a path in the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Given a belief J ⊆ I and a state-action-state pair s a−→ s′, then we define Up(J, s, a, s′) = {i ∈ J | pi(s, a, s′) > 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', the subset of environments in which the transition exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a path π ∈ Path, we define its corresponding belief B(π) ⊆ I recursively as: B(s0) = I and B(π · sas′)) = Up(B(π · s), s, a, s′) The belief in a MEMDP monotonically decreases along a path, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', if we know that we are not in a particular environment, this remains true indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We aim to use a model where memoryless policies suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' To that end, we cast MEMDPs into the exponentially larger belief-observation MDPs [16]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 6 (BOMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a MEMDP N = ⟨S, A, ιinit, {pi}i∈I⟩, we define its belief-observation MDP (BOMDP) as a POMDP GN = ⟨⟨S′, A, ι′ init, p′⟩, Z, O⟩ with states S′ = S × I × P (I), initial distribution ι′ init(⟨s, j, I⟩) = ιinit(s) · |I|−1, transition relation p′(⟨s, j, J⟩, a)(⟨s′, j, J′⟩) = pj(s, a, s′) with J′ = Up(J, s, a, s′), observations Z = S × P (I), and observation function O(⟨s, j, J⟩) = ⟨s, J⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Compared to the union-POMDP, BOMDPs also track the belief by updating it accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We clarify the correspondence between paths of the BOMDP and the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a path π through the MEMDP, we can mimic this path exactly in the MDPs Nj for j ∈ Bπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' As we track Bπ in the state, we can deduce from the BOMDP state in which environments we can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For MEMDP N and the path ⟨s1, j, J1⟩a1⟨s2, j, J2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ⟨sn, j, Jn⟩ of the BOMDP GN , let j ∈ J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Then: Jn ̸= ∅ and the path (s1, v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' sn exists in MDP Ni iff i ∈ J1 ∩ Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Consequently, the belief of a path can be uniquely determined by the observation of the last state reached, hence the name belief-observation MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For every pair of paths π, π′ in a BOMDP, we have: B(π) = B(π′) implies O(last(π)) = O(last(π′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For notation, we define SJ = {⟨s, j, J⟩ | j ∈ J, s ∈ S}, and analogously write ZJ = {⟨s, J⟩ | s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We lift the target states T to states in the BOMDP: TGN = {⟨s, j, J⟩ | s ∈ T, J ⊆ I, j ∈ J} and define target observations TZ = O(TGN ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 7 (Winning in a BOMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let GN be a BOMDP with target observations TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An observation-based policy σ is winning from some observation z ∈ Z, if for all s ∈ O−1(z) it holds that PrGN (s → O−1(TZ) | σ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 2 This translation is notationally simpler than going via the union-POMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 7 Furthermore, a policy σ is winning if it is winning for the initial distribution ιinit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An observation z is winning if there exists a winning policy for z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The winning region WinT GN is the set of all winning observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Almost-sure winning in the BOMDP corresponds to winning in the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' There exists a winning policy for a MEMDP N with target states T iff there exists a winning policy in the BOMDP GN with target states TGN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Intuitively, the important aspect is that for almost-sure reachability, observation- based memoryless policies are sufficient [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For any such policy, the induced Markov chains on the union-POMDP and the BOMDP are bisimilar [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' BOMDPs make policy search conceptually easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' First, as memoryless poli- cies suffice for almost-sure reachability, winning regions are independent of fixed policies: For policies σ and σ′ that are winning in observation z and z′, re- spectively, there must exist a policy ˆσ that is winning for both z and z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Sec- ond, winning regions can be determined in polynomial time in the size of the BOMDP [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='3 Fragments of BOMDPs To avoid storing the exponentially sized BOMDP, we only build fragments: We may select any set of observations as frontier observations and make the states with those observations absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We later discuss the selection of frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 8 (Sliced BOMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a BOMDP GN = ⟨⟨S, A, ιinit, p⟩, Z, O⟩ and a set of frontier observations F ⊆ Z, we define a BOMDP GN |F = ⟨⟨S, A, ιinit, p′⟩, Z, O⟩ with: ∀s ∈ S, a ∈ A: p′(s, a) = � dirac(s) if O(s) ∈ F, p(s, a) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We exploit this sliced BOMDP to derive constraints on the set of winning states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For every BOMDP GN with states S and targets T and for all fron- tier observations F ⊆ Z it holds that: WinT GN |F ⊆ WinT GN ⊆ WinT ∪F GN |F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Making (non-target) observations absorbing extends the set of losing observa- tions, while adding target states extends the set of winning observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 4 Computational Complexity The BOMDP above yields an exponential time and space algorithm via Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We can avoid the exponential memory requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This section shows the PSPACE-completeness of deciding whether a winning policy exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The almost-sure reachability decision problem is PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The result follows from Lemmas 11 and 10 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='3, we show that representing the winning policy itself may however require exponential space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges {1, 2, 3} {2, 3} {1, 2} {1} {2} {3} {1, 3} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 2: The environment graph for our running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='1 Deciding Almost-Sure Winning for MEMDPs in PSPACE We develop an algorithm with a polynomial memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The algorithm exploits locality of cyclic behavior in the BOMDP, as formalized by an acyclic environment graph and local BOMDPs that match the nodes in the environment graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The algorithm recurses on the environment graph while memorizing re- sults from polynomially many local BOMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The graph-structure of BOMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' First, along a path of the MEMDP, we will only gain information and are thus able to rule out certain environments [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Due to the monotonicity of the update operator, we have for any BOMDP that ⟨s, j, J⟩ ∈ Reachable(⟨s′, j, J′⟩) implies J ⊆ J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We define a graph over environ- ment sets that describes how the belief-support can update over a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 9 (Environment graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let N be a MEMDP and p the tran- sition function of GN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The environment graph GE N = (VN , EN ) for N is a directed graph with vertices VN = P (I) and edges EN = {⟨J, J′⟩ | ∃s, s′ ∈ S, a ∈ A, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='p(⟨s, j, J⟩, a, ⟨s′, j, J′⟩) > 0 and J ̸= J′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Figure 2 shows the environment graph for the MEMDP in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' It consists of the different belief-supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For example, the transition from {1, 2, 3} to {2, 3} and to {1} is due to the action q1 in state s0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ■ Paths in the environment graph abstract paths in the BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Path fragments where the belief-support remains unchanged are summarized into one step, as we do not create edges of the form ⟨J, J⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We formalize this idea: Let π = ⟨s1, j, J1⟩a1⟨s2, j, J2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ⟨sn, j, Jn⟩ be a path in the BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For any J ⊆ I, we call π a J-local path, if Ji = J for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a MEMDP N with environment graph GE N , there is a path J1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jn iff there is a path π = π1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' πn in GN s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' every πi is Ji-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The shape of the environment graph is crucial for the algorithm we develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let GE N = (VN , EN ) be an environment graph for MEMDP N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' First, EN (J, J′) implies J′ ⊊ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Thus, G is acyclic and has maximal path length |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The maximal outdegree of the graph is |S|2|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The monotonicity regarding J, J′ follows from definition of the belief update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The bound on the outdegreee is a consequence from Lemma 9 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 9 Local belief-support BOMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Before we continue, we remark that the (future) dynamics in a BOMDP only depend on the current state and set of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' More formally, we capture this intuition as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let GN be a BOMDP with states S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For any state ⟨s, j, J⟩ ∈ S′, let N ′ = ReachFragment(N↓J, dirac(s)) and Y = � i∈J⟨s, i, J⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Then: ReachFragment(GN , unif(Y )) = GN ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The key insight is that restricting does not change the transition functions for the environments j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Furthermore, using monotonicity of the update, we only reach BOMDP-states whose behavior is determined by the environments in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This intuition allows us to analyze the BOMDP locally and lift the results to the complete BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We define a local BOMDP as the part of a BOMDP starting in any state in SJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' All states not in SJ are made absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 10 (Local BOMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Given a MEMDP N with BOMDP GN and a set of environments J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The local BOMDP for environments J is the fragment LocG(J) = ReachFragment(GN↓J |F, unif(SJ)) where F = Z \\ ZJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This definition of a local BOMDP coincides with a fragment of the complete BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We then mark exactly the winning observations restricted to the envi- ronment sets J′ ⊊ J as winning in the local BOMDP and compute all winning observations in the local BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' These observations are winning in the com- plete BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The following concretization of Lemma 4 formalizes this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Consider a MEMDP N and a subset of environments J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Win T ′ GN LocG(J) ∩ ZJ = Win TGN GN ∩ ZJ with T ′ GN = TGN ∪ (Win TGN GN \\ ZJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Furthermore, local BOMDPs are polynomially bounded in the size of the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let N be a MEMDP with states S and actions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' LocG(J) has at most O(|S|2 · |A| · |J|) states and O(|S|2 · |A| · |J|2) transitions3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A PSPACE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We present Algorithm 1 for the MEMDP decision problem, which recurses depth-first over the paths in the environment graph4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We first state the correctness and the space complexity of this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ASWinning in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1 solves the decision problem in PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' To prove correctness, we first note that Search(N, J, T ) computes Win TGN GN ∩ZJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We show this by induction over the structure of the environment graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For all J without outgoing edges, the local BOMDP coincides with a BOMDP just for 3 The number of transitions is the number of nonzero entries in p 4 In contrast to depth-first-search, we do not memorize nodes we visited earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges Algorithm 1 Search algorithm 1: function Search(MEMDP N = ⟨S, A, {pi}i∈I, ιinit⟩, J ⊆ I, T ⊆ S) 2: T ′ ← {⟨s, j, J⟩ | j ∈ J, s ∈ T } 3: for J′ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' EN (J, J′) do ⊲ Consider the edges in the env.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' graph (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 9) 4: WJ′ ← Search(N, J′, T ) ⊲ Recursion!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5: T ′ ← T ′ ∪ {⟨s, j, J′⟩ | j ∈ J, ⟨s, J′⟩ ∈ WJ′} 6: return WinT ′ LocG(J) ∩ ZJ ⊲ Construct BOMDP as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 10, then model check 7: 8: function ASWinning(MEMDP N = ⟨S, A, {pi}i∈I, ιinit⟩, T ⊆ S) 9: return O(Supp(ιinit)) ⊆ Search(N, I, T ) x M1 : x⊥ x⊤ α⊗ 1 2 1 2 y W F α⊗ α⊗ ⊤ ⊥ x M2 : x⊤ x⊥ α⊗ 1 2 1 2 y W F α⊗ α⊗ ⊤ ⊥ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 3: Constructed MEMDP for the QBF formula ∀x∃y � (x ∨ y) ∧ (¬x ∨ ¬y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' environments J (Lemma 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Otherwise, observe that T ′ in l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5 coincides with its definition in Lemma 8 and thus, by the same lemma, we return Win TGN GN ∩ZJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' To finalize the proof, a winning policy exists in the MEMDP if the observation of the initial states of the BOMDP are winning (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The algorithm must terminate as it recurses over all paths of a finite acyclic graph, see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Following Lemma 9, the number of frontier states is then bounded by |S|2 · |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The main body of the algorithm therefore requires polynomial space, and the maximal recursion depth (stack height) is |I| (Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Together, this yields a space complexity in O(|S|2 · |A| · |I|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 Deciding Almost-Sure Winning for MEMDPs Is PSPACE-hard It is not possible to improve the algorithm beyond PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The MEMDP decision problem is PSPACE-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Hardness holds even for acyclic MEMDPs and uses the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' On winning acyclic MEMDPs deterministic winning policies exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In particular, almost-sure reachability coincides with avoiding the sink states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This is a safety property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For safety, deterministic policies are sufficient, as ran- domization visits only additional states, which is not beneficial for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Regarding Lemma 11, we sketch a polynomial-time reduction from the PSPACE- complete TQBF problem [20] problem to the MEMDP decision problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let Ψ be a QBF formula, Ψ = ∃x1∀y1∃x2∀y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ∃xn∀yn � Φ � with Φ a Boolean formula in conjunctive normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The problem is to decide whether Ψ is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 11 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Consider the QBF formula Ψ = ∀x∃y � (x ∨ y) ∧ (¬x ∨ ¬y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We construct a MEMDP with an environment for every clause, see Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The state space consists of three states for each variable v ∈ V : the state v and the states v⊤ and v⊥ that encode their assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Additionallly, we have a dedicated target W and sink state F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We consider three actions: The actions true (⊤) and false (⊥) semantically describe the assignment to existentially quantified variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The action any α⊗ is used for all other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Every environment reaches the target state iff one literal in the clause is assigned true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In the example, intuitively, a policy should assign the negation of x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Formally, the policy σ, characterized by σ(π · y) = ⊤ iff x⊥ ∈ π, is winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ■ As a consequence of this construction, we may also deduce the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Deciding whether a memoryless winning policy exists is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The proof of NP hardness uses a similar construction for the propositional SAT fragment of QBF, without universal quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Additionally, the problem for memoryless policies is in NP, because one can nondeterministically guess a (poly- nomially sized) memoryless policy and verify in each environment independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='3 Policy Problem Policies, mapping histories to actions, are generally infinite objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' However, we may extract winning policies from the BOMDP, which is (only) exponential in the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Finite state controllers [34] are a suitable and widespread represen- tation of policies that require only a finite amount of memory, formally defined in ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='. Intuitively, the number of memory states reflects the number of equivalence classes of histories that a policy can distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In general, we cannot hope to find smaller policies than those obtained via a BOMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' There is a family of MEMDPs {N n}n≥1 where for each n, N n has 2n environments and O(n) states and where every winning policy for N n requires at least 2n memory states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We illustrate the witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Consider a family of MEMDPs {N n}n, where N n has 2n MDPs, 4n states partitioned into two parts, and at most 2n outgoing actions per state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We outline the MEMDP N in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In the first part, there is only one action per state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The notation is as follows: in state s0 and MDP N n 1 , we transition with probability one to state a0, whereas in N n 2 we transition with probability one to state b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In every other MDP, we transition with probability one half to either state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In state s1, we do the analogous construction for envi- ronments 3, 4, and all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A path s0b0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' is thus consistent with every MDP except N n 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The first part ends in state sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' By construction, there are 2n paths ending in sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Each of them is (in)consistent with a unique set of n environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In the second part, a policy may guess n times an environment by selecting an action αi for every i ∈ [2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Only in MDP N n i , action αi leads to a target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5 We depict a slightly simplified MEMDP for conciseness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges s0 a1 b1 i = 1 i = 2 i ̸= 1 i ̸= 2 1 2 1 2 s1 a2 b2 i = 3 i = 4 i ̸= 3 i ̸= 4 1 2 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' sn an bn i = n − 1 i = n i ̸= n − 1 i ̸= n 1 2 1 2 g1 W g2 gn αi αi αi gn+1 αj αj αj Mi : i ∈ [2n] j ̸= i 1 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 4: Witness for exponential memory requirement for winning policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In all other MDPs, the transition leads from state gj to gj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The state gn+1 is absorbing in all MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Importantly, after taking an action αi and arriving in gj+1, there is (at most) one more MDP inconsistent with the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Every MEMDP N n in this family has a winning policy which takes σ(π·gi) = α2i−1 if ai−1 ∈ π and σ(π · gi) = α2i otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Furthermore, when arriving in state sn, the state of a finite memory controller must reflect the precise set of environments consistent with the history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' These are 2n many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The proof shows that if we store less information, two paths will lead to the same memory state, but with different sets of environments being consistent with these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' As we can rule out only n environments using the n actions in the second part of the MEMDP, we cannot ensure winning in every environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5 A Partial Game Exploration Algorithm In this section, we present an algorithm for the policy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We tune the algorithm towards runtime instead of memory complexity, but aim to avoid running out of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We use several key ingredients to create a pragmatic variation of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 1, with support for extracting the winning policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' First, we use an abstraction from BOMDPs to a belief stochastic game (BSG) similar to [45] that reduces the number of states and simplifies the iterative construction6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Second, we tailor and generalize ideas from bounded model check- ing [6] to build and model check only a fragment of the BSG, using explicit partial exploration approaches as in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', [33,9,42,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Third, our exploration does not continuously extend the fragment, but can also prune this fragment by using the model checking results obtained so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The structure of the BSG as captured by the environment graph makes the approach promising and yields some natural heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Fourth, the structure of the winning region allows to generalize results to unseen states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We thereby operationalize an idea from [26] in a partial exploration context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Finally, we analyze individual MDPs as an efficient and significant preprocessing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In the following we discuss these ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 6 At the time of writing, we were unaware of a polytime algorithm for BOMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 13 Algorithm 2 Policy finding algorithm 1: function FindPolicy(MEMDP N = ⟨S, A, {pi}i∈I, ιinit⟩, targets T ⊆ S) 2: W ← {⟨s, J⟩ | s ∈ T, J ⊆ I};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' L ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' i ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Sinit ← Supp(ιinit) × {I} 3: while Sinit ̸⊆ (W ∪ L) do 4: ⟨B, F⟩ ← GenerateGameSlice(N, W, L, i) 5: W ← W ∪ WinW B 6: L ← L ∪ S \\ WinW ∪F B 7: i ← i + 1 8: if Sinit ⊆ W then return ExtractPolicy(W ) else return ⊥ Abstraction to Belief Support Games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We briefly recap stochastic games (SG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' See [38,17] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Definition 11 (SG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A stochastic game is a tuple B = ⟨M, S1, S2⟩, where M = ⟨S, A, ιinit, p⟩ is an MDP and (S1, S2) is a partition of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' S1 are Player 1 states, and S2 are Player 2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' As common, we also ‘partition’ (memoryless deterministic) policies into two functions σ1 : S1 → A and σ1 : S2 → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A Player 1 policy σ1 is winning for state s if Pr(T | σ1, σ2) for all σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We (re)use WinT BN to denote the set of states with a winning policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We apply a game-based abstraction to group states that have the same ob- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Player 1 states capture the observation in the BOMDP, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', tuples ⟨s, J⟩ of MEMDP states s and subsets J of the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Player 1 selects the action a, the result is Player 2 state ⟨⟨s, J⟩, a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Then Player 2 chooses an environment j ∈ J, and the game mimicks the outgoing transition from ⟨s, j, J⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', it mimicks the transition from s in Nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Formally: Definition 12 (BSG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Let GN be a BOMDP with GN = ⟨⟨S, A, ιinit, p⟩, Z, O⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A belief support game BN for GN is an SG BN = ⟨⟨S′, A′, ι′ init, p⟩, S1, S2⟩ with S′ = S1 ∪ S2 as usual, Player 1 states S1 = Z, Player 2 states S2 = Z × A, actions A′ = A ∪ I, initial distribution ι′ init(⟨s, I⟩) = � i∈I ιinit(⟨s, i, I⟩), and the (partial) transition function p defined separately for Player 1 and 2: p′(z, a) = dirac(⟨z, a⟩) (Player 1) p′(⟨z, a⟩, j, z′) = p(⟨s, j, J⟩, a)(⟨s′, j, J′⟩) with z = ⟨s, J⟩, z′ = ⟨s′, J′⟩ (Player 2) Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' An (acylic) MEMDP N with target states T is winning if(f) there exists a winning policy in the BSG BN with target states TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Thus, on acyclic MEMDPs, a BSG-based algorithm is sound and complete, how- ever, on cyclic MDPs, it may not find the winning policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The remainder of the algorithm is formulated on the BSG, we use sliced BSGs as the BSG of a sliced BOMDP, or equivalently, as a BSG with some states made absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Main algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We outline Algorithm 2 for the policy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We track the sets of almost-sure observations and losing observations (states in the BSG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges Algorithm 3 Game generation algorithm 1: function GenerateGameSlice(MEMDP N, W , L, i) 2: Q ← {sι};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' E = {sι} 3: while s ∈ Q and |E| ≤ Bound[i] exists do 4: E ← E ∪ {s} ⊲ Mark s as explored 5: B ← BN |(S \\ E) ⊲ Extend game, cut-off everything not explored 6: Q ← Reachable(B) \\ (E ∪ W ∪ L) ⊲ Add newly reached states 7: return B, Q Initially, target states are winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Furthermore, via a simple preprocessing, we determine some winning and losing states on the individual MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We iterate until the initial state is winning or losing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Our algorithm constructs a sliced BSG and decides on-the-fly whether a state should be a frontier state, returning the sliced BSG and the used frontier states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We discuss the implemen- tation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For the sliced BSG, we compute the winning region twice: Once assuming that the frontier states are winning, once assuming they are loosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This yields an approximation of the winning and losing states, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Soundness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The algorithm is sound, assuming that the BN is indeed a sliced BSG with frontier F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In particular, then the following invariant holds: W ⊆ WinT BN and L ∩ WinT BN = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This invariant exploits that from a sliced BSG we can (implicitly) slice the complete BSG while preserving the winning status of every state, formalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In future iterations we only explore the implicitly sliced BSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Given W ⊆ Win TBN BN and L ⊆ S \\ Win TBN BN : Win TBN BN = Win TBN ∪W BN |W∪L Termination depends on the sliced game generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' It suffices to ensure that in the long run, either W or L grow as there are only finitely many states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' If W and L remain the same longer than some number of iterations,W ∪ L will be used as frontier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Then, the new game will suffice to determine if s ∈ W in one shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Generating the sliced BSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Algorithm 3 outlines the generation of the sliced BSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In particular, we explore the implicit BSG from the initial state but make every state that we do not explicitly explore absorbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In every iteration, we first check if there are states in Q left to explore and if the number of explored states in E is below a threshold Bound[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Then, we take a state from the priority queue and add it to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We find new reachable states7 and add them to the queue Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Generalizing the winning and losing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We can determine that a state in the game BN is winning without ever exploring it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Therefore, observe the following natural property of MEMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 7 In l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5 we do not rebuild the game B from scratch but incrementally construct the data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Likewise, reachable states are a direct byproduct of this construc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 15 Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A winning policy in MEMDP N is winning in N↓J for any J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A direct consequence is the following statement for two environments J1 ⊆ J2: ⟨s, J2⟩ ∈ WinT BN implies ⟨s, J1⟩ ∈ WinT BN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Consequently, we can store W (and symmetrically, L) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For every MEMDP state s ∈ S, Ws = {J | ⟨s, J⟩ ∈ W} is downward closed on the partial order P = (I, ⊂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This allows for efficient storage: We only have to store the set of pairwise maximal elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', the antichain, W max s = {J ∈ Ws | ∀J′ ∈ Ws with J ̸⊆ J′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' To determine whether ⟨s, J⟩ is winning, we check whether J ⊆ J′ for some J′ ∈ W max s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Adding J to W max s requires removing all J′ ⊆ J and then adding J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Note, however, that |W max s | is still exponential in |I| in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Selection of heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The algorithm allows some degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We evaluate the following aspects empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' (1) The maximal size bound[i] of a sliced BSG at iteration i is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' If it is too small, the sets W and L will grow slowly in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The trade-off is further complicated by the fact that the sets W and L may generalize to unseen states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' (2) For a fixed bound[i], it is unclear how to prioritize the exploration of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The PSPACE algorithm suggests that going deep is good, whereas the potential for generalization to unseen states is largest when going broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' (3) Finally, there is overhead in com- puting both W and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' If there is a winning policy, we only need to compute W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' However, computing L may ensure that we can prune parts of the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A similar observation holds for computing W on unsatisfiable instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Algorithm 2 can be mildly tweaked to meet the PSPACE algorithm in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The priority queue must ensure to always include complete (reach- able) local BSGs and to explore states ⟨s, J⟩ with small J first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Furthermore, W and L require regular pruning, and we cannot extract a policy if we prune W to a polynomial size bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Practically, we may write pruned parts of W to disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Extracting policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A (randomized) winning policy can be extracted from the final set W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Any policy σ with Supp(σ(π)) = W for any path π is winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For a deterministic winning policy, we additionally store the policies after model checking a fragment of the game, and apply the translation from Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 6 Experiments We highlight two aspects: (1) A comparison of our prototype to existing baselines for POMDPs, and (2) an examination of the exploration heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The appendix contains details on the implementation, the benchmarks, and more results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We provide a novel PArtial Game Exploration (PaGE) proto- type, based on Algorithm 2, on top of the probabilistic model checker Storm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges 1 9 90 900 1 9 90 900 TO MO TO MO PaGE (default) POMDP-bel 1 9 90 900 1 9 90 900 TO MO TO MO PaGE (default) POMDP-SAT 1 9 90 900 1 9 90 900 TO MO TO MO PaGE (default) PaGE (pos entrpy) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5: Performance of baselines and novel PaGE algorithm We represent MEMDPs using the Prism language with integer constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Every assignment to these constants induces an explicit MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' SGs are constructed and solved using existing data structures and graph algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We create a set of benchmarks inspired by the POMDP and MEMDP literature [26,12,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We consider a combination of satisfiable and unsatisfiable benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In the latter case, a winning policy does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We construct POMDPs from MEMDPs as in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' As baselines, we use the follow- ing two existing POMDP algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For almost-sure properties, a belief-MDP construction [7] acts similar to an efficiently engineered variant of our game- construction, but tailored towards more general quantitative properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' A SAT- based approach [26] aims to find increasingly larger policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We evaluate all benchmarks on a system with a 3GHz Intel Core i9-10980XE processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We use a time limit of 30 minutes and a memory limit of 32 GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Figure 5 shows the (logscale) performance comparisons between differ- ent configurations8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Green circles reflect satisfiable and red crosses unsatisfiable benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' On the x-axis is PaGE in its default configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The first plot compares to the belief-MDP construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The tailored heuristics and represen- tation of the belief-support give a significant edge in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The few points below the line are due to a higher exploration rate when building the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The second plot compares to the SAT-based approach, which is only suitable for finding policies, not for disproving their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This ap- proach implicitly searches for a particular class of policies, whose structure is not appropriate for some MEMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' The third plot compares PaGE in the de- fault configuration – with negative entropy as priority function – with PaGE using positive entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' As expected, different priorities have a significant impact on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Table 1 shows an overview of satisfiable and unsatisfiable benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Each table shows the number of environments, states, and actions-per-state in the MEMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For PaGE, we include both the default configuration (negative en- tropy) and variation (positive entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For both configurations, we provide columns with the time and the maximum size of the BSG constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We also 8 Every point ⟨x, y⟩ in the graph reflects a benchmarks which was solved by the con- figuration on the x-axis in x time and by the configuration on the y-axis in y time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Points above the diagonal are thus faster for the configuration on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Robust Almost-Sure Reachability in Multi-Environment MDPs 17 Table 1: Satisfiable and unsatisfiable benchmark results PaGE(posentr) PaGE(negentr) Belief SAT |I| |S| |A| t n t n t t Grid 19 132 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 3002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 3002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='6 3.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='6 448443 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='6 TO 24 51 25 TO MO MO TO Frogger 10 1200 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 1200 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='4 20 1200 4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='1 8001 TO TO PaGE(posentr) PaGE(negentr) Belief |I| |S| |A| t n t n t MMind 16 21 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='1 1003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='2 1445 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='3 27 17 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='5 5167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='5 7579 2.' metadata={'source': 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+page_content='8 9005 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='8 388127 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='1 24 50 25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='3 41022 MO MO 32 66 33 347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='7 337177 MO MO include the time for the two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Unsurprisingly, the number of states to be explored is a good predictor for the performance and the relative performance is as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 7 Conclusion This paper considers multi-environment MDPs with an arbitrary number of en- vironments and an almost-sure reachability objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' We show novel and tight complexity bounds and use these insights to derive a new algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' This algo- rithm outperforms approaches for POMDPs on a broad set of benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' For future work, we will apply an algorithm directly on the BOMDP [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 18 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' van der Vegt, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Jansen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Junges Data-Availability Statement Supplementary material related to this paper is openly available on Zenodo at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content='7560675 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Roman Andriushchenko, Milan Ceska, Sebastian Junges, Joost-Pieter Katoen, and Simon Stupinsk´y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' PAYNT: A tool for inductive synthesis of probabilistic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In CAV, volume 12759 of LNCS, pages 856–869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Sebastian Arming, Ezio Bartocci, Krishnendu Chatterjee, Joost-Pieter Katoen, and Ana Sokolova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Parameter-independent strategies for pmdps via pomdps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' In QEST, volume 11024 of LNCS, pages 53–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' 3.' 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Katoen, and Bernd Becker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Strategy synthesis for pomdps in robot planning via game-based abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=' Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} +page_content=', 66(3):1040– 1054, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNFIT4oBgHgl3EQfjSvo/content/2301.11296v1.pdf'} diff --git a/ZNAyT4oBgHgl3EQfvvli/vector_store/index.pkl 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CHOCANO♭, ANA LUZ´ON*, MANUEL A. MOR´ON♮, L. FELIPE PRIETO MART´INEZ† +Abstract. We point out how to use the classical characteristic method, that is used to +solve quasilinear PDE’s, to obtain the matrix exponential of some lower triangle infinite +matrices. We use the Lie Frechet structure of the Riordan group described in [4]. After +that we describe some linear dynamical systems in K[[x]] with a concrete involution being +a symmetry or a time-reversal symmetry for them. We take this opportunity to assign +some dynamical properties to the Pascal Triangle. +1. Introduction +Several phenomena may be modelled using systems of differential equations and a key +notion to get their solutions is the exponential of (finite) square matrices. Therefore, the +problem of computing these exponentials is relevant. Moreover, some crucial developments +of Linear Algebra have been motivated by the study of the matrix exponential. We recom- +mend [13] for a survey on this topic. +Furthermore, the matrix exponential lies in the core of Lie Theory and connects the +natural Lie algebra of all n × n matrices with the general linear group GL(n, K). We think +that [15] is a very good introductory text for this point of view. +By allowing concepts as manifolds modelled in infinite dimensional spaces (e.g., Banach +spaces, Frechet spaces etc.), the theory of Lie groups has been extended to the infinite +dimensional framework [12]. See [5] for some coherence relationships between infinite di- +mensional Lie group structures and pro-Lie group structures when both are shared by a +group. This is the case of the Riordan group. The previous facts motivated us to introduce +and develop in [4] a structure of infinite dimensional Lie group on the Riordan group. +In the following section we recall some basic facts about the Riordan group and its +Lie group and pro-Lie group structures described in [4]. Notice that although this group +appeared under this name more or less recently, the group structure and many of its +elements are latent in many developments of classical mathematics (special sequences of +numbers and polynomials, Umbral Calculus and much more). It is clear that, historically, +the first (and surely the best) known Riordan matrix is Pascal’s Triangle. Besides this, +now, there is a lot of historical names of mathematicians related to some Riordan matrices. +1 + +The aim of this note is to describe the matrix exponential, for some infinite lower triangu- +lar matrices, from the solution of certain partial differential equations, which we find using +the method of characteristics (see [7]). We do all of this within the framework of the Rior- +dan Lie group and the corresponding Lie algebra. Then, motivated by general symmetry +properties of dynamical systems ([8] and [14]), we describe some of those systems in K[[x]] +that come from the Riordan group with a very special Riordan involution as a symmetry or +time-reversal symmetry. We consider a sequence of linear ordinary differential equations in +Euclidean spaces as problems approaching certain partial differential equations, using for +that the pro-structures of both: the Riordan group and the corresponding Lie algebra. We +propose a geometric analysis of these problems and we enumerate symmetric properties of +a special sequence related to the Pascal Triangle. +Apart from the general description of the Riordan group contained in Section 2, we also +need, along the paper, some specific results from [4]. This is the reason why in Section 3 +and 4 we recall some particularly related results from [4] to make this note as self-contained +as possible. +Since our first approach to the Riordan group, in [11], is different from the usual one in +the literature, we have also different notation. This is the reason why we recommend our +previous works [11], [9], [10] and, of course, [4] (ordered chronologically) for information +about basic results and notation used herein. We still maintain the notation K for a field +although in this paper we are only considering K as the real numbers. This is because +many of the results and ideas can be translated, at least, to the case K = C, the field of +complex numbers. +2. Basic facts on the differentiable structure of the Riordan group +The results of this section can be found in [11], [9], [10] and [4]. +2.1. Riordan matrices and the Riordan group. The definition of Riordan matrix +and the related concept of Riordan group appeared in the foundational paper [16] due to +Shapiro, Getu, Woan, and Woodson. The original definition of a Riordan matrix given in +[16] is more restrictive than that used currently in the literature, which is precisely the one +that we are going to use herein. +A Riordan matrix is an infinite matrix D = (di,j)i,j∈N whose columns are the coefficients +of successive terms of a geometric progression in K[[x]] where the initial term is a formal +power series of order 0 and the common ratio is a formal power series of order 1 (so D +needs to be lower triangular and its diagonal needs to be a geometric progression in K). +We represent a Riordan matrix D by T(f | g), where f(x) = �∞ +k=0 fkxk and g(x) = +�∞ +k=0 gkxk are formal power series in K[[x]] with f(0) ̸= 0 and g(0) ̸= 0, so that di,j = [xi] xjf(x) +gj+1(x). +2 + +Consequently, the first term of the geometric progression is f(x) +g(x) and the common ratio is +x +g(x). In this terms, Pascal’s triangle is T(1 | 1 − x). +The above definition can be reinterpreted saying that the generating function of the j-th +column (starting at j = 0) of D is the formal power series xjf(x) +gj+1(x), which makes sense because +g(0) ̸= 0. Hence, D is a lower triangular matrix and it is invertible because f(0) ̸= 0. +In [16] it was stated one of the main results about Riordan matrices. Currently many +authors call it the Fundamental Theorem for Riordan matrices (FTRM). Let D = T(f | g) +be a Riordan matrix and let γ(x) = �∞ +k=0 γkxk be a power series in K[[x]]. Consider the +column vector c = (γ0, γ1, γ2, · · · )T. Then, the generating function of the matrix product +Dc is f(x) +g(x)γ( +x +g(x)). +This fact is represented by T(f | g)(γ) = f(x) +g(x)γ( +x +g(x)). A proof of this +result, using a special ultrametric space (K[[x]], d) can be found in [11, Proposition 19]. +The Riordan group (i.e., the set of all Riordan matrices with the usual product of matri- +ces), denoted by R(K) or shortly R, is a subgroup of the group of invertible infinite lower +triangular matrices with the usual product of matrices as the operation. The product is +given by +T(f | g)T(l | m) = T +� +fl +�x +g +� ��gm +�x +g +�� +, +where fl +�x +g +� +≡ f(x) · l +� x +g(x) +� +and analogously for the second term, and the inverse is +given by +(T(f | g))−1 ≡ T −1(f | g) = T +� +1 +f( x +A) +���A +� +, +where +� x +A +� +◦ +�x +g +� += +�x +g +� +◦ +� x +A +� += x. See [11, Proposition 20] for more details. +The sequence of the coefficients of the previous formal power series, denoted by A, is the +so-called A-sequence of T(f | g). Obviously, the A-sequence of T(f | g) depends only on +the power series g. Moreover, if A = � +k≥0 akxk, then +di,j = +i−j +� +k=0 +akdi−1,j−1+k +i, j ≥ 1. +2.2. The Lie and pro-Lie group structures on the Riordan group. Suppose that +K is the field of real or complex numbers, denoted by R and C respectively. +Let us +consider a natural way to give a completely metrizable topology in K[[x]], by means of +the identification KN ≡ K[[x]] obtained by passing from sequences to ordinary generating +functions and vice versa. +3 + +The topology considered in KN is always the product topology for the usual topology +in K. Therefore, we convert K[[x]] into a Frechet space, that is, a completely metrizable +locally convex linear topological space. +This is the starting point to describe a natural Frechet Lie group structure on the Riordan +group. Beside this, the Riordan group can be described as the inverse limit of an inverse +sequence of groups of finite matrices obtaining a pro-Lie group structure on the Riordan +group. +It is well known that any Riordan matrix is completely determined by its first column and +its A-sequence. In this way, any Riordan matrix D = (di,j)i,j∈N is defined by a sequence +u = (uk)k∈N with u0 ̸= 0, u1 ̸= 0, and u2k = xk, u2k+1 = ak, being xk = dk,0, A(x) = +� +n≥0 anxn and di,j = �i−j +k=0 akdi−1,j−1+k for j ≥ 1. We denote the matrix D described +above by ϕ∞(u). +Let us consider K with the usual Euclidean topology, the product topology in KN and +the basic open set +U∞ = +� +u = (uk)k∈N ∈ KN | u0 ̸= 0, u1 ̸= 0 +� +in KN. Set +ϕ∞ +: +U∞ +−→ +R(K) +u +�−→ +ϕ∞(u). +Obviously, ϕ∞ is a bijective function. So, we consider the unique topology on R(K), that +makes ϕ∞ a homeomorphism. Note that the topological space R(K), the locally convex +vector space KN and the map ϕ∞ : U∞ → R(K) fit all conditions to get: +Theorem 1. (R(K), (U∞, ϕ∞)) is a smooth manifold modelled on the locally convex +vector space KN. Moreover, R(K) with this smooth structure is a Lie group. +One of the main tools that we have used to get results on the Riordan group is to consider +it as the inverse limit of an inverse sequence of groups of finite matrices. A natural way +to do this is as follows. For every n ∈ N, consider the general linear group GL(n + 1, K) +formed by all (n+1)×(n+1) invertible matrices with coefficients in K. Since every Riordan +matrix is lower triangular, we can define a natural homomorphism Πn : R → GL(n + 1, K) +given by +Πn((di,j)i,j∈N) = (di,j)i,j=0,··· ,n. +For obvious reasons, we will refer to this homomorphism as the projection of the corre- +sponding Riordan matrix. +To describe the Riordan group as an inverse limit of an inverse sequence of groups of +finite matrices we use the results of [10]. We first consider the subgroup of GL(n + 1, K) +defined by Rn = Πn(R). +4 + +Definition 2. Let D = (di,j)i,j=0,···,n+1 ∈ Rn+1. We define Pn : Rn+1 → Rn by +Pn((di,j)i,j=0,1,···,n+1) = (di,j)i,j=0,···,n. +Pn(D) is obtained from D by deleting its last row and its last column. Pn is a group ho- +momorphism for every n because the matrices are lower triangular. Moreover, the diagram +below is commutative +R +Πn +� +Πn+1 +�● +● +● +● +● +● +● +● +● +Rn +Rn+1. +Pn +� +From this, we get +Theorem 3. The Riordan group R is isomorphic to lim +←−{(Rn)n∈N, (Pn)n∈N}. Conse- +quently, R is a pro-Lie group. +See [10] for more details and notation. +2.3. The Lie Algebra of the Riordan group. Again, using the results of [4] we obtain +a full and faithful representation of the Lie algebra L(R(K)) of the Lie group R(K). We +have that L = (ℓi,j)i,j∈N ∈ L(R(K)) if and only if L is lower triangular and there are two +sequences (χi)i∈N and (αi)i∈N such that +L = + + + + + + + + + + + + + + +χ0 +χ1 +χ0 + α0 +χ2 +χ1 + α1 +χ0 + 2α0 +... +... +... +... +χn−1 +χn−2 + αn−2 +χn−3 + 2αn−3 +· · · +χ0 + (n − 1)α0 +χn +χn−1 + αn−1 +χn−2 + 2αn−2 +· · · +χ1 + (n − 1)α1 +χ0 + nα0 +... +... +... +· · · +... +... +... + + + + + + + + + + + + + + +with the usual sum of matrices and the usual product by scalars in K. The Lie bracket is +[L1, L2] = L1L2 − L2L1. +We denote the matrix L above as L(χ, α), where χ(x) = +� +n≥0 +χnxn, α(x) = +� +n≥0 +αnxn, and +L(R(K)) = {L(χ, α) | χ, α ∈ K[[x]]} . +Proposition 4. Any L = L(χ, α) ∈ L(R(K)) induces a linear continuous map, denoted +again by L : KN → KN, given by L(h) = χ(x)h(x) + xα(x)h′(x) where h(x) = � +n≥0 hnxn. +5 + +The continuous linear map L(χ, α) can be viewed as a C∞ vector field in the Frechet +space KN under the canonical identification ThKN = KN in the tangent space at any h. +From this point of view we have the following proposition. +Proposition 5. The initial value problem +� +γ′(t) = L(χ, α)(γ(t)) +γ(0) = h +in KN has a unique solution given by +γ(t) = etL(χ,α)(h). +Consequently, there is a one-parameter group of Riordan matrices T(f(x, t) | g(x, t)) such +that +γ(t) = T(f(x, t) | g(x, t))(h) = f(x, t) +g(x, t)h +� +x +g(x, t) +� +. +In particular, {T(f(x, t) | g(x, t))}t∈R is an abelian subgroup of the Riordan group, because +it defines a continuous dynamical system (or flow) in K[[x]]. +Due to the special patterns followed by the matrices in the Lie Algebra and those in the +Riordan group, we named the corresponding section in [4] as Arithmetic vector fields and +geometric flows. +2.4. The partial differential equation induced by an element in the Lie Algebra. +For any t ∈ R and L ∈ L(R(K)) we have a well-defined Riordan matrix etL. Note that for +any t ∈ R, γ(t) ∈ K[[x]], where γ satisfies the conditions of Proposition 5. With this in +mind, another way to interpret the above proposition is as follows. +Corollary 6. Let χ, α ∈ K[[x]]. Then the unique solution of the initial value problem +� +∂u +∂t = χ(x)u(x, t) + xα(x) ∂u +∂x +u(x, 0) = h(x) +in K[[x, t]] is given by +u(x, t) = etL(χ,α)(h(x)) = f(x, t) +g(x, t)h +� +x +g(x, t) +� +. +3. From the matrix exponential etL to the solutions of the PDE +∂u +∂t = χ(x)u(x, t) + xα(x) ∂u +∂x and back +Once recalled some basic results from [4], we propose the following strategy. +Given any element L = L(χ, α) in the Lie Algebra of the Riordan group, we associate to +it the partial differential equation ∂u +∂t = χ(x)u(x, t) + xα(x) ∂u +∂x. At this point, we run into +the following dichotomy. +6 + +(a) If we are able to compute the one-parameter group etL and to recognize etL = +T(f(x, t) | g(x, t)) as a Riordan matrix for any t ∈ R, then this allows us to solve +in K[[x, t]] the initial value problem +� +∂u +∂t = χ(x)u(x, t) + xα(x) ∂u +∂x +u(x, 0) = h(x) +for any h ∈ K[[x]] because the solution is given by +u(x, t) = etL(χ(x),α(x))(h(x)) = f(x, t) +g(x, t)h +� +x +g(x, t) +� +. +(b) If, on the contrary, we are able to solve in K[[x, t]] the initial value problem +� +∂u +∂t = χ(x)u(x, t) + xα(x) ∂u +∂x +u(x, 0) = h(x) +for any h ∈ K[[x]], then we get the one-parameter subgroup etL(χ(x),α(x)) = T(f(x, t) | +g(x, t)). +We can compute both parameters f(x, t) and g(x, t), because +f(x,t) +g(x,t) is +the unique solution for the initial condition h(x) ≡ 1 and f(x,t) +g(x,t) +x +g(x,t) is the unique +solution for the initial condition h(x) = x. Evaluating now at t = 1, we have the +corresponding matrix exponential. +3.1. From the matrix exponential to the solution of the PDE. +Example 7. +(i) Consider the matrix +D = + + + + + + + + +1 +0 +0 +0 +· · · +0 +2 +0 +0 +· · · +0 +0 +3 +0 +· · · +0 +0 +0 +4 +· · · +... +... +... +... +... + + + + + + + + +. +Obviously, D = L(1, 1) is in L(R(K)). The partial differential equation associated +to the matrix D is +∂u +∂t = u(x, t) + x∂u +∂x +The solution of the corresponding initial value problem +� +∂u +∂t = u(x, t) + x∂u +∂x +u(x, 0) = h(x) +7 + +is given by u(x, t) = etD(h). It is clear, by definition, that +etD = + + + + + + + + +et +0 +0 +0 +· · · +0 +e2t +0 +0 +· · · +0 +0 +e3t +0 +· · · +0 +0 +0 +e4t +· · · +... +... +... +... +... + + + + + + + + +. +For any t ∈ R, we have that etD = T(1 | e−t) as a Riordan matrix. Therefore, the +solution is u(x, t) = T(1 | e−t)(h) = eth(xet). +(ii) In this more interesting example, we consider the matrix +H = + + + + + + + + +0 +0 +0 +0 +· · · +1 +0 +0 +0 +· · · +0 +2 +0 +0 +· · · +0 +0 +3 +0 +· · · +... +... +... +... +... + + + + + + + + +. +Note that H = L(x, x) and so H ∈ L(R(K)). The PDE induced by H is +∂u +∂t − x2∂u +∂x = xu(x, t) +The solution of the corresponding initial value problem +� +∂u +∂t = xu(x, t) + x2 ∂u +∂x +u(x, 0) = h(x) +is given by u(x, t) = etH(h). To compute this solution we are going to take advantage +of some previous work of another authors. Particularly, we are going to follow [1], +where the matrix H is called the creation matrix. Formula (9) in [1, Page 233] +computes etΛn−1(H). Moreover, it is obvious that Pn−1(etΛn(H)) = etΛn−1(H) for any +t ∈ R and any integer n ≥ 2 (see [4, Page 542] for the notation concerning Λn). +Hence, in our Riordan matrix notation, we get etH = T(1 | 1 − xt) and the solution +of the corresponding initial value problem is given by +u(x, t) = etH(h) = +1 +1 − xth +� +x +1 − xt +� +. +Evaluating at t = 1, we obtain eH = T(1 | 1 − x), which is the Pascal triangle. +8 + +3.2. From the solution of a PDE to the matrix exponential: +the Method of +Characteristics. To point out how to go from the solution of a PDE to the exponential +map of the Lie algebra to the Riordan group, we are going to deal with the following family +of significant examples. +For any couple of real numbers a, b and any non-negative integer number n, consider +the infinite lower triangular matrix La,b +n += (d(n) +i,j )i,j∈N, where d(n) +i,j = 0 if i − j ̸= n and +d(n) +i,j = a + jb if i − j = n. How can we compute or describe the exponential of each of the +matrices La,b +n ? +Note that La,b +n += L(axn, bxn). This means that they belong to the Lie algebra of the +Riordan group. Moreover, +La,b +0 += + + + + + + + + +a +0 +0 +0 +· · · +0 +a + b +0 +0 +· · · +0 +0 +a + 2b +0 +· · · +0 +0 +0 +a + 3b +· · · +... +... +... +... +... + + + + + + + + +, +La,b +1 += + + + + + + + + +0 +0 +0 +0 +· · · +a +0 +0 +0 +· · · +0 +a + b +0 +0 +· · · +0 +0 +a + 2b +0 +· · · +... +... +... +... +... + + + + + + + + +, +and so on. +Observe that D = L1,1 +0 +and H = L1,1 +1 , where D and H are the matrices +considered in the previous examples. The PDE induced by La,b +n +is +∂u +∂t − bxn+1∂u +∂x = axnu(x, t). +Note that if b = 0; then, etLa,0 +n += T(eatxn | 1) is a Toeplitz matrix (in the Riordan group) +for any t ∈ R. Consequently, the solution of the corresponding initial value problem +� +∂u +∂t = axnu(x, t) +u(x, 0) = h(x) +is given by u(x, t) = etLa,0 +n (h) = eatxnh(x). +From now on, assume b ̸= 0. Consider the matrix La,b +n +and the corresponding initial +value problem +(1) +� +∂u +∂t − bxn+1 ∂u +∂x = axnu(x, t) +u(x, 0) = h(x). +9 + +Let us use the Method of Characteristics to solve it (see [7], Chapter I). Suppose t = t(r, s), +x = x(r, s) and z = z(r, s). So we consider the following system of ODEs + + + + + +dt +ds = 1 +dx +ds = −bxn+1 +dz +ds = axnz +with initial conditions + + + + + +t(r, 0) = 0 +x(r, 0) = r +z(r, 0) = h(r) +Integrating the system and imposing the initial conditions we get + + + + + +t = s +1 +xn = 1+nbsrn +rn +z = (1 + nbsrn) +a +nbh(r) +and + + + + + + + +s = t +r = +x +n√ +1−nbtxn +z = +� +1 +1−bntxn +� a +nb h +� +x +n√ +1−nbtxn +� , +which yields that the solution of (1) is given by +u(x, t) = +� +1 +1 − bntxn +� a +nb +h +� +x +n√ +1 − nbtxn +� +or +u(x, t) = +n +�� +1 +1 − bntxn +� a +b +h +� +x +n√ +1 − nbtxn +� +. +All above in this subsection may be summarized as +Theorem 8. Suppose that a and b are two real numbers and that n is a non-negative +integer number. +Consider the infinite lower triangular matrix La,b +n += (d(n) +i,j )i,j∈N, where +d(n) +i,j = 0 if i − j ̸= n and d(n) +i,j = a + jb if i − j = n. Then, La,b +n +is in the Lie algebra of the +Riordan group and for any t ∈ R the Riordan matrix etLa,b +n +is given by +etLa,b +n = + + + +T +� +n� +(1 − bntxn) +b−a +b +| +n√ +1 − bntxn +� +if b ̸= 0 +T +� +eatxn | 1 +� +if b=0. +10 + +4. A special involution as symmetry or time-reversal symmetry for flows +in K[[x]] related to the Riordan group. +There is an involution M in the Riordan group playing a special role in such group. This +matrix is +M = T(−1| − 1) = + + + + + + + + +1 +0 +0 +0 +· · · +0 +−1 +0 +0 +· · · +0 +0 +1 +0 +· · · +0 +0 +0 +−1 +· · · +... +... +... +... +... + + + + + + + + +. +Considered as an endomorphism in K[[x]], M is continuous and has exactly two eigen- +values 1 and −1. The corresponding eigenspaces are E(1) = {h ∈ K[x]]/h(x) = h(−x)} +and E(−1) = {h ∈ K[x]]/ − h(x) = h(−x)}, i.e., they are the linear subspaces of even, +respectively odd, formal power series. Both of them are closed subspaces in K[[x]] and +we get K[[x]] = E(1) ⊕ E(−1) and (M − I) ◦ (M + I) = (M + I) ◦ (M − I) ≡ 0, where +I = T(1 | 1) is the identity. +The reason of the interest about M is because it is used for defining what is known as a +pseudo-involution in the Riordan group. Following [3], we say that a Riordan matrix R is +a pseudo-involution if the product RM is an involution, that is, RMRM = I. +In Group Theory we have the definitions of reversible and strongly reversible elements +(see [14]). We recall it here for completeness. +Definition 9. +(i) An element g of a group G is said to be reversible in G if there is +another element h of G such that +hgh−1 = g−1. +In this situation we also say that h reverses g and h is a reverser of g. +(ii) An element g of a group G is said to be strongly reversible in G if there is an +involution h of G such that +hgh−1 = g−1. +Proposition 10. If R is a pseudo-involution, then R is strongly reversible in the Riordan +group and is the product of two involutions. +Proof. Since RMRM = I we directly obtain, multiplying on the left by the inverse R−1, +that MRM = R−1 so R is strongly reversible. Moreover, R−1M is an involution because +(R−1M)−1 = MR = MMR−1M = R−1M. Consequently, we have that R = MR−1M. +□ +The above proposition tells us that the pseudo-involutions are particular examples of +strongly reversible elements in the Riordan group and M is a reverser for any of them. +11 + +Consider the left and right translations in R(K) given, respectively, by +LD +: +R(K) +−→ +R(K) +X +�−→ +LD(X) = DX, +RD +: +R(K) +−→ +R(K) +X +�−→ +RD(X) = XD. +Since the product is a C∞-function both LD and RD are diffeomorphisms. We need to +recall the following facts from [4]. +Proposition 11. Let T(f | g) be a Riordan matrix. The tangent space TT(f|g)R(K) to +R(K) at T(f | g) is given by +TT(f|g)R(K) = {T(f | g)L(χ, α)|χ, α ∈ K[[x]]} = {L(χ, α)T(f | g)|χ, α ∈ K[[x]]}. +Moreover, the conjugation by T(f | g) defined by +conjT(f|g) +: +R(K) +−→ +R(K) +X +�−→ +T(f | g)XT −1(f | g) +is a C∞-diffeomorphism and its tangent (or differential) map at the identity +DconjT(f|g)(I) +: +L(R(K)) +−→ +L(R(K)) +is given by +DconjT(f|g)(I)(L(χ, α)) = T(f | g)L(χ, α)T −1(f | g). +Finally, for any t ∈ K we have +etDconjT (f|g)(I)(L(χ,α)) = conjT (f|g)(etL(χ,α)). +From this, we deduce +Corollary 12. Given T(f | g) ∈ R(K) and L(χ, α) ∈ L(R(K)), there exists a unique +L(˜χ, ˜α) ∈ L(R(K)) such that +T(f | g)L(χ, α) = L(˜χ, ˜α)T(f | g) +or equivalently +DconjT(f|g)(I)(L(χ, α)) = L(˜χ, ˜α). +Moreover, +(2) +˜χ = +χ +� +x +g +� +(g − xg′)f − xα +� +x +g +� +(f ′g − g′f) +f(g − xg′) +, +˜α = +gα +� +x +g +� +g − xg′ . +As a consequence, we obtain +12 + +Proposition 13. Consider the involution M and the C∞-diffeomorphism +conjM +: +R(K) +−→ +R(K) +X +�−→ +MXM. +Then +(i) the tangent (or differential) map at identity +DconjM(I) +: +L(R(K)) +−→ +L(R(K)) +is a linear involution +(ii) for any couple of real numbers a and b such that a ̸= 0 and b ̸= 0 the matrix La,b +n +is +an eigenvector of DconjM(I) corresponding to the eigenvalue −1 if n is odd and an +eigenvector of DconjM(I) corresponding to the eigenvalue 1 if n is even (including +n = 0). +Proof. It is clear that DconjM(I) ◦ DconjM(I) = IL(R(K)) because M is an involution in +R(K), where IL(R(K)) represents the identity map in L(R(K)). To prove (ii), recall that +M = T(−1 | −1) and that La,b +n = L(axn, bxn). From the previous corollary we get +DconjM(I)(L(χ, α)) = L(˜χ, ˜α) +where ˜χ and ˜α are given by (2). In this case, f = g = −1 and χ(x) = axn, α(x) = bxn. For +the case that n even, the announced result is obvious. The same also holds for the case n +is odd because −L(χ, α) = L(−χ, −α) for any χ, α in K[[x]]. +□ +Finally, we obtain the following (here we use [8] for some of the definitions appearing +below). +Theorem 14. Let a and b two real numbers such that a = b = 0 does not hold. Suppose +also that n is a non-negative integer number. Consider the infinite lower triangular matrix +La,b +n . Then we have the following. +(i) If n is even, the one-parameter subgroup {etLa,b +n }t∈R is contained in the centralizer +of the involution M. In other words, M is a symmetry for the flow {etLa,b +n }t∈R in +K[[x]]. +(ii) If n is odd, any element in the one-parameter subgroup {etLa,b +n }t∈R is a pseudo- +involution and the involution M is a time-reversal symmetry of the flow {etLa,b +n }t∈R +in K[[x]]. +Proof. Suppose n is even. Using Proposition 13, we have DconjM(I)(La,b +n ) = La,b +n . Hence +etLa,b +n = etDconjM (I)(La,b +n ) = MetLa,b +n M. +The last equality above is a consequence of Proposition 11. Consequently, etLa,b +n +is in the +centralizer of the involution M for every t ∈ R. +13 + +Suppose now that n is odd. By analogous arguments, we first have that DconjM(I)(La,b +n ) = +−La,b +n +and then +e−tLa,b +n = etDconjM(I)(La,b +n ) = MetLa,b +n M. +Therefore, the Riordan matrix etLa,b +n +is reversible for any t ∈ R and the involution M is a +reverser for all of them. This fact implies that etLa,b +n is strongly reversible and that etLa,b +n is a +pseudo-involution because etLa,b +n M is an involution (etLa,b +n MetLa,b +n M = etLa,b +n e−tLa,b +n = I). +□ +Corollary 15. The same as in the above theorem is true, word by word, changing the +involution M by the involution −M = T(1 | −1). +5. On some dynamical properties of Pascal Triangle +Using some of the results obtained herein we can get some new information about the +Riordan group. For example, in [4] we recognized as a subgroup of the Riordan group, the +substitution group of formal power series. This group was introduced in [6] (see also [2] for +a good survey about it) where results about topological generation are established. One +can notice at once that some of our one-parameter groups {etLa,b +n }t∈R are involved in [6, +Section 3]. In this section, we focus only on the Pascal Triangle and assign to it others of +the many properties related to patterns and symmetries that it has. +We will use the pro-Lie group structure of the Riordan group. +So, we will consider +elements of the Riordan group and of the Lie algebra as approximated by the components +of the corresponding points in the inverse limit interpretation of both R(K) and L(R(K)) +(see [4] for the pro-structure of L(R(K))). +To clarify the above approaching process recall that related to any problem +(3) +γ′(t) = L(γ(t)) +with L ∈ L(R(K)) or, equivalently +∂u +∂t = χ(x)u(x, t) + xα(x)∂u +∂x, +if L = L(χ(x), α(x)), we have a sequence of finite dimensional problems, denoted by +{(3)n}n∈N. We call this sequence as the sequence of approaching problems of (3). +Problem (3)n: Approaching problems. +Consider the Euclidean space Rn+1. For any +x ∈ Rn+1 denote by x = (x0, x1, · · · , xn) to its usual components. Suppose xT represents +the transpose matrix of x. Let x : R −→ Rn+1 be any derivable curve given by x(t) = +(x0(t), x1(t), · · · , xn(t)). We denote by x′(t) = (x′ +0(t), x′ +1(t), · · · , x′ +n(t)) the derivative of x +at t, where x′ +i is the usual derivative of a real function with real variable for any i = 0, · · · , n. +When we refer to the approaching problem (3)n, we refer to the linear differential equation +x′T = DΠn(I)(L)xT . +14 + +In the above equation DΠn(I) represents the differential at the identity I in the Riordan +group R(K) of the projection +Πn : R(K) −→ Rn(K) +in the pro-Lie group structure of R(K), see again [4] if needed. +5.1. The approaching problems related to Pascal Triangle. Recall that, Pascal Tri- +angle, is the time 1 map of the dynamical system generated by the linear differential +equation in K[[x]] +(4) +γ′(t) = L1,1 +1 (γ(t)), +where L1,1 +1 += L(x, x), or, equivalently, +∂u +∂t = xu + x2∂u +∂x +in K[[x, t]]. Recall also that γ(t) ∈ R[[x]] for any t ∈ R and that L1,1 +1 (γ(t)) = xγ(t)+x2 dγ(t) +dx , +where +d +dx is the formal derivative in R[[x]]. While γ : R −→ R[[x]] is a curve and γ′(t) is the +usual derivative in t, when we consider R[[x]] identified with RN with the product topology. +Using the pro-Lie group structure in R(K), the corresponding pro-Lie algebra structure in +L(R(K)) and recalling that +L1,1 +1 += + + + + + + + + +0 +0 +0 +0 +· · · +1 +0 +0 +0 +· · · +0 +2 +0 +0 +· · · +0 +0 +3 +0 +· · · +... +... +... +... +... + + + + + + + + +. +We can associate to problem (4) a countably infinite family of finite dimensional problems +(4)n in the euclidean space Rn+1 for any non-negative integer n. As we said before, we will +interpret the family of problems {(4)n}n∈N as approaching the problem (4) when n tends +to ∞. For the first few values of n we have: +(4)0 for n = 0, we consider the differential equation x′ +0 = 0 in the one dimensional +euclidean space R. +(4)1 for n = 1, we consider the differential equation in R2 +� +x′ +0(t) +x′ +1(t) +� += +� +0 +0 +1 +0 +� � +x0(t) +x1(t) +� +(4)2 for n = 2, we consider the differential equation in R3 + + + +x′ +0(t) +x′ +1(t) +x′ +2(t) + + + = + + + +0 +0 +0 +1 +0 +0 +0 +2 +0 + + + + + + +x0(t) +x1(t) +x2(t) + + + +15 + +(4)3 for n = 3, we consider the differential equation in R4 + + + + + +x′ +0(t) +x′ +1(t) +x′ +2(t) +x′ +3(t) + + + + + = + + + + + +0 +0 +0 +0 +1 +0 +0 +0 +0 +2 +0 +0 +0 +0 +3 +0 + + + + + + + + + + +x0(t) +x1(t) +x2(t) +x3(t) + + + + + +and so on. +The problem (4)0 is very easy to analyze. Any point in the phase space R is an equilibrium +point. The phase flow is trivial and nothing is moving under it. +Let us now consider (4)1. The equilibrium points in this case are just the points in the +x1-axe, which means that they are of the form (0, b), where b ∈ R. All of them are unstable +in the Liapunov sense. The rest of the orbits are the straight lines x0 = a for a fix non-null +a ∈ R. Through any orbit the motion is uniform. In the semiplane x0 > 0 the sense of +the motion is increasing, respect to the x1-axe as t increases, i.e., the particle comes from +the −∞ part respect to the x1-axe and goes to (positive) ∞ of such axe as t increases. In +the semiplane x0 < 0 the sense of the motion is the opposite one. The constant speed of +the motion in the orbit x0 = a is | a |, the absolute value of a. Particles move quickly for +large values of | a | and slowly for small values | a | and they do not move in the x1-axe. +Therefore, there seems to be something in the x1-axe slowing down the motion. We can +deduce all above only knowing that the corresponding velocity vector field for the equation +(4)1, in the euclidean plane, is given by X(a, b) = (0, a). We can also compute, quickly and +easily the solution of problem (4)1 with initial condition x0 = a and x1 = b because the +corresponding matrix is nilpotent. It is the curve x(t) = (a, at + b). +From Theorem 14 and Corollary 15, we get that Π1(M) = +� +1 +0 +0 +−1 +� +and Π1(−M) = +� +−1 +0 +0 +1 +� +are time-reversal symmetries for the flow generated by the equation (4)1. Con- +sider the orbit θ(a, b) = {(a, at + b), a ̸= 0, b, t ∈ R} of (4)1. Then it is symmetric respect +to the involution Π1(M), see Definition 4.1 in [8]. This, in particular, means that if we +transform the orbit θ(a, b) by Π1(M) we get again θ(a, b) but the parametrization obtained +by means of t is not a solution. However, if we finally change t by −t in such obtained +curve we have a solution of (4)1; in this case the initial value at t = 0 is (a, −b). Of course, +θ(a, b) = θ(a, −b). +What is the behaviour under the action of the involution Π1(−M)? if we transform +the orbit θ(a, b) by means of Π1(−M), we get another different orbit of (4)1. In fact, we +obtain Π1(−M)(θ(a, b)) = θ(−a, b) and θ(a, b) ̸= θ(−a, b). Again, the parametrization so +obtained by means of t is not a solution of (4)1. But, again, if we change t by −t we get +16 + +another solution but with different orbit. Finally see that the composition of both time +reversal symmetries is really a symmetry for (4)1. Then we obtain that −I is a symmetry +and it implies that if x(t) = (a, at + b) is a solution of the problem with initial condition +x(0) = (a, b), then −x(t) = −I(x(t)) is a solution with initial condition (−a, −b). +Remark 16. Note that any orbit x0 = a ̸= 0 of (4)1 is symmetric respect to the time +reversal symmetry Π1(M). On the contrary, no orbit (except for equilibrium points) is +symmetric respect to the time reversal symmetry Π1(−M). +We propose to the reader the beautiful exercise of analysing the problem (4)2 and the +role of the involutions Π2(M) = + + + +1 +0 +0 +0 +−1 +0 +0 +0 +1 + + + and Π2(−M) = + + + +−1 +0 +0 +0 +1 +0 +0 +0 +−1 + + + as time +reversal symmetries for (4)2. The geometric analysis of this problem is, obviously, richer +than that of (4)1. In the problem (4)1 the involutions Π1(M) and Π1(−M) are conjugated +and they represent reflections about different axes. But in (4)2, Π2(M) is a reflection about +the plane x1 = 0, which is a plane of fixed points of Π2(M), and Π2(−M) is a rotation of +angle π around the x1-axe. Of course they are not conjugated. Analysing this case, one +can see first that the equilibrium points are those of the form (0, 0, c) and that under the +flow generated by (4)2, particles are moving within affine hyperplanes x0 = a. When a = 0 +we obtain the motion described in (4)1 in this hyperplane. For a ̸= 0 the orbits of points +are parabolas. In this case, the velocity vector field is given by X(a, b, c) = (0, a, 2b) and +the solution of (4)2 with initial conditions x0(0) = a, x1(0) = b and x2(0) = c is given +by x(t) = (a, at + b, at2 + 2bt + c). As in the previous case, any orbit, which is not an +equilibrium point, is symmetric with respect to the involution Π2(M), but any non-trivial +orbit is transformed by Π2(−M) into another different orbit. In both cases, if after the +transformation, we change t by −t we obtain new solutions of the problem (4)2. Finally +−I : R3 −→ R3 is a symmetry for the problem. +5.2. Facts and/or conjectures and/or speculations on the problems (4)n and (4). +The flow induced by problem (4) in K[[x]] is given by +Φ : K[[x]] × R −→ K[[x]] +where Φ(h, t) = etL1,1 +1 (h) = +1 +1−xth +� +x +1−xt +� +, while the flow generated by the problem (4)n in +Rn+1 is +Φn : Rn+1 × R −→ Rn+1 +whose matrix expression is Φn(x, t) = Πn(etL1,1 +1 )xT. +We now are going to state, without proofs, properties related to problems (4)n and (4). +This is the reason why we entitled this subsection as we did. +17 + +Proposition 17. (Dynamical properties related to problems (4)n) Let n be a +non-negative integer number, then we have the following properties. +(i) The orbit of any point a = (a0, a1, · · · , an) in Rn+1 is contained in the affine hy- +perplane x0 = a0. Moreover if a0 = 0 and if one consider Rn = {(x0, x1, · · · , xn) ∈ +Rn+1/x0 = 0}, the motion induced by Φn in Rn is Φn−1. +(ii) The equilibrium points in (4)n are those in the xn-axis and if n > 0 all of them are +unstable in the Liapunov sense. +(iii) The non-trivial orbits in (4)n, i.e., those which are not equilibrium points, are related +to the so called moment curve in the corresponding hyperplane. In particular the +solution of (4)n with initial condition (1, 0, · · · , 0) ∈ Rn+1 is x(t) = (1, t, t2 · · · , tn) +which is a copy of the corresponding moment curve in the hyperplane x0 = 1. +(iv) Any non-trivial orbit in (4)n is symmetric with respect to the time-reversal symmetry +Πn(M) and no one of them is symmetric respect to Πn(−M). Anyway, if we have a +non-trivial solution of (4)n, i.e., a non-constant one, and we transform it by any of +the involutions Πn(M) or Πn(−M) and then change t by −t we get another solution +of (4)n. +(v) −I : Rn+1 −→ Rn+1 is a symmetry for the equation (4)n. +Motivated by the previous result and knowing that Pascal triangle is the time one map +of the flow Φ, we can state: +Proposition 18. (Some dynamical properties of Pascal Triangle) +(i) The flow Φ has not equilibrium points up the null power series, that is, all the +coefficients of the power series are null. +(ii) The orbit of the formal power series constantly 1, by means of Φ, is the set of +geometric progressions { +1 +1−tx}t∈R. Then one can think about 1 moving through the +set of the germs of analytic functions, at x = 0, because for any t ∈ R the function +ft(x) = +1 +1−tx for x ∈ (− 1 +|t|, 1 +|t|) is analytic at x = 0. Moreover, this orbit can be seen +as the asymptotic behaviour, as n goes to ∞, of the moment curves. +(iii) The Riordan involution M and −M are time-reversal symmetries for the flow Φ. +Any orbit of problem (4) is symmetric respect to M and no orbit, except for the +unique equilibrium point, is symmetric respect to −M. Anyway, if we transform +any solution of (4) by means of M or −M and then change t by −t we get another +solution of the problem. +(iv) −I : K[[x]] −→ K[[x]] is a symmetry for the equation (4). +To finish this paper note the following. +Claim: Any non-trivial orbit of the problem (4)n has empty α-limit set and ω-limit set. +On the other hand, the problem (4)n has time reversal symmetries. This fact allows us to +18 + +think that there should be relationships between both limit sets. We then decided to force +the corresponding flows, by means of considering a compactification of the correspond- +ing phase spaces, to get non-empty α-limit and ω-limit sets and to look for relationships +between them. We proceed as follows. +Consider the one point (or Alexandroff) compactification of Rn+1 which is, topologically, +the n + 1-dimensional sphere Sn+1. Let us denote by ∞ the added point. Note that any +t-map of the flow Φn, Φt +n, can be continuously extended to a map � +Φtn : Sn+1 −→ Sn+1 +defining only � +Φt +n(∞) = ∞. In this way we get a dynamical system +� +Φn : Sn+1 × R −→ Sn+1. +For every non-negative integer n we can also extend the time reversal symmetries Πn(M) +and Πn(−M) to continuous maps � +Πn(M) and +� +Πn(−M) from Sn+1 onto itself imposing that +the point ∞ is a fixed point for both of them. We can identify, topologically, Rn+1 with +Sn+1\{∞} (which is a dense subset of Sn+1). With all these constructions we have +Proposition 19. Let n be a non-negative integer number. Then we have the following. +(i) The maps � +Πn(M) and +� +Πn(−M) are continuous involutions in Sn+1 and they are +time-reversal symmetries for the dynamical system � +Φn. +(ii) The orbits of the dynamical system � +Φn are those of Φn (after the mentioned identi- +fication) plus {∞} which is an equilibrium point. Moreover, every non-trivial orbit +of � +Φn is homoclinic, being the point ∞ an attractor and a repeller of all of them. +Consequently, the α-limit and the ω-limit sets of any non-trivial orbit coincide. +References +[1] L. Aceto, D.Trigiante. The matrices of Pascal and other greats. Amer. Math. Monthly +108(3) (2001) 232–245. +[2] I.K. Babenko. Algebra, geometry and topology of the substitution group of formal power +series. Russian Math. Surveys 68 (1) (2013) 1-68. +[3] N.T. Cameron. and A. Nkwanta. On some (pseudo) involutions in the Riordan Group. +Journal of Integer Sequences. Vol. 8 (2005) Article 05.3.7. +[4] G.-S. Cheon, A. Luz´on, M. A. Mor´on, L. F. Prieto-Martinez and M. Song Finite and +infinite dimensional Lie group structures on Riordan groups. Adv. Math. 319 (2017) +522-566. +[5] K.H. Hofmann and K.-H. Neeb. Pro-Lie groups which are infinite dimensional Lie +groups. Math. Proc. Camb. Phil. Soc. 146 (2009) 351-378. +[6] S.A. Jennings. Substitution groups of formal power series Canadian J. Math. 6 (1954) +325-340 +19 + +[7] F. John, Partial Differential Equations, Third Edition. Applied Mathematical Sciences, +1. Springer-Verlag, New York, 1980. ISBN: 0-387-90327-5 +[8] J. Lamb, J.A.G. Roberts. +Time-reversal symmetry in dynamical systems: a survey +Phys. D 112 (1-2) (1998) 1-39. +[9] A. Luz´on. +Iterative processes related to Riordan arrays: The reciprocation and the +inversion of power series. Discrete Math. 310 (2010) 3607-3618. +[10] A. Luz´on, D. Merlini, M. A. Mor´on, L. F. Prieto-Martinez and R. Sprugnoli. Some +inverse limit approaches to the Riordan group. +Linear Algebra Appl. +491 +(2016) +239-262. +[11] A. Luz´on and M. A. Mor´on. Ultrametrics, Banach’s fixed point theorem and the Rior- +dan group. Discrete Appl. Math. 156 (2008) 2620-2635. +[12] J. Milnor, Remarks on infinite-dimensional Lie groups. pp. 1007-1057, In: B. DeWitt, +R. Stora (eds), “Relativit´e, groupes et topologie II” (Les Houches, 1983), North Hol- +land, 1984. +[13] C. Moler, C. Van Loan. Nineteen Dubious Way to Compute the Exponential of a +Matrix,Twenty-Five Years Later. Siam Rev. 45 (2003) 3-49. +[14] A.G. O’Farrell and I. Short. Reversibility in dynamics and group theory. London Math- +ematical Society Lecture Note Series, 416. Cambridge University Press, Cambridge, +2015. ISBN: 978-1-107-44288-7 +[15] W. Rossmann. Lie groups. An introduction through linear groups. Oxford Graduate +text in Mathematics. Oxford University Press, Oxford. 2002 . +[16] L. W. Shapiro, S. Getu, W.J. Woan and L. Woodson. The Riordan group. Discrete +Appl. Math. 34 (1991) 229-239. +♭ Departamento de Matem´atica Aplicada, Ciencia e Ingenier´ıa de los Materiales y Tec- +nolog´ıa Electr´onica, ESCET Universidad Rey Juan Carlos, 28933 M´ostoles (Madrid), +Spain +Email address: pedro.chocano@urjc.es +*Departamento de Matem´atica Aplicada. Universidad Polit´ecnica de Madrid (Spain). +Email address: anamaria.luzon@upm.es +♮ Departamento de Algebra, Geometr´ıa y Topolog´ıa. +Universidad Complutense de +Madrid and Instituto de Matem´atica Interdisciplinar (IMI)(Spain). +Email address: mamoron@mat.ucm.es +† Departamento de Matem´atica Aplicada, ETS Arquitectura, Universidad Polit´ecnica +de Madrid (Madrid). +Email address: luisfelipe.prieto@upm.es +20 + diff --git a/aNAyT4oBgHgl3EQfW_dN/content/tmp_files/load_file.txt b/aNAyT4oBgHgl3EQfW_dN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbbd3ee69b19570362537e428d1e11d275e31888 --- /dev/null +++ b/aNAyT4oBgHgl3EQfW_dN/content/tmp_files/load_file.txt @@ -0,0 +1,609 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf,len=608 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='00173v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='DS] 31 Dec 2022 CHARACTERISTIC CURVES AND THE EXPONENTIATION IN THE RIORDAN LIE GROUP: A CONNECTION THROUGH EXAMPLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' PEDRO J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' CHOCANO♭, ANA LUZ´ON*, MANUEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' MOR´ON♮, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' FELIPE PRIETO MART´INEZ† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We point out how to use the classical characteristic method, that is used to solve quasilinear PDE’s, to obtain the matrix exponential of some lower triangle infinite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We use the Lie Frechet structure of the Riordan group described in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' After that we describe some linear dynamical systems in K[[x]] with a concrete involution being a symmetry or a time-reversal symmetry for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We take this opportunity to assign some dynamical properties to the Pascal Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Introduction Several phenomena may be modelled using systems of differential equations and a key notion to get their solutions is the exponential of (finite) square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Therefore, the problem of computing these exponentials is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, some crucial developments of Linear Algebra have been motivated by the study of the matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We recom- mend [13] for a survey on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Furthermore, the matrix exponential lies in the core of Lie Theory and connects the natural Lie algebra of all n × n matrices with the general linear group GL(n, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We think that [15] is a very good introductory text for this point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' By allowing concepts as manifolds modelled in infinite dimensional spaces (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=', Banach spaces, Frechet spaces etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' ), the theory of Lie groups has been extended to the infinite dimensional framework [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' See [5] for some coherence relationships between infinite di- mensional Lie group structures and pro-Lie group structures when both are shared by a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This is the case of the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The previous facts motivated us to introduce and develop in [4] a structure of infinite dimensional Lie group on the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In the following section we recall some basic facts about the Riordan group and its Lie group and pro-Lie group structures described in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Notice that although this group appeared under this name more or less recently, the group structure and many of its elements are latent in many developments of classical mathematics (special sequences of numbers and polynomials, Umbral Calculus and much more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' It is clear that, historically, the first (and surely the best) known Riordan matrix is Pascal’s Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Besides this, now, there is a lot of historical names of mathematicians related to some Riordan matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 1 The aim of this note is to describe the matrix exponential, for some infinite lower triangu- lar matrices, from the solution of certain partial differential equations, which we find using the method of characteristics (see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We do all of this within the framework of the Rior- dan Lie group and the corresponding Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then, motivated by general symmetry properties of dynamical systems ([8] and [14]), we describe some of those systems in K[[x]] that come from the Riordan group with a very special Riordan involution as a symmetry or time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We consider a sequence of linear ordinary differential equations in Euclidean spaces as problems approaching certain partial differential equations, using for that the pro-structures of both: the Riordan group and the corresponding Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We propose a geometric analysis of these problems and we enumerate symmetric properties of a special sequence related to the Pascal Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Apart from the general description of the Riordan group contained in Section 2, we also need, along the paper, some specific results from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This is the reason why in Section 3 and 4 we recall some particularly related results from [4] to make this note as self-contained as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Since our first approach to the Riordan group, in [11], is different from the usual one in the literature, we have also different notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This is the reason why we recommend our previous works [11], [9], [10] and, of course, [4] (ordered chronologically) for information about basic results and notation used herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We still maintain the notation K for a field although in this paper we are only considering K as the real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This is because many of the results and ideas can be translated, at least, to the case K = C, the field of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Basic facts on the differentiable structure of the Riordan group The results of this section can be found in [11], [9], [10] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Riordan matrices and the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The definition of Riordan matrix and the related concept of Riordan group appeared in the foundational paper [16] due to Shapiro, Getu, Woan, and Woodson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The original definition of a Riordan matrix given in [16] is more restrictive than that used currently in the literature, which is precisely the one that we are going to use herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' A Riordan matrix is an infinite matrix D = (di,j)i,j∈N whose columns are the coefficients of successive terms of a geometric progression in K[[x]] where the initial term is a formal power series of order 0 and the common ratio is a formal power series of order 1 (so D needs to be lower triangular and its diagonal needs to be a geometric progression in K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We represent a Riordan matrix D by T(f | g), where f(x) = �∞ k=0 fkxk and g(x) = �∞ k=0 gkxk are formal power series in K[[x]] with f(0) ̸= 0 and g(0) ̸= 0, so that di,j = [xi] xjf(x) gj+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2 Consequently, the first term of the geometric progression is f(x) g(x) and the common ratio is x g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this terms, Pascal’s triangle is T(1 | 1 − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The above definition can be reinterpreted saying that the generating function of the j-th column (starting at j = 0) of D is the formal power series xjf(x) gj+1(x), which makes sense because g(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Hence, D is a lower triangular matrix and it is invertible because f(0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In [16] it was stated one of the main results about Riordan matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Currently many authors call it the Fundamental Theorem for Riordan matrices (FTRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let D = T(f | g) be a Riordan matrix and let γ(x) = �∞ k=0 γkxk be a power series in K[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the column vector c = (γ0, γ1, γ2, · · · )T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then, the generating function of the matrix product Dc is f(x) g(x)γ( x g(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This fact is represented by T(f | g)(γ) = f(x) g(x)γ( x g(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' A proof of this result, using a special ultrametric space (K[[x]], d) can be found in [11, Proposition 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The Riordan group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=', the set of all Riordan matrices with the usual product of matri- ces), denoted by R(K) or shortly R, is a subgroup of the group of invertible infinite lower triangular matrices with the usual product of matrices as the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The product is given by T(f | g)T(l | m) = T � fl �x g � ��gm �x g �� , where fl �x g � ≡ f(x) · l � x g(x) � and analogously for the second term, and the inverse is given by (T(f | g))−1 ≡ T −1(f | g) = T � 1 f( x A) ���A � , where � x A � �x g � = �x g � � x A � = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' See [11, Proposition 20] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The sequence of the coefficients of the previous formal power series, denoted by A, is the so-called A-sequence of T(f | g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Obviously, the A-sequence of T(f | g) depends only on the power series g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, if A = � k≥0 akxk, then di,j = i−j � k=0 akdi−1,j−1+k i, j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The Lie and pro-Lie group structures on the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Suppose that K is the field of real or complex numbers, denoted by R and C respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let us consider a natural way to give a completely metrizable topology in K[[x]], by means of the identification KN ≡ K[[x]] obtained by passing from sequences to ordinary generating functions and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 3 The topology considered in KN is always the product topology for the usual topology in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Therefore, we convert K[[x]] into a Frechet space, that is, a completely metrizable locally convex linear topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This is the starting point to describe a natural Frechet Lie group structure on the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Beside this, the Riordan group can be described as the inverse limit of an inverse sequence of groups of finite matrices obtaining a pro-Lie group structure on the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' It is well known that any Riordan matrix is completely determined by its first column and its A-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this way, any Riordan matrix D = (di,j)i,j∈N is defined by a sequence u = (uk)k∈N with u0 ̸= 0, u1 ̸= 0, and u2k = xk, u2k+1 = ak, being xk = dk,0, A(x) = � n≥0 anxn and di,j = �i−j k=0 akdi−1,j−1+k for j ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We denote the matrix D described above by ϕ∞(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let us consider K with the usual Euclidean topology, the product topology in KN and the basic open set U∞ = � u = (uk)k∈N ∈ KN | u0 ̸= 0, u1 ̸= 0 � in KN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Set ϕ∞ : U∞ −→ R(K) u �−→ ϕ∞(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Obviously, ϕ∞ is a bijective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' So, we consider the unique topology on R(K), that makes ϕ∞ a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that the topological space R(K), the locally convex vector space KN and the map ϕ∞ : U∞ → R(K) fit all conditions to get: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (R(K), (U∞, ϕ∞)) is a smooth manifold modelled on the locally convex vector space KN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, R(K) with this smooth structure is a Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' One of the main tools that we have used to get results on the Riordan group is to consider it as the inverse limit of an inverse sequence of groups of finite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' A natural way to do this is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For every n ∈ N, consider the general linear group GL(n + 1, K) formed by all (n+1)×(n+1) invertible matrices with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Since every Riordan matrix is lower triangular, we can define a natural homomorphism Πn : R → GL(n + 1, K) given by Πn((di,j)i,j∈N) = (di,j)i,j=0,··· ,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For obvious reasons, we will refer to this homomorphism as the projection of the corre- sponding Riordan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' To describe the Riordan group as an inverse limit of an inverse sequence of groups of finite matrices we use the results of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We first consider the subgroup of GL(n + 1, K) defined by Rn = Πn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 4 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let D = (di,j)i,j=0,···,n+1 ∈ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We define Pn : Rn+1 → Rn by Pn((di,j)i,j=0,1,···,n+1) = (di,j)i,j=0,···,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Pn(D) is obtained from D by deleting its last row and its last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Pn is a group ho- momorphism for every n because the matrices are lower triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, the diagram below is commutative R Πn � Πn+1 �● Rn Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Pn � From this, we get Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The Riordan group R is isomorphic to lim ←−{(Rn)n∈N, (Pn)n∈N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Conse- quently, R is a pro-Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' See [10] for more details and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The Lie Algebra of the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Again, using the results of [4] we obtain a full and faithful representation of the Lie algebra L(R(K)) of the Lie group R(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We have that L = (ℓi,j)i,j∈N ∈ L(R(K)) if and only if L is lower triangular and there are two sequences (χi)i∈N and (αi)i∈N such that L = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed χ0 χ1 χ0 + α0 χ2 χ1 + α1 χ0 + 2α0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' χn−1 χn−2 + αn−2 χn−3 + 2αn−3 · · χ0 + (n − 1)α0 χn χn−1 + αn−1 χn−2 + 2αn−2 · · χ1 + (n − 1)α1 χ0 + nα0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 with the usual sum of matrices and the usual product by scalars in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The Lie bracket is [L1, L2] = L1L2 − L2L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We denote the matrix L above as L(χ, α), where χ(x) = � n≥0 χnxn, α(x) = � n≥0 αnxn, and L(R(K)) = {L(χ, α) | χ, α ∈ K[[x]]} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Any L = L(χ, α) ∈ L(R(K)) induces a linear continuous map, denoted again by L : KN → KN, given by L(h) = χ(x)h(x) + xα(x)h′(x) where h(x) = � n≥0 hnxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 5 The continuous linear map L(χ, α) can be viewed as a C∞ vector field in the Frechet space KN under the canonical identification ThKN = KN in the tangent space at any h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From this point of view we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The initial value problem � γ′(t) = L(χ, α)(γ(t)) γ(0) = h in KN has a unique solution given by γ(t) = etL(χ,α)(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consequently, there is a one-parameter group of Riordan matrices T(f(x, t) | g(x, t)) such that γ(t) = T(f(x, t) | g(x, t))(h) = f(x, t) g(x, t)h � x g(x, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In particular, {T(f(x, t) | g(x, t))}t∈R is an abelian subgroup of the Riordan group, because it defines a continuous dynamical system (or flow) in K[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Due to the special patterns followed by the matrices in the Lie Algebra and those in the Riordan group, we named the corresponding section in [4] as Arithmetic vector fields and geometric flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The partial differential equation induced by an element in the Lie Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For any t ∈ R and L ∈ L(R(K)) we have a well-defined Riordan matrix etL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that for any t ∈ R, γ(t) ∈ K[[x]], where γ satisfies the conditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' With this in mind, another way to interpret the above proposition is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let χ, α ∈ K[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then the unique solution of the initial value problem � ∂u ∂t = χ(x)u(x, t) + xα(x) ∂u ∂x u(x, 0) = h(x) in K[[x, t]] is given by u(x, t) = etL(χ,α)(h(x)) = f(x, t) g(x, t)h � x g(x, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From the matrix exponential etL to the solutions of the PDE ∂u ∂t = χ(x)u(x, t) + xα(x) ∂u ∂x and back Once recalled some basic results from [4], we propose the following strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Given any element L = L(χ, α) in the Lie Algebra of the Riordan group, we associate to it the partial differential equation ∂u ∂t = χ(x)u(x, t) + xα(x) ∂u ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' At this point, we run into the following dichotomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 6 (a) If we are able to compute the one-parameter group etL and to recognize etL = T(f(x, t) | g(x, t)) as a Riordan matrix for any t ∈ R, then this allows us to solve in K[[x, t]] the initial value problem � ∂u ∂t = χ(x)u(x, t) + xα(x) ∂u ∂x u(x, 0) = h(x) for any h ∈ K[[x]] because the solution is given by u(x, t) = etL(χ(x),α(x))(h(x)) = f(x, t) g(x, t)h � x g(x, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (b) If, on the contrary, we are able to solve in K[[x, t]] the initial value problem � ∂u ∂t = χ(x)u(x, t) + xα(x) ∂u ∂x u(x, 0) = h(x) for any h ∈ K[[x]], then we get the one-parameter subgroup etL(χ(x),α(x)) = T(f(x, t) | g(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We can compute both parameters f(x, t) and g(x, t), because f(x,t) g(x,t) is the unique solution for the initial condition h(x) ≡ 1 and f(x,t) g(x,t) x g(x,t) is the unique solution for the initial condition h(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Evaluating now at t = 1, we have the corresponding matrix exponential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From the matrix exponential to the solution of the PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (i) Consider the matrix D = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 · · 0 2 0 0 · · 0 0 3 0 · · 0 0 0 4 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Obviously, D = L(1, 1) is in L(R(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The partial differential equation associated to the matrix D is ∂u ∂t = u(x, t) + x∂u ∂x The solution of the corresponding initial value problem � ∂u ∂t = u(x, t) + x∂u ∂x u(x, 0) = h(x) 7 is given by u(x, t) = etD(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' It is clear, by definition, that etD = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed et 0 0 0 · · 0 e2t 0 0 · · 0 0 e3t 0 · · 0 0 0 e4t · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For any t ∈ R, we have that etD = T(1 | e−t) as a Riordan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Therefore, the solution is u(x, t) = T(1 | e−t)(h) = eth(xet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (ii) In this more interesting example, we consider the matrix H = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 · · 1 0 0 0 · · 0 2 0 0 · · 0 0 3 0 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that H = L(x, x) and so H ∈ L(R(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The PDE induced by H is ∂u ∂t − x2∂u ∂x = xu(x, t) The solution of the corresponding initial value problem � ∂u ∂t = xu(x, t) + x2 ∂u ∂x u(x, 0) = h(x) is given by u(x, t) = etH(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' To compute this solution we are going to take advantage of some previous work of another authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Particularly, we are going to follow [1], where the matrix H is called the creation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Formula (9) in [1, Page 233] computes etΛn−1(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, it is obvious that Pn−1(etΛn(H)) = etΛn−1(H) for any t ∈ R and any integer n ≥ 2 (see [4, Page 542] for the notation concerning Λn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Hence, in our Riordan matrix notation, we get etH = T(1 | 1 − xt) and the solution of the corresponding initial value problem is given by u(x, t) = etH(h) = 1 1 − xth � x 1 − xt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Evaluating at t = 1, we obtain eH = T(1 | 1 − x), which is the Pascal triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From the solution of a PDE to the matrix exponential: the Method of Characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' To point out how to go from the solution of a PDE to the exponential map of the Lie algebra to the Riordan group, we are going to deal with the following family of significant examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For any couple of real numbers a, b and any non-negative integer number n, consider the infinite lower triangular matrix La,b n = (d(n) i,j )i,j∈N, where d(n) i,j = 0 if i − j ̸= n and d(n) i,j = a + jb if i − j = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' How can we compute or describe the exponential of each of the matrices La,b n ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that La,b n = L(axn, bxn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This means that they belong to the Lie algebra of the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, La,b 0 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed a 0 0 0 · · 0 a + b 0 0 · · 0 0 a + 2b 0 · · 0 0 0 a + 3b · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , La,b 1 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 · · a 0 0 0 · · 0 a + b 0 0 · · 0 0 a + 2b 0 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Observe that D = L1,1 0 and H = L1,1 1 , where D and H are the matrices considered in the previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The PDE induced by La,b n is ∂u ∂t − bxn+1∂u ∂x = axnu(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that if b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' then, etLa,0 n = T(eatxn | 1) is a Toeplitz matrix (in the Riordan group) for any t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consequently, the solution of the corresponding initial value problem � ∂u ∂t = axnu(x, t) u(x, 0) = h(x) is given by u(x, t) = etLa,0 n (h) = eatxnh(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From now on, assume b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the matrix La,b n and the corresponding initial value problem (1) � ∂u ∂t − bxn+1 ∂u ∂x = axnu(x, t) u(x, 0) = h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 9 Let us use the Method of Characteristics to solve it (see [7], Chapter I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Suppose t = t(r, s), x = x(r, s) and z = z(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' So we consider the following system of ODEs \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 dt ds = 1 dx ds = −bxn+1 dz ds = axnz with initial conditions \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 t(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 0) = 0 x(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 0) = r z(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 0) = h(r) Integrating the system and imposing the initial conditions we get \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 t = s 1 xn = 1+nbsrn rn z = (1 + nbsrn) a nbh(r) and \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 s = t r = x n√ 1−nbtxn z = � 1 1−bntxn � a nb h � x n√ 1−nbtxn � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' which yields that the solution of (1) is given by u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' t) = � 1 1 − bntxn � a nb h � x n√ 1 − nbtxn � or u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' t) = n �� 1 1 − bntxn � a b h � x n√ 1 − nbtxn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' All above in this subsection may be summarized as Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Suppose that a and b are two real numbers and that n is a non-negative integer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the infinite lower triangular matrix La,b n = (d(n) i,j )i,j∈N, where d(n) i,j = 0 if i − j ̸= n and d(n) i,j = a + jb if i − j = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then, La,b n is in the Lie algebra of the Riordan group and for any t ∈ R the Riordan matrix etLa,b n is given by etLa,b n = \uf8f1 \uf8f2 \uf8f3 T � n� (1 − bntxn) b−a b | n√ 1 − bntxn � if b ̸= 0 T � eatxn | 1 � if b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' A special involution as symmetry or time-reversal symmetry for flows in K[[x]] related to the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' There is an involution M in the Riordan group playing a special role in such group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This matrix is M = T(−1| − 1) = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 · · 0 −1 0 0 · · 0 0 1 0 · · 0 0 0 −1 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Considered as an endomorphism in K[[x]], M is continuous and has exactly two eigen- values 1 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The corresponding eigenspaces are E(1) = {h ∈ K[x]]/h(x) = h(−x)} and E(−1) = {h ∈ K[x]]/ − h(x) = h(−x)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=', they are the linear subspaces of even, respectively odd, formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Both of them are closed subspaces in K[[x]] and we get K[[x]] = E(1) ⊕ E(−1) and (M − I) ◦ (M + I) = (M + I) ◦ (M − I) ≡ 0, where I = T(1 | 1) is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The reason of the interest about M is because it is used for defining what is known as a pseudo-involution in the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Following [3], we say that a Riordan matrix R is a pseudo-involution if the product RM is an involution, that is, RMRM = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In Group Theory we have the definitions of reversible and strongly reversible elements (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We recall it here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (i) An element g of a group G is said to be reversible in G if there is another element h of G such that hgh−1 = g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this situation we also say that h reverses g and h is a reverser of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (ii) An element g of a group G is said to be strongly reversible in G if there is an involution h of G such that hgh−1 = g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' If R is a pseudo-involution, then R is strongly reversible in the Riordan group and is the product of two involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Since RMRM = I we directly obtain, multiplying on the left by the inverse R−1, that MRM = R−1 so R is strongly reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, R−1M is an involution because (R−1M)−1 = MR = MMR−1M = R−1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consequently, we have that R = MR−1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' □ The above proposition tells us that the pseudo-involutions are particular examples of strongly reversible elements in the Riordan group and M is a reverser for any of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 11 Consider the left and right translations in R(K) given, respectively, by LD : R(K) −→ R(K) X �−→ LD(X) = DX, RD : R(K) −→ R(K) X �−→ RD(X) = XD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Since the product is a C∞-function both LD and RD are diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We need to recall the following facts from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let T(f | g) be a Riordan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The tangent space TT(f|g)R(K) to R(K) at T(f | g) is given by TT(f|g)R(K) = {T(f | g)L(χ, α)|χ, α ∈ K[[x]]} = {L(χ, α)T(f | g)|χ, α ∈ K[[x]]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, the conjugation by T(f | g) defined by conjT(f|g) : R(K) −→ R(K) X �−→ T(f | g)XT −1(f | g) is a C∞-diffeomorphism and its tangent (or differential) map at the identity DconjT(f|g)(I) : L(R(K)) −→ L(R(K)) is given by DconjT(f|g)(I)(L(χ, α)) = T(f | g)L(χ, α)T −1(f | g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Finally, for any t ∈ K we have etDconjT (f|g)(I)(L(χ,α)) = conjT (f|g)(etL(χ,α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From this, we deduce Corollary 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Given T(f | g) ∈ R(K) and L(χ, α) ∈ L(R(K)), there exists a unique L(˜χ, ˜α) ∈ L(R(K)) such that T(f | g)L(χ, α) = L(˜χ, ˜α)T(f | g) or equivalently DconjT(f|g)(I)(L(χ, α)) = L(˜χ, ˜α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, (2) ˜χ = χ � x g � (g − xg′)f − xα � x g � (f ′g − g′f) f(g − xg′) , ˜α = gα � x g � g − xg′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' As a consequence, we obtain 12 Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the involution M and the C∞-diffeomorphism conjM : R(K) −→ R(K) X �−→ MXM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then (i) the tangent (or differential) map at identity DconjM(I) : L(R(K)) −→ L(R(K)) is a linear involution (ii) for any couple of real numbers a and b such that a ̸= 0 and b ̸= 0 the matrix La,b n is an eigenvector of DconjM(I) corresponding to the eigenvalue −1 if n is odd and an eigenvector of DconjM(I) corresponding to the eigenvalue 1 if n is even (including n = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' It is clear that DconjM(I) ◦ DconjM(I) = IL(R(K)) because M is an involution in R(K), where IL(R(K)) represents the identity map in L(R(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' To prove (ii), recall that M = T(−1 | −1) and that La,b n = L(axn, bxn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From the previous corollary we get DconjM(I)(L(χ, α)) = L(˜χ, ˜α) where ˜χ and ˜α are given by (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this case, f = g = −1 and χ(x) = axn, α(x) = bxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For the case that n even, the announced result is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The same also holds for the case n is odd because −L(χ, α) = L(−χ, −α) for any χ, α in K[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' □ Finally, we obtain the following (here we use [8] for some of the definitions appearing below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let a and b two real numbers such that a = b = 0 does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Suppose also that n is a non-negative integer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the infinite lower triangular matrix La,b n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (i) If n is even, the one-parameter subgroup {etLa,b n }t∈R is contained in the centralizer of the involution M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In other words, M is a symmetry for the flow {etLa,b n }t∈R in K[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (ii) If n is odd, any element in the one-parameter subgroup {etLa,b n }t∈R is a pseudo- involution and the involution M is a time-reversal symmetry of the flow {etLa,b n }t∈R in K[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Suppose n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Using Proposition 13, we have DconjM(I)(La,b n ) = La,b n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Hence etLa,b n = etDconjM (I)(La,b n ) = MetLa,b n M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The last equality above is a consequence of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consequently, etLa,b n is in the centralizer of the involution M for every t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 13 Suppose now that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' By analogous arguments, we first have that DconjM(I)(La,b n ) = −La,b n and then e−tLa,b n = etDconjM(I)(La,b n ) = MetLa,b n M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Therefore, the Riordan matrix etLa,b n is reversible for any t ∈ R and the involution M is a reverser for all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This fact implies that etLa,b n is strongly reversible and that etLa,b n is a pseudo-involution because etLa,b n M is an involution (etLa,b n MetLa,b n M = etLa,b n e−tLa,b n = I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' □ Corollary 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The same as in the above theorem is true, word by word, changing the involution M by the involution −M = T(1 | −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' On some dynamical properties of Pascal Triangle Using some of the results obtained herein we can get some new information about the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For example, in [4] we recognized as a subgroup of the Riordan group, the substitution group of formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This group was introduced in [6] (see also [2] for a good survey about it) where results about topological generation are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' One can notice at once that some of our one-parameter groups {etLa,b n }t∈R are involved in [6, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this section, we focus only on the Pascal Triangle and assign to it others of the many properties related to patterns and symmetries that it has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We will use the pro-Lie group structure of the Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' So, we will consider elements of the Riordan group and of the Lie algebra as approximated by the components of the corresponding points in the inverse limit interpretation of both R(K) and L(R(K)) (see [4] for the pro-structure of L(R(K))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' To clarify the above approaching process recall that related to any problem (3) γ′(t) = L(γ(t)) with L ∈ L(R(K)) or, equivalently ∂u ∂t = χ(x)u(x, t) + xα(x)∂u ∂x, if L = L(χ(x), α(x)), we have a sequence of finite dimensional problems, denoted by {(3)n}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We call this sequence as the sequence of approaching problems of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Problem (3)n: Approaching problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the Euclidean space Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For any x ∈ Rn+1 denote by x = (x0, x1, · · · , xn) to its usual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Suppose xT represents the transpose matrix of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let x : R −→ Rn+1 be any derivable curve given by x(t) = (x0(t), x1(t), · · · , xn(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We denote by x′(t) = (x′ 0(t), x′ 1(t), · · · , x′ n(t)) the derivative of x at t, where x′ i is the usual derivative of a real function with real variable for any i = 0, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' When we refer to the approaching problem (3)n, we refer to the linear differential equation x′T = DΠn(I)(L)xT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 14 In the above equation DΠn(I) represents the differential at the identity I in the Riordan group R(K) of the projection Πn : R(K) −→ Rn(K) in the pro-Lie group structure of R(K), see again [4] if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The approaching problems related to Pascal Triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Recall that, Pascal Tri- angle, is the time 1 map of the dynamical system generated by the linear differential equation in K[[x]] (4) γ′(t) = L1,1 1 (γ(t)), where L1,1 1 = L(x, x), or, equivalently, ∂u ∂t = xu + x2∂u ∂x in K[[x, t]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Recall also that γ(t) ∈ R[[x]] for any t ∈ R and that L1,1 1 (γ(t)) = xγ(t)+x2 dγ(t) dx , where d dx is the formal derivative in R[[x]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' While γ : R −→ R[[x]] is a curve and γ′(t) is the usual derivative in t, when we consider R[[x]] identified with RN with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Using the pro-Lie group structure in R(K), the corresponding pro-Lie algebra structure in L(R(K)) and recalling that L1,1 1 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 · · 1 0 0 0 · · 0 2 0 0 · · 0 0 3 0 · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We can associate to problem (4) a countably infinite family of finite dimensional problems (4)n in the euclidean space Rn+1 for any non-negative integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' As we said before, we will interpret the family of problems {(4)n}n∈N as approaching the problem (4) when n tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For the first few values of n we have: (4)0 for n = 0, we consider the differential equation x′ 0 = 0 in the one dimensional euclidean space R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (4)1 for n = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' we consider the differential equation in R2 � x′ 0(t) x′ 1(t) � = � 0 0 1 0 � � x0(t) x1(t) � (4)2 for n = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' we consider the differential equation in R3 \uf8eb \uf8ec \uf8ed x′ 0(t) x′ 1(t) x′ 2(t) \uf8f6 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ed 0 0 0 1 0 0 0 2 0 \uf8f6 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ed x0(t) x1(t) x2(t) \uf8f6 \uf8f7 \uf8f8 15 (4)3 for n = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' we consider the differential equation in R4 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed x′ 0(t) x′ 1(t) x′ 2(t) x′ 3(t) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed x0(t) x1(t) x2(t) x3(t) \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The problem (4)0 is very easy to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Any point in the phase space R is an equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The phase flow is trivial and nothing is moving under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let us now consider (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The equilibrium points in this case are just the points in the x1-axe, which means that they are of the form (0, b), where b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' All of them are unstable in the Liapunov sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The rest of the orbits are the straight lines x0 = a for a fix non-null a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Through any orbit the motion is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In the semiplane x0 > 0 the sense of the motion is increasing, respect to the x1-axe as t increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=', the particle comes from the −∞ part respect to the x1-axe and goes to (positive) ∞ of such axe as t increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In the semiplane x0 < 0 the sense of the motion is the opposite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The constant speed of the motion in the orbit x0 = a is | a |, the absolute value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Particles move quickly for large values of | a | and slowly for small values | a | and they do not move in the x1-axe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Therefore, there seems to be something in the x1-axe slowing down the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We can deduce all above only knowing that the corresponding velocity vector field for the equation (4)1, in the euclidean plane, is given by X(a, b) = (0, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We can also compute, quickly and easily the solution of problem (4)1 with initial condition x0 = a and x1 = b because the corresponding matrix is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' It is the curve x(t) = (a, at + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' From Theorem 14 and Corollary 15, we get that Π1(M) = � 1 0 0 −1 � and Π1(−M) = � −1 0 0 1 � are time-reversal symmetries for the flow generated by the equation (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Con- sider the orbit θ(a, b) = {(a, at + b), a ̸= 0, b, t ∈ R} of (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then it is symmetric respect to the involution Π1(M), see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='1 in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This, in particular, means that if we transform the orbit θ(a, b) by Π1(M) we get again θ(a, b) but the parametrization obtained by means of t is not a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' However, if we finally change t by −t in such obtained curve we have a solution of (4)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' in this case the initial value at t = 0 is (a, −b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Of course, θ(a, b) = θ(a, −b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' What is the behaviour under the action of the involution Π1(−M)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' if we transform the orbit θ(a, b) by means of Π1(−M), we get another different orbit of (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In fact, we obtain Π1(−M)(θ(a, b)) = θ(−a, b) and θ(a, b) ̸= θ(−a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Again, the parametrization so obtained by means of t is not a solution of (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' But, again, if we change t by −t we get 16 another solution but with different orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Finally see that the composition of both time reversal symmetries is really a symmetry for (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then we obtain that −I is a symmetry and it implies that if x(t) = (a, at + b) is a solution of the problem with initial condition x(0) = (a, b), then −x(t) = −I(x(t)) is a solution with initial condition (−a, −b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Remark 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that any orbit x0 = a ̸= 0 of (4)1 is symmetric respect to the time reversal symmetry Π1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' On the contrary, no orbit (except for equilibrium points) is symmetric respect to the time reversal symmetry Π1(−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We propose to the reader the beautiful exercise of analysing the problem (4)2 and the role of the involutions Π2(M) = \uf8eb \uf8ec \uf8ed 1 0 0 0 −1 0 0 0 1 \uf8f6 \uf8f7 \uf8f8 and Π2(−M) = \uf8eb \uf8ec \uf8ed −1 0 0 0 1 0 0 0 −1 \uf8f6 \uf8f7 \uf8f8 as time reversal symmetries for (4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The geometric analysis of this problem is, obviously, richer than that of (4)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In the problem (4)1 the involutions Π1(M) and Π1(−M) are conjugated and they represent reflections about different axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' But in (4)2, Π2(M) is a reflection about the plane x1 = 0, which is a plane of fixed points of Π2(M), and Π2(−M) is a rotation of angle π around the x1-axe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Of course they are not conjugated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Analysing this case, one can see first that the equilibrium points are those of the form (0, 0, c) and that under the flow generated by (4)2, particles are moving within affine hyperplanes x0 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' When a = 0 we obtain the motion described in (4)1 in this hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For a ̸= 0 the orbits of points are parabolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this case, the velocity vector field is given by X(a, b, c) = (0, a, 2b) and the solution of (4)2 with initial conditions x0(0) = a, x1(0) = b and x2(0) = c is given by x(t) = (a, at + b, at2 + 2bt + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' As in the previous case, any orbit, which is not an equilibrium point, is symmetric with respect to the involution Π2(M), but any non-trivial orbit is transformed by Π2(−M) into another different orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In both cases, if after the transformation, we change t by −t we obtain new solutions of the problem (4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Finally −I : R3 −→ R3 is a symmetry for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Facts and/or conjectures and/or speculations on the problems (4)n and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The flow induced by problem (4) in K[[x]] is given by Φ : K[[x]] × R −→ K[[x]] where Φ(h, t) = etL1,1 1 (h) = 1 1−xth � x 1−xt � , while the flow generated by the problem (4)n in Rn+1 is Φn : Rn+1 × R −→ Rn+1 whose matrix expression is Φn(x, t) = Πn(etL1,1 1 )xT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We now are going to state, without proofs, properties related to problems (4)n and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This is the reason why we entitled this subsection as we did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 17 Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (Dynamical properties related to problems (4)n) Let n be a non-negative integer number, then we have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (i) The orbit of any point a = (a0, a1, · · · , an) in Rn+1 is contained in the affine hy- perplane x0 = a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover if a0 = 0 and if one consider Rn = {(x0, x1, · · · , xn) ∈ Rn+1/x0 = 0}, the motion induced by Φn in Rn is Φn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (ii) The equilibrium points in (4)n are those in the xn-axis and if n > 0 all of them are unstable in the Liapunov sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (iii) The non-trivial orbits in (4)n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=', those which are not equilibrium points, are related to the so called moment curve in the corresponding hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In particular the solution of (4)n with initial condition (1, 0, · · · , 0) ∈ Rn+1 is x(t) = (1, t, t2 · · · , tn) which is a copy of the corresponding moment curve in the hyperplane x0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (iv) Any non-trivial orbit in (4)n is symmetric with respect to the time-reversal symmetry Πn(M) and no one of them is symmetric respect to Πn(−M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Anyway, if we have a non-trivial solution of (4)n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=', a non-constant one, and we transform it by any of the involutions Πn(M) or Πn(−M) and then change t by −t we get another solution of (4)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (v) −I : Rn+1 −→ Rn+1 is a symmetry for the equation (4)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Motivated by the previous result and knowing that Pascal triangle is the time one map of the flow Φ, we can state: Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (Some dynamical properties of Pascal Triangle) (i) The flow Φ has not equilibrium points up the null power series, that is, all the coefficients of the power series are null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (ii) The orbit of the formal power series constantly 1, by means of Φ, is the set of geometric progressions { 1 1−tx}t∈R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then one can think about 1 moving through the set of the germs of analytic functions, at x = 0, because for any t ∈ R the function ft(x) = 1 1−tx for x ∈ (− 1 |t|, 1 |t|) is analytic at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, this orbit can be seen as the asymptotic behaviour, as n goes to ∞, of the moment curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (iii) The Riordan involution M and −M are time-reversal symmetries for the flow Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Any orbit of problem (4) is symmetric respect to M and no orbit, except for the unique equilibrium point, is symmetric respect to −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Anyway, if we transform any solution of (4) by means of M or −M and then change t by −t we get another solution of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (iv) −I : K[[x]] −→ K[[x]] is a symmetry for the equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' To finish this paper note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Claim: Any non-trivial orbit of the problem (4)n has empty α-limit set and ω-limit set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' On the other hand, the problem (4)n has time reversal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' This fact allows us to 18 think that there should be relationships between both limit sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We then decided to force the corresponding flows, by means of considering a compactification of the correspond- ing phase spaces, to get non-empty α-limit and ω-limit sets and to look for relationships between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consider the one point (or Alexandroff) compactification of Rn+1 which is, topologically, the n + 1-dimensional sphere Sn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let us denote by ∞ the added point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Note that any t-map of the flow Φn, Φt n, can be continuously extended to a map � Φtn : Sn+1 −→ Sn+1 defining only � Φt n(∞) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' In this way we get a dynamical system � Φn : Sn+1 × R −→ Sn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' For every non-negative integer n we can also extend the time reversal symmetries Πn(M) and Πn(−M) to continuous maps � Πn(M) and � Πn(−M) from Sn+1 onto itself imposing that the point ∞ is a fixed point for both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' We can identify, topologically, Rn+1 with Sn+1\\{∞} (which is a dense subset of Sn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' With all these constructions we have Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Let n be a non-negative integer number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (i) The maps � Πn(M) and � Πn(−M) are continuous involutions in Sn+1 and they are time-reversal symmetries for the dynamical system � Φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' (ii) The orbits of the dynamical system � Φn are those of Φn (after the mentioned identi- fication) plus {∞} which is an equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Moreover, every non-trivial orbit of � Φn is homoclinic, being the point ∞ an attractor and a repeller of all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Consequently, the α-limit and the ω-limit sets of any non-trivial orbit coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Aceto, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='Trigiante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The matrices of Pascal and other greats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Amer.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Oxford University Press, Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 2002 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Shapiro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Getu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Woan and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Woodson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' The Riordan group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' 34 (1991) 229-239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' ♭ Departamento de Matem´atica Aplicada, Ciencia e Ingenier´ıa de los Materiales y Tec- nolog´ıa Electr´onica, ESCET Universidad Rey Juan Carlos, 28933 M´ostoles (Madrid), Spain Email address: pedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='chocano@urjc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='es Departamento de Matem´atica Aplicada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Universidad Polit´ecnica de Madrid (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Email address: anamaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='luzon@upm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='es ♮ Departamento de Algebra, Geometr´ıa y Topolog´ıa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Universidad Complutense de Madrid and Instituto de Matem´atica Interdisciplinar (IMI)(Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content=' Email address: mamoron@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAyT4oBgHgl3EQfW_dN/content/2301.00173v1.pdf'} +page_content='es † Departamento de Matem´atica Aplicada, ETS Arquitectura, Universidad Polit´ecnica de Madrid (Madrid).' 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+Quantitative determination of minimum spanning tree structures: +Using the pulsar tree for analyzing the appearance of new classes of pulsars +C. R. García1,3★, Diego F. Torres1,2,3† +1Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Barcelona, Spain +2Institució Catalana de Recerca i Estudis Avançats (ICREA), E-08010 Barcelona, Spain +3Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain +16 January 2023 +ABSTRACT +In this work, we introduce a quantitative methodology to define what is the main trunk and what are the significant branches +of a minimum spanning tree (MST). We apply it to the pulsar tree, i.e. the MST of the pulsar population constructed upon a +Euclidean distance over the pulsar’s intrinsic properties. Our method makes use of the betweenness centrality estimator, as well +as of non-parametric tests to establish the distinct character of the defined branches. Armed with these concepts, we study how +the pulsar population has evolved throughout history, and analyze how to judge whether a new class of pulsars appears in new +data, future surveys, or new incarnations of pulsar catalogs. +Key words: pulsars: general, stars: neutron, methods: data analysis +1 INTRODUCTION +In a recent work (García et al. 2022), we have applied a princi- +pal components analysis, e.g., Pearson (1901); Shlens (2014), over +magnitudes depending on the intrinsic pulsar’s timing properties +(considered as proxies to relevant physical pulsar features, e.g., the +magnetic surface field, the spin-down power, etc.), to analyze whether +the information contained by the pulsar’s period and period deriva- +tive are enough to describe the variety of the pulsar population. We +showed that 𝑃 �𝑃 are not principal components and do not contain the +full variance of the pulsar population. Thus, any distance ranking or +visualization based only on 𝑃 and �𝑃 is potentially misleading. Subse- +quently, we have introduced the use of graph theory to the problem, +and in particular, presented the Pulsar Tree. This is the minimum +spanning tree (MST, see e.g., Gower & Ross (1969); Kruskal (1956)) +of the pulsar population constructed upon a Euclidean distance over +the pulsar’s intrinsic properties. We prepared as well an online tool +http://www.pulsartree.ice.csic.es, a site which contains vi- +sualization tools and data to allow users to gather information in terms +of the MST and the distance ranking. Here, we shall build upon Gar- +cía et al. (2022) and we shall take for granted that the reader is aware +of its introductory appendices accounting for the needed conceptual +ingredients from graph theory. This work has the following aims: +• We want to establish a quantitative methodology to define what +is the main trunk and what are the significant branches of a pulsar +tree (or any MST, in general). In particular, we want to introduce a +qualifier to consider whether the branches of the tree have statistically +significant differences among themselves. +• Once we establish what is the trunk and the significant branches, +★ E-mail: crodriguez@ice.csic.es +† E-mail: dtorres@ice.csic.es +we want to develop a methodology to tell us whether these structures +have grown in earlier incarnations of the pulsar population. +• As a spinoff of the latter study, we analyze how to judge whether +a new class of pulsars appears in new data, future surveys, or new +incarnations of pulsar catalogs. +We shall use v1.67 of the ATNF catalog (Manchester et al. 2005), +collecting pulsars entered in the database until March 2022. This +version contains 2509 pulsars, of which 2242 are isolated pulsars and +267 are pulsars belonging to binary systems having period derivatives +greater than zero. Using the label ’date’ of the ATNF catalog, which +refers to the date of the pulsar discovery, we shall also consider the +historical evolution of the pulsar population. +2 IDENTIFYING THE MAIN TRUNK AND BRANCHES OF +THE TREE +2.1 Betweenness centrality +Betweenness centrality defines how central a node is, or put other- +wise, how many times a given node of a graph is in between any +two others (see Freeman (1977), see also Moxley & Moxley (1974)). +More precisely, it measures centrality from the ratio between the +number of times a node 𝑣 appears on the shortest path, also known +as geodesic distance, between any two other nodes (𝑠, 𝑡), and the +number of possible shortest paths that could occur between them +(e.g., see Brandes (2001)), +𝐶𝐵 = +2 +(𝑁 − 1)(𝑁 − 2) +∑︁ +𝑠≠𝑣≠𝑡 ∈𝑉 +𝜎𝑠,𝑡 (𝑣) +𝜎𝑠,𝑡 +. +(1) +Here, 𝜎𝑠,𝑡 (𝑣) can be equal to 1 or to 0. 𝜎𝑠,𝑡 (𝑣) = 1 only when any +geodesic distance between the nodes 𝑠, 𝑡 pass through the node 𝑣. In +a general graph, 𝜎𝑠,𝑡 is the total number of shortest paths between +© 2023 The Authors +arXiv:2301.05408v1 [astro-ph.HE] 13 Jan 2023 + +2 +García & Torres +two nodes (𝑠, 𝑡). As imposed by the inexistence of closed loops in +an MST, i.e., the MST is acyclic (see eg., Tarjan (1983)), only one +path is possible between any two nodes of an MST (see eg., Wilson +(2010)). Thus, the denominator within the sum of Eq. (1) equals +unity, 𝜎𝑠,𝑡 = 1, for any two pairs of nodes, since there is only one +path connecting two nodes in the MST. To avoid duplicities, and +resorting to the non-directionality of the MST, 𝜎𝑠,𝑡 = 𝜎𝑡,𝑠, and will +be taken into account only once in the sum Eq. (1) is normalized by +multiplying it by 2/(𝑁 − 1)(𝑁 − 2) where 𝑁 = |𝑉| is the number of +nodes we have in the set 𝑉. This is done to compare results obtained +between graphs of different sizes regarding the number of nodes. +An explicit example with an MST, and the graph from which we +obtain that MST, of a few nodes, will help clarify the computation, +and we provide such an example in the Appendix. Betweenness +centrality is in fact helpful in formalizing an obvious mental idea +of how central a node is for a given graph, allowing us to have +mathematical definitions based on its distribution. Fig. 1 shows the +MST of the current pulsar population as presented in our previous +work (García et al. 2022) after the application of Eq. (1). Fig. 2 shows +the distribution of the betweenness centrality values just computed. +Outliers are clearly appearing in this asymmetric distribution. +2.2 Definition of the main trunk +To establish a central interval for the distribution of betweenness +centrality we shall use an appropriate technique for asymmetric dis- +tributions, based on quartiles. The central interval will be defined as +(see, e.g., Tukey (1977)), +[𝑄1 − 𝑘 × 𝐼𝑄𝑅]; [𝑄3 + 𝑘 × 𝐼𝑄𝑅] , +(2) +where 𝑘 is a coefficient typically taken equal to 3. The use of quartiles +(𝑄1, 𝑄3) to identify central values and outliers are validated for +both symmetric and asymmetric distributions: they do not assume +anything about the mean or the standard deviation, and their use is +compatible with distributions of positive rightward skew, as ours. +Furthermore, any asymmetric distribution is more robustly defined +by the median as a measure of central tendency and the interquartile +range (IQR= 𝑄3 −𝑄1) as a measure of its dispersion, as both are less +sensitive to extreme values. For our case, the right-hand side of the +central interval shown in Eq. (2), together with the usual value of 3 +for 𝑘 sets the condition for a node to be considered central (i.e., an +outlier of the betweenness centrality distribution). This condition is +to be taken as necessary, but not sufficient for a node to be considered +part of the main trunk. In addition to being formed by outliers of the +betweenness centrality distribution (something that relates to a bare- +eye identification as the main trunk in Fig. 1) we need to provide a +criterion to establish where it starts and ends. To that aim, we shall +request a topological condition: As the trunk is a path, i.e., a sequence +of consecutive nodes containing no duplicates, we shall require that +it must be starting and ending at nodes whose degree must be greater +than 2. In this way, we ensure that at the terminations of the trunk, +there are nodes that give rise to significant substructure, i.e., to relate +to the mind-image of branches opening up from the trunk (branches +are to be defined more precisely below) with a possible physical +significance. The term significant used above –which quantitative +meaning is discussed next – is herein added to explicitly avoid ending +the tree in a degree 3 node but from where the branches departing +from it are formed just by a few nodes (noise). Thus, we define the +main trunk as the longest substructure of the MST formed by outliers +of the betweenness centrality distribution, starting and ending in +nodes of degree 3 or higher, and giving rise to significant branches. +2.3 Significant branches +We adopt a general conceptual definition (that is useful for our pulsar +tree case, as well as for any other MST): a relevant branch is defined +as a group of nodes departing from the main trunk that contains at +least a minimum percentage of the total population of nodes in the +graph and can be quantitatively distinguished from other branches. +To impose that a branch contains at least a minimum percentage of +the total number of nodes in the graph (to be referred to as signifi- +cance threshold) is done to avoid having branches with just a small +collection of nodes, usually departing briefly from other substruc- +tures. These are to be considered noise of the latter. The significance +threshold will be empirically found directly from the MST being +studied. Note that the larger this threshold is (i.e., the more nodes +will be assigned to branches), the smaller the main trunk will be. For +instance, if this threshold is 10%, i.e., each branch should contain at +least 10% of the total number of nodes, there will be fewer branches +(and with more nodes each), and a smaller main trunk, than when +it is fixed at 5%. To define the significance threshold and measure +at once whether one branch distinguishes from others we shall use +the Kolmogorov-Smirnov (KS) statistics. The KS test compares two +distributions under study via the distance between their empirical +cumulative distribution functions, e.g., see Wolfe (2012); Lehmann +(2012); Yadolah (2008). The KS test does not assume any form of +distribution beforehand, which makes it a non-parametric test some- +thing that allows its use for any type of distribution. The aim here will +be to see if we can reject the null hypothesis (𝐻0): the distribution of +the properties of the nodes of two given branches are consistent with +them coming from the same parent distribution. We will be seeking +to reject this null hypothesis at 95% confidence level (CL) or better. +When this happens, we shall be establishing that whatever distance +is used to compute the weights between nodes and form the MST +is separating branches whose nodes are drawn from statistically dis- +tinct parent populations. Specifically for the case of the pulsar tree, +we shall recall the results of our PCA analysis (García et al. 2022). It +defined that two principal components were needed for the set of vari- +ables studied (spin period, spin period derivative, surface magnetic +flux density at the equator, the magnetic field at the light cylinder, +spin-down energy loss rate, characteristic age, surface electric volt- +age, and Goldreich-Julian charge density). The distributions for the +principal components of the pulsar population 𝑃𝐶1 and 𝑃𝐶2 are +skewed, asymmetric distributions with pronounced rightward and +leftward shifts. The explained variance by the first principal compo- +nent, 𝑃𝐶1, reaches ∼ 72% in the current, most numerous incarnation +of the catalog. Thus, we shall request that the KS test rejects that +any two branches have a 𝑃𝐶1 distribution that could come from the +same parent population. The threshold value from which branches +are defined can then be iteratively fixed to the minimum value pos- +sible at which all branches are distinct under the KS test applied to +their 𝑃𝐶1 distribution. If the threshold would be smaller, the number +of branches increases, the average branch size decreases, and not all +the smaller sub-structures represent distinct populations. +2.3.1 Other tests and principal components +One can entertain that several variations on the methodology above +are possible in the general case. For instance, one can ask why not +requesting that both 𝑃𝐶1 and 𝑃𝐶2 distributions are distinguished for +all branches via the KS test. One can also ask why not using a dif- +ferent test from the KS. We briefly comment on these aspects below +and justify the proposed approach as the most reasonable. When a +principal component represents low values of explained variance, it +MNRAS 000, 1–12 (2023) + +Classes of pulsars and the MST +3 +0 +10−2 +10−1 +Figure 1. A color-coded representation of MST(2509, 2508) according to the values of the betweenness centrality coefficient after application of Eq. (1). The +color intensity varies from zero (with the strongest color), for those nodes located at the termination of the graph being nodes of 1 degree, to the nodes located +in the central part of the MST where 𝐶𝐵 reaches its highest values around 0.6. +means that the sample presents incomplete partial information in this +dimension so all associated distributions will be of lower relevance +for representing the nodes. Nevertheless, we find that our results are +stable if we were to use both 𝑃𝐶1 and 𝑃𝐶2 to define the minimum +threshold. We have checked that under 𝑃𝐶2, our branches distin- +guish themselves in all cases and that the KS test with 𝑃𝐶2 rejects +the null hypothesis at smaller significant thresholds than 𝑃𝐶1. This +justifies our choice of using the most relevant principal component +only. Regarding the second question posed, we find the KS test es- +pecially suited to the kind of null hypothesis we want to reject and +simple enough for our aims. Other usual tests in astronomy, such as +the F-test, e.g., see Jones (1994), are not really applicable as if a +distribution is skewed, one could argue that the variance is not an op- +timum dispersion measure. Other non-parametric tests for comparing +two populations, such as the Anderson-Darling (e.g., see Scholz & +Stephens (1987)) are also based on empirical distribution functions. +We have checked that in all cases where the KS test rejects the null +hypothesis, also the Anderson-Darling test does. We prefer the KS +test over the Anderson-Darling – apart from its simplicity – also be- +cause the KS test loses rejection power the smaller the samples being +compared are. This is reasonable for our case since it does not make +sense to define branches formed by just a few pulsars. +2.4 The pulsar tree main trunk and branches +Taking into the betweenness centrality distribution of Fig. 2 and +Eq. (2), we obtain that 10% of the nodes are outliers of the between- +ness centrality distribution. These nodes fulfill the necessary condi- +tions to be part of the main trunk. The use of the iterative method +described provides a significance threshold of 3.8%, i.e., each of the +branches has to have at least 3.8% of the total number of nodes, or +at least a hundred pulsars. The significant branches so identified are +shown in Fig. 3. The branches identified bear resemblance with the +ones one would simply point by hand if looking at the MST. Some of +these branches were already used as examples in Section 4 of our ear- +lier work (García et al. 2022) when commenting on the pulsar tree as +a descriptive tool. The top and bottom branches of Fig. 3 roughly cor- +respond to binary pulsars, and to the more energetic isolated pulsars +of the sample (the youngest and those with the highest light cylinder +magnetic field pulsars are located towards the end of this branch), +respectively. Rightwards departing branches characterize by increas- +ing values of surface magnetic fields, ending with magnetars at their +extremes. The outgoing leftward branch contains the oldest isolated +pulsars. Fig. 4 shows examples of the distribution of variables for +some of the significant branches of the pulsar tree, as can be extracted +MNRAS 000, 1–12 (2023) + +4 +García & Torres +0.0 +0.2 +0.4 +0.6 +CB +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Counts +0.000 +0.005 +0.010 +0.015 +0.020 +CB +0 +200 +400 +600 +800 +Counts +Figure 2. Distribution of the betweenness centrality values seen after applying +Eq. (1) on the MST of the pulsar population (ATNF v.1.67, shown in Fig. 1). +In the upper right corner, the distribution of about 70% of the data is zoomed +in. +from the online tool at http://www.pulsartree.ice.csic.es. +It is clear that the significant branches separate different physical +properties. To emphasize this point, Fig. 5 compares the principal +components corresponding to each branch, as defined by García et al. +(2022), showing differences in their distribution. We also note the +case of the fifth panel from the left in Fig. 5, showing the distribution +of the branch containing the binary pulsars. This branch is large and +mixes pulsars in binaries with isolated ones, and as such, it shows a +double peak structure in the distribution, which happens in no other +branch of the MST. Looking at the latter, the bottom part of the +branch of this large structure (where we find a degree 4 node and +two leftward departing branches) is where the long-period, isolated +pulsars are located. This can also be clearly seen using the Pulsar +Tree Web. One can also gather that these two branches contain 118 +pulsars in total (73, and 45 in each). Thus none of these branches is in +excess of the significance threshold, and according to the definition, +they cannot be classified as individually significant as of yet. We find +however that by doing the KS test to test the null hypothesis between +the corresponding distribution of the whole structure, of the individ- +ual branches, and all other branches we determine the null hypothesis +is rejected. This is then a case where we foresee that an increased +number of pulsars will likely join these branches into one generat- +ing one additional significant branch of the MST, or increase each +of the branches’ number of nodes to make them individually signifi- +cant. Fig. 6 shows the distributions of each of the underlying physical +magnitudes considered to construct the principal components. In this +latter case, some of the individual variables seem to have similar dis- +tributions in the different branches. However, according to the KS +test, the branches (that were chosen from 𝑃𝐶1) also show rejection +of the null hypothesis when considering the individual magnitudes +directly. All the information separately hosted in these variables is +captured by 𝑃𝐶1 itself. The use of 𝑃𝐶1, therefore, simplifies and +accelerates the comparison. This seems to be a general result along +the MSTs we have analyzed, although we cannot rule out a priori +that some branches could be similar under the KS test of a particular +intrinsic magnitude despite being dissimilar (always in terms of the +null hypothesis) under the KS test of the principal components. We +note the bimodality appearing in one of the top branches of the MST. +This happens because most of the binary pulsars are located in that +part (see right panel of Fig. 4) and therefore, the analyzed variables +and thus the PCs are affected by the behavior of these binary systems. +2.4.1 Simulations +To give further credit to the branch separation achieved we have +carried out simulations. Particularly, we have considered a random +separation of the pulsar population into groups corresponding to the +number of branches we identify, each with the respective number of +pulsars. We thus created fake, random branches, not motivated by the +MST. We have done this exercise more than 105 times, and in each +instance, we computed the KS test among the random ’branches’ so +created. Statistical separating fake branches prove very difficult: Not +in a single case of our simulations, we find that all 7 branches can be +separated. In fact, in 55% of the simulations, the test is not able to +reject the null hypothesis for not even 2 random branches. +3 EVOLUTION OF THE MST WITH THE PULSAR +POPULATION +We now turn to consider how our methods apply to the evolution of +the pulsar population throughout history. This analysis is illustrative +of its application, but we remark that as the pulsar population builds +up in history with the number of pulsars known, thus a larger sam- +ple must necessarily be more significant for population analysis. We +shall consider the pulsars known until the years 1978, 1988, 1998, +2008, and 2018, and the current population as described above, 2022. +These sets contain 147, 439, 662, 1660, 2267, and 2509 pulsars, re- +spectively. Application of Eq. (2) establishes a maximum percentage +of values that can be considered outliers for each of these samples, +resulting in 0%, 7%, 9%, 9%, 13%, and 10%, respectively. We recall +that the nodes conforming to the trunk in each of the catalogs must +have a 𝐶𝐵 value that makes them outliers of the corresponding be- +tweenness centrality distribution. Fig. 7 shows the pulsar tree along +history, marking the betweenness centrality values in the same scale +of Fig. 1. Note that due to the topological conditions imposed for the +definition of the main trunk (see §2.2; i.e., that the trunk is a path, +and especially, that the trunk must start and end in nodes of degree 3 +or more that give rise to branches with at least a given a number of +nodes) the size of the branches that can be a priori reached also has +a maximum value beyond which these trunk conditions cannot be +fulfilled. This maximum value is 9.7%, 9.3%, 8.9%, 7%, and 14.5% +respectively from 1988 onwards; the significance threshold obtained +is consistently lower than them in all epochs. Interestingly, due to +the small size of the sample, the MST of the population of pulsars +known until 1978 does not provide any outlier in the distribution +of betweenness centrality. Correspondingly, that MST is less struc- +tured. The significance thresholds are 0%, 6.2%, 5.8%, 8.6%, 2.8%, +and 3.8%, respectively. The number of significant branches along +the history is 0, 5, 5, 4, 9, and 7, respectively. It is interesting to see +that the number of branches is similar throughout history, despite the +introduction of up to 20 times more pulsars than those known up to +1978. These are shown in Fig. 8. We emphasize, as stated before, +that these MSTs are still frames of the knowledge gathered from the +MNRAS 000, 1–12 (2023) + +Classes of pulsars and the MST +5 +Figure 3. Significant branches of v.1.67 (March 2022) are shown after applying the techniques described in the text. These have at least 3.8% of the total +population size, following the branches clockwise (from 12:00) they have the following number of pulsars: 585, 364, 157, 224, 444, 427, and 119, respectively. +The MST nodes depicted in the main part are 64 and represent the trunk. Note that the remaining light blue (125) nodes belong to non-significant branches and +are considered to be the noise of the main trunk. +Figure 4. Three examples of the distribution of variables for some of the significant branches of the pulsar tree, as can be extracted from the online tool at +http://www.pulsartree.ice.csic.es. We suggest looking at these examples directly there to have access to additional functionalities, being able to zoom +in and see individual values for each pulsar. The left panel shows the distribution of the surface magnetic field in the middle branches, increasing towards the +outskirts of the MST, ending in low-field magnetars (light orange) and classical magnetars (orange). The middle panel shows the spin-down power distribution +of the bottom branch, with those having a pulsar wind nebula detected in TeV (H. E. S. S. Collaboration et al. 2018) (grey) noted. The right panel shows the +distribution of the period in the upper branches of the MST where binaries (purple) and long-period pulsars are located. +MNRAS 000, 1–12 (2023) + +BSURF<1E9 +1E9 a, in which case π(c) ≤ c and [a + u, b] ∩ ↓π(c) = [a + u, b] ∩ ↓c. +Thus M(π(c)) → M(c) is an isomorphism by the definition of [a + u, b]-determined +modules, and M ∼= resπ res[a,b] M. +To prove the converse, assume that Supp(M) ⊆ ↑a and M has an encoding by +the closed interval [a, b] with the encoding convex projection π: Zn → [a, b]. Let +c := (c1, . . . , cn) ∈ Zn. Suppose that ci < ai for some i ∈ {1, . . . , n}. From the +condition Supp(M) ⊆ ↑a, we see that M(c) = 0. Since M is finitely determined, we +also have M(π(c)) = M(c) = 0. Thus M(c) = 0 if ci ≤ ai for some i ∈ {1, . . . , n}. +If this is not the case, we have c ≥ a + u. Let c ≤ d in C such that a + u ≤ c ≤ d +and [a + u, b] ∩ ↓ c = [a + u, b] ∩ ↓ d. This implies that π(c) = π(d), so M(c ≤ d) is +an isomorphism. +□ +To proceed, we have to shift our focus to RZ +n-modules. Let a ≤ b in Zn. With +the notation from section 3, we will view the case S = [a, b]. In particular, we +have α = α[a,b] and β = β[a,b]. Proposition 3.4 gives us formulas for α and β. If +c := (c1, . . . , cn) ∈ Z +n and d := (d1, . . . , dn) ∈ [a, b], then +α(c) = (α1(c1), . . . , αn(cn)) +and +β(d) = (β1(d1), . . . , βn(dn)). +Here αi := αSi and βi := βSi for all i ∈ {1, . . . , n}. Explicitly, +αi(ci) = + + + + + +−∞, if ci < ai; +ci, if ai ≤ ci ≤ bi; +bi, if ci > bi +and +βi(di) = +�ai, if di = −∞; +di, otherwise +for every i ∈ {1, . . . , n}. The next proposition shows us that the composition β ◦ α +is an extension of the convex projection π from Zn to Z +n. +Proposition 4.3. Let π: Zn → [a, b] be the convex projection. +Then for any +c := (c1, . . . , cn) ∈ Zn, +π(c) = (β ◦ α)(c). +Proof. Suppose first that n = 1. Recall that π(c) = max(a, min(c, b)). Now there +are three cases: +• If c ∈ [a, b], then (β ◦ α)(c) = β(c) = c = π(c); +• If c < a, then (β ◦ α)(c) = β(−∞) = a = π(c); +• If c > b, then (β ◦ α)(c) = β(b) = b = π(c). +Suppose next that n > 1. Using Proposition 3.4, we may write +α(c) = (α1(c1), . . . , αn(cn)) +and +β(d) = (β1(d1), . . . , βn(dn)) + +FINITE PRESENTATION OF FINITELY DETERMINED MODULES +11 +for all d ∈ [a, b]. Similarly, recall that +π(c) = (π1(c1), . . . , πn(cn)). +It now follows from the case n = 1 that +(β ◦ α)(c) = β(α1(c1), . . . , αn(cn)) += ((β1 ◦ α1)(c1), . . . , (βn ◦ αn)(cn)) += (π1(c1), . . . , πn(cn)) += π(c). +□ +Remark 4.4. In an effort to keep the notation simpler, we only defined β for the +elements in the image of α. Of course, we could have defined β in a fully dual +fashion to α, starting from posets that are strongly bounded from below, adding +the point ∞ to Z, and defining a set S dually to S. This would have resulted in +the situation where +(β|S ◦ α)(c) = (α|S ◦ β)(c) = π(c) +for all c ∈ Zn. In other words, the same result would have been achieved. +We saw in Remark 4.1 that if M is encoded by a closed interval [a, b] with the +convex projection π: Zn → [a, b], we have M(c) ∼= M(π(c)) for all c ∈ Zn. On +the other hand, by Theorem 2.5, we have M(α(c)) ∼= M(c) for all c ∈ Z +n if M is +S-determined and S ⊆ Z +n is finite. In preparation for the proof of Theorem 4.7, +we will now show that a similar result applies to β in both cases. +Proposition 4.5. Set u := (1, 1, . . . , 1) ∈ Zn. Let M be an RZn-module, and let +c ∈ [a, b]. +1) If M has an encoding by the closed interval [a, b] with the convex projection +π: Zn → [a, b], then M(c) ∼= M(β(c)). +2) If M is [a + u, b]-determined, then M(c) ∼= M(β(c)). +Proof. To show 1), suppose that M has an encoding by the closed interval [a, b] +with the convex projection π: Zn → [a, b]. Then, by the definition of M, +M(c) = +lim +d≥c, d∈Zn M(d). +The encoding gives us M(d) ∼= M(π(d)) for all d ∈ Zn. This implies that +M(c) ∼= +lim +d≥c, d∈Zn M(π(d)). +We may now apply Corollary 3.6 to see that M(c) ∼= M(β(α(c))). +Note that +c ∈ [a, b] implies α(c) = c. Thus M(c) ∼= M(β(c)). +Next, to prove 2), let M be [a + u, b]-determined. Since c ≤ β(c), it is then +enough to show that [a + u, b]∩↓c = [a + u, b]∩↓β(c). We instantly have ↓c ⊆ ↓β(c). +For the other direction, let d := (d1, . . . , dn) ∈ [a + u, b] ∩ ↓β(c). We want to show +that d ≤ c. Recall that we may write β(c) = (β1(c1), . . . , βn(cn)), where +βi(ci) = +�ai, if ci = −∞; +ci, otherwise. + +12 +EERO HYRY AND MARKUS KLEMETTI +for all i ∈ {1, . . . , n}. Suppose that i ∈ {1, . . . , n}. If βi(ci) = ci, we have di ≤ +βi(ci) = ci. +Otherwise, if βi(ci) = ai, we must have di = ci = −∞, because +di, ci ∈ [ai + 1, bi]. We conclude that d ≤ c. +□ +Remark 4.6. Let M be a pointwise finitely presented RZn-module and let c ∈ Z +n. +If M is finitely determined with the convex projection π: Zn → [a, b], then from +the proof of Proposition 4.5, we have M(c) ∼= M((β ◦α)(c)), so that M is pointwise +finitely presented. +We are now ready to state +Theorem 4.7. Let M be a pointwise finitely presented RZn-module. +Then the +following are equivalent: +1) M is finitely determined; +2) M is S-determined for some finite S ⊆ Z +n; +3) M is finitely presented. +Proof. We will first show the equivalence of 1) and 2). Note that for any finite +subset S ⊆ Z +n, we can always find a, b ∈ Zn such that S ⊆ [a + u, b]. Consider the +functor +α′ = α[a+u,b] : Z +n → [a + u, b], +and denote its restriction to Zn by α. By Theorem 2.5, M is [a + u, b]-determined if +and only if M is encoded by α′. That is, M ∼= resα′ N for some R[a + u, b]-module +N. By restricting to Zn, we see that +M ∼= resZn resα′ N = resα N, +so α encodes M. Conversely, if M is encoded by α, then M has an obvious encoding +by α′, because α is a surjection on objects. Next, we note that the restriction of β +to [a + u, b], +β : [a + u, b] → [a, b]. +is an isomorphism of posets. Therefore β ◦ α is an encoding of M if and only if α is +an encoding of M. These conditions are equivalent to M being finitely determined, +because β ◦ α = π. Namely, for all c ∈ Zn, we have (β ◦ α)(c) = (β ◦ α′)(c), where +(β ◦ α′)(c)i = +� +βi(−∞), if ci = ai, +βi(αi(ci)), else. += +�ai, if ci = ai, +πi(ci), else. += π(c)i +for all i ∈ {1, . . . , n}. +Finally, we observe that the equivalence of 2) and 3) follows from the main result +of our previous paper, [4, p. 25, Thm. 4.15]. For Z +n-modules, it states that being +pointwise finitely presented and S-determined for some finite S ⊆ Z +n is equivalent +to being finitely presented. +Also note Remark 4.6, which shows us that M is +pointwise finitely presented. +□ +We are now able to give a “sharpened” version of Proposition 4.2. + +FINITE PRESENTATION OF FINITELY DETERMINED MODULES +13 +Corollary 4.8. If M is an RZn-module and a, b ∈ Zn such that a ≤ b, then the +following are equivalent: +1) M is encoded by the convex projection π: Zn → [a, b]; +2) M is [a + u, b]-determined, where u := (1, . . . , 1) ∈ Zn. +Proof. We showed in the proof of Theorem 4.7 that 2) implies 1). +Conversely, +suppose that 1) holds. Let c ≤ d in Z +n such that [a + u, b] ∩ ↓c = [a + u, b] ∩ ↓d. +Coordinatewise, for i = {1, . . . , n}, this implies that either ci = di, bi ≤ ci < di or +ci < di ≤ ai. In any case, (β ◦ α)(c) = (β ◦ α)(d), so that +M(c) ∼= M((β ◦ α)(c)) = M((β ◦ α)(d)) ∼= M(d). +Thus M(c ≤ d) is an isomorphism, and M is [a + u, b]-determined. +□ +To demonstrate Theorem 4.7 and Corollary 4.8, it is convenient to take the +point of view of topological data analysis, and consider the births and deaths of +elements of a module. Given an RZn module M, one can track how an element +x ∈ M(c), where c ∈ Z +n, evolves when mapped with the homomorphisms M(c ≤ c′), +(c, c′ ∈ Z +n). We say that the element x is born at c if it is not in the image of any +morphism M(c′ ≤ c), where c′ < c. On the other hand, the element x dies at c′′ if +M(c ≤ c′′)(x) = 0, but M(c ≤ c′)(m) ̸= 0 for all c ≤ c′ < c′′. +Consider now an RZ2-module M that is finitely determined, and let π: Z2 → +[a, b] be the accompanying convex projection. Note that no new elements are born +or die in the leftmost edge or the bottom edge of the box [a, b]. +This follows +from the fact that every element on these two edges has already appeared infinite +times before, and was born at some infinitary point. Let us write a = (a1, a2). +For example, if an element, say x ∈ M((a1, c)), maps to zero on the leftmost +edge of [a, b], in M((a1, c + 1)), then x ∈ M((−∞, c)) will also map to zero in +M((−∞, c + 1)). Thus x does not “die” at the point (a1, c + 1), but rather at the +infinitary point (−∞, c + 1) ∈ [a + u, b]. +Remark 4.9. Let M be an RZn-module and c ∈ Z +n. Consider the natural homo- +morphism +λM,c : +colim +d≤c,d∈[a+u,b] +M(d) → M(c). +Following [4, p. 15, Def. 3.6], we say that c is a birth if λM,c is a non-epimorphism, +and a death if λM,c is a non-monomorpism. +Furthermore, suppose that M is +[a + u, b]-determined, and the births are “well-behaved” enough. That is, for any +birth c, the module M(c)/ Im λM,c is projective. The latter of course holds if R is +a field. Then, as we discussed in [4, p. 21, Remark 3.27], births and deaths show +the positions of the minimal generators and relations of M. +In the next example, we will demonstrate how, for a finitely determined module +M, the extension M has births and deaths at infinitary points that guarantee the +existence of a finite presentation of M. +Example 4.10. Let M be an RZ2-module that is defined on objects by +M(c) = +�R, if c ≤ (0, 0); +0, otherwise, + +14 +EERO HYRY AND MARKUS KLEMETTI +for all c ∈ Z2, and where a morphism R → R is always idR. Then M is finitely +determined with the convex projection π: Z2 → [(0, 0), (1, 1)]. Now, by Remark +4.8, M is [(1, 1), (1, 1)]-determined. Here [(1, 1), (1, 1)] is the set +{(−∞, −∞), (1, −∞), (−∞, 1), (1, 1)}. +In particular, we have M((−∞, −∞)) = R, and +M((−∞, 1)) = M((1, −∞)) = M((1, 1)) = 0. +Furthermore, by Theorem 4.7, M is now finitely presented. In more concrete terms, +we have an exact sequence of RZ +2-modules +K → N → M → 0, +where +N = R[MorZ +2((−∞, −∞), −)] +and +K = R[MorZ +2((1, −∞), −)] ⊕ R[MorZ +2((−∞, 1), −)]. +Here (−∞, −∞) is the only birth of M, while (1, −∞) and (−∞, 1) are the deaths. +Example 4.11. If k is a field, then it is well known that finitely generated kZn- +modules are finitely presented. +This result, however, does not apply to kZ +n- +modules. For a counterexample, consider a kZ2-module M, where +M((x, y)) = +�k, if x + y < 0; +0, otherwise. +Clearly M is finitely generated with its only birth in (−∞, −∞). It is not finitely +presented, since the deaths happen at points (n, −n) for all n ∈ Z. +Finally, we want to relate Theorem 4.7 to the work of Perling ([8]). Recall that +a subset L ⊆ Z +n is a join-sublattice if mub(S) ∈ L for every finite subset S ⊆ L. +Note that this is equivalent to the condition that L = ˆL. Given a join-sublattice +L ⊆ Z +n, following Perling in [8, pp. 16-19, Ch. 3.1], we define the zip-functor +zipL : RZn-Mod → RL-Mod +and the unzip-functor +unzipL : RL-Mod → RZ +n-Mod. +Contrary to Perling, we do not assume that R is a field. The zip-functor maps +an RZn-module M to the RL-module resL M, whereas he unzip-functor maps an +RL-module N to an RZ +n-module unzipL N defined by +(unzipL N)(c) = +�N(mub(L ∩ ↓c)), if L ∩ ↓c ̸= ∅; +0, otherwise +for all c ∈ Z +n. Note that Supp(unzipL N) ⊆ ↑L. +Remark 4.12. It turns out that unzipL is essentially the same thing as resα, when +L is finite and α := αL. There is the slight complication that unzipL is defined for +RL-modules, while resα is defined for R˜L-modules. We may, however, extend an +RL-module N to an R˜L-module ˜N by setting +˜N((−∞, . . . , −∞)) = 0, + +FINITE PRESENTATION OF FINITELY DETERMINED MODULES +15 +if (−∞, . . . , −∞) /∈ L, and ˜N(c) = N(c), otherwise. Having defined the module ˜N +in this way, we see that unzipL N ∼= resα ˜N. +Given an RZn-module M, the join-sublattice L is called M-admissible in [8, p. 18, +Def. 3.4] if the condition M ∼= unzipL zipL M is satisfied. This leads us to the +following proposition. +Proposition 4.13. Let M be an RZn-module, and L a finite join-sublattice. Then +L is M-admissible if and only if M is L-determined. +Proof. Let c ∈ Z +n. With the earlier notation, we see that +unzipL zipL M = unzipL resL M ∼= resα � +resL M, +where +(resα � +resL M)(c) = +� +(resα res˜L M)(c), if L ∩ ↓c ̸= ∅; +0, otherwise. +Assume first that M ∼= unzipL zipL M. If L ∩ ↓c = ∅, we have M(c) = 0 by the +definition of the functor unzipL. But in this case α(c) ≤ c, so that L ∩ ↓α(c) = ∅. +Using the definition of unzipL again, we get +(resα res˜L M)(c) = M(α(c)) = 0. +On the other hand, if there is an element d ∈ L ∩ ↓c, then, by the above formula, +M(c) ∼= (resα res˜L M)(c). Thus, +M ∼= resα res˜L M +and Supp(M) ⊆ ↑L, so M is L-determined by Theorem 2.5. +Conversely, suppose that M is L-determined. By Theorem 2.5, we have M ∼= +resα res˜L M and Supp(M) ⊆ ↑L. The above formula shows us that +(unzipL zipL M)(c) = (resα res˜L M)(c) +for all c ∈ ↑L. If c /∈ ↑L, then c /∈ Supp(M), which means that M(c) = 0. In this +case, we also have (unzipL zipL M)(c) = 0 by the definition of the functor unzipL. +Thus we have an isomorphism +M ∼= unzipL zipL M. +□ +References +[1] M. Brun and G. Fløystad, The auslander–reiten translate on monomial rings, Advances in +Mathematics 226 (2011), no. 1, 952–991. +[2] G. Carlsson and A. Zomorodian, The theory of multidimensional persistence, Discrete & +Computational Geometry 42 (2009), no. 1, 71–93. +[3] A. Djament, Des propri´et´es de finitude des foncteurs polynomiaux, Fundamenta Mathemat- +icae 233 (2016), 197–256. +[4] E. Hyry and M. Klemetti, Generalized persistence and graded structures, Homology, Homo- +topy and Applications 24 (2022), no. 1, 27–53. +[5] W. L¨uck, Transformation groups and algebraic k-theory, Springer, 1989. +[6] E. Miller, The alexander duality functors and local duality with monomial support, Journal +of Algebra 231 (2000), no. 1, 180–234. +[7] E. +Miller, Homological +algebra +of +modules +over +posets, +arXiv e-prints +(July +2020), +arXiv:2008.00063, available at 2008.00063. +[8] M. Perling, Resolutions and cohomologies of toric sheaves: the affine case, International +Journal of Mathematics 24 (2013), no. 09, 1350069. + +16 +EERO HYRY AND MARKUS KLEMETTI +[9] N. Popescu, Abelian categories with applications to rings and modules, Vol. 3, Academic +Press, 1973. +[10] T. tom Dieck, Transformation groups and representation theory, Vol. 766, Springer, 2006. +Email address: eero.hyry@tuni.fi +Faculty of Information Technology and Communication Sciences, Tampere Univer- +sity, Kanslerinrinne 1 (Pinni B), Tampere, 33100, Finland +Email address: markus.o.klemetti@gmail.com +Faculty of Information Technology and Communication Sciences, Tampere Univer- +sity, Kanslerinrinne 1 (Pinni B), Tampere, 33100, Finland + diff --git a/cdFAT4oBgHgl3EQfYR04/content/tmp_files/load_file.txt b/cdFAT4oBgHgl3EQfYR04/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbf8b879e96f3106f5cdb633ca34763aacffa586 --- /dev/null +++ b/cdFAT4oBgHgl3EQfYR04/content/tmp_files/load_file.txt @@ -0,0 +1,848 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf,len=847 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='08538v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='AT] 20 Jan 2023 FINITE PRESENTATION OF FINITELY DETERMINED MODULES EERO HYRY AND MARKUS KLEMETTI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this article we study certain notions of ‘tameness’ for the per- sistence modules studied in topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In particular, we show that after adding infinitary points the so called finitely determined modules become finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Introduction This article is motivated by topological data analysis, which is a recent field of mathematics studying the shape of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' One of the main methods of topological data analysis is persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In persistent homology one studies the data by associating a filtered topological space to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' By taking homology with coefficients in a field, one obtains a diagram of vector spaces and linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This diagram is called a persistence module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In the standard case, the filtration is indexed by Z or R, but the indexing set can by any poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Carlsson and Zomorodian realized that one can consider persistence modules indexed by Zn as Zn-graded modules over a polynomial ring of n variables (see [2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 78, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This opened the way for methods of commutative algebra and algebraic geometry in topological data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' However, it is important to consider also more general indexing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' More formally, a persistence module indexed by a poset C with coefficients in a field k is a functor from C, interpreted as a category, to the category of k-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For the sake of generality, instead of a field k, we prefer in this article to work with any commutative ring R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Following the terminology of representation theory, we call a functor C → R-Mod an RC-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this terminology, a persistence module is then a kC-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Persistence modules need not be finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For computational reasons, one has therefore introduced several notions of ‘tameness’ for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this context, Miller defines in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 24, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1] an encoding of an RC-module M by a poset D to be a poset morphism f : C → D with an RD-module N such that the restriction resf N ∼= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We define in [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 22, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1] an RC-module M to be S-determined if there exists a subset S ⊆ C such that Supp(M) ⊆ ↑S, and for every c ≤ d in C the implication S ∩ ↓c = S ∩ ↓d ⇒ M(c ≤ d) is an isomorphism holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For any T ⊆ C, we use the usual notations ↑T := {c ∈ C | t ≤ c for some t ∈ T } Date: January 19, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 55N31, 13E15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Persistence module, Finitely determined, Finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 1 2 EERO HYRY AND MARKUS KLEMETTI and ↓T := {c ∈ C | c ≤ t for some t ∈ T } for the upset generated and the downset cogenerated by T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This is a straightforward generalization of the notion of a ‘positively a-determined’ Nn- graded module, where a ∈ Nn, as defined in [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 186, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' One can look at them as modules determined by their restriction to the interval [0, a] ⊆ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' They are finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Positively a-determined modules have been much studied by commutative algebraists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' [1] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose now that S ⊆ C is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We will consider the set ˜S of all minimal upper bounds of the subsets of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We are going to define a functor α: C → ˜S by mapping an element of C to the unique minimal upper bound of the elements of S below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In our main result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, we will prove that M is S-determined for some finite S ⊆ C if and only if α is an encoding of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The definition given by Miller in [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 186, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1] includes also the so called ‘finitely determined’ modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' They are further studied in the context of topological data-analysis in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Finitely determined modules are Zn-graded modules fully determined by their restriction to an interval [a, b] ⊆ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='They are not finitely presented in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' However, as a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5 we can show in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7 that after adding infinitary points to Zn, finitely determined modules in fact become finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We follow here an idea due to Perling (see [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We also show in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='13 that our terminology is compatible with that of admissible posets used in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Preliminaries Throughout this article we use the terminology of category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We will always assume that C is a small category and R a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For any set X, we denote by R[X] the free R-module generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' An RC-module is a functor from C to the category of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' A morphism between RC-modules is a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For more details on RC-modules, we refer to [5] and [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Recall first that an RC-module M is called finitely generated if there exists an epimorphism � i∈I R[MorC(ci, −)] → M, where I is a finite set, and ci ∈ C for all i ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' finitely presented if there exists an exact sequence � j∈J R[MorC(dj, −)] → � i∈I R[MorC(ci, −)] → M → 0, where I and J are finite sets, and ci, dj ∈ C for all i ∈ I and j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' See, for example, [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let ϕ: S → C be a functor between small categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Recall that the restriction resϕ : RC-Mod → RS-Mod is the functor defined by precomposition with ϕ, and the induction indϕ : RS-Mod → RC-Mod is its left Kan extension along ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The induction is the left adjoint of the restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The counit of this adjunction gives us for every RC-module M the canonical morphism µM : indϕ resϕ M → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' FINITE PRESENTATION OF FINITELY DETERMINED MODULES 3 More explicitly, for any RC-module M and RS-module N, we have the pointwise formulas (resϕ M)(s) = M(ϕ(s)) and (indϕ N)(c) = colim (t,u)∈(ϕ/c)N(t) for all s ∈ S and c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here (ϕ/c) denotes the slice category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Its objects are pairs (s, u), where s ∈ S and u: ϕ(s) → c is a morphism in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For (s, u), (t, v) ∈ Ob(ϕ/c), a morphism (s, u) → (t, v) is a morphism f : s → t in S with vϕ(f) = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We will typically assume that S is a full subcategory of C and that ϕ is the inclusion functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this case, we use the notations resS and indS instead of resϕ and indϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If C is also a poset, the latter formula yields (indS N)(c) = colim t∈S, t≤cN(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let C be a small category and S ⊆ C a full subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' An RC-module M is said to be S-generated if the natural morphism ρM : � s∈S M(s)[MorC(s, −)] → M is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here M(s)[MorC(s, −)] := M(s) ⊗R R[MorC(s, −)], where the tensor product is taken pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since the morphism ρM factors through the canonical morphism µM, we see that M is S-generated if and only if µM is an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Following [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='14], we say that M is S-presented if it is S-generated and the following condition holds: Given an exact sequence of RC-modules 0 → L → N → M → 0, where N is S-generated, then L is S-generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' It is shown in [3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 13, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='14], that M is S-presented if and only if µM is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Modules over strongly bounded posets In order to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, we need to recall some order theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In the following, C always denotes a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let S ⊆ C be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We denote the set of minimal upper bounds of S by mub(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If S is finite, we set ˆS := � ∅̸=S′⊆S mub(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In other words, ˆS is the set of minimal upper bounds of non-empty subsets of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We say that the poset C is strongly bounded from above if every finite S ⊆ C has a unique minimal upper bound in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If C is strongly bounded from above, then ˆS is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The condition of C being strongly bounded from above is equivalent to C being a bounded join-semilattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Also note that if C is strongly bounded from above, then C is weakly bounded from above and mub-complete, as defined in [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 23, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let C be strongly bounded from above, and let S ⊆ C be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' From now on, we consider mub(S) as an element of C, and not as a (one element) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4 EERO HYRY AND MARKUS KLEMETTI In particular, every element of ˆS is then of the form mub(S′), where S′ ⊆ S is a non-empty subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Viewing C as a join-semilattice, we have the join-operation a ∨ b := mub(a, b) := mub({a, b}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Extending this operation to finite sets, we get an operation that coincides with taking minimal upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let C be strongly bounded from above, and let S ⊆ C be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then ˆˆS = ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' An element s ∈ ˆˆS may be written as s = mub(mub(S1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , mub(Sn)), where S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , Sn are (finite) non-empty subsets of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since the join-operation is associative in join-semilattices, we see that s = n� i=1 ( � Si) = � ( n� i=1 Si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This implies that s = mub(S1 ∪ · · · ∪ Sn), which belongs to ˆS by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Assume that C is strongly bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then C has a minimum element min(C) = mub(∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let S ⊆ C be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Denote ˜S := ˆS ∪ {min(C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We define a poset morphism αS : C → ˜S by setting αS(c) = mub(S ∩ ↓c) for every c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In other words, αS maps each c ∈ C to the minimal upper bound of the elements of S below it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' To show that αS actually is a poset morphism, suppose that c ≤ d in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then S ∩ ↓c ⊆ S ∩ ↓d, which implies that αS(c) ≤ αS(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let C be strongly bounded from above, and let S ⊆ C be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then αS = α ˆS = α ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2, we first note that ˜ˆS = ˜S and ˜˜S = ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We claim that mub(S ∩ ↓c) = mub( ˆS ∩ ↓c) = mub( ˜S ∩ ↓c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The latter equation follows from the fact that for all subsets T ⊆ C, we have mub(T ) = mub(T ∪ {min(C)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In particular, mub(T ) = min(C), if T = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For the first equation, since S ⊆ ˆS, we have mub(S ∩ ↓c) ≤ mub( ˆS ∩ ↓c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' On the other hand, ˆS ∩↓c is a subset of ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus mub( ˆS ∩↓c) ∈ ˆˆS = ˆS, where the equation follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' By the definition of ˆS, we may now write mub( ˆS ∩ ↓c) = mub(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , sn), where s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , sn ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Furthermore, mub( ˆS ∩↓c) ≤ c, so we also have s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , sn ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This implies that mub(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , sn) ≤ mub(S ∩ ↓c), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ FINITE PRESENTATION OF FINITELY DETERMINED MODULES 5 Encouraged by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3, we will just write α instead of αS, if there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Before moving on to the main theorem of this section, we require one more lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let C be strongly bounded from above, and let S ⊆ C be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then ˆS ∩ ↓α(c) = ˆS ∩ ↓c for all c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We immediately see that ˆS ∩ ↓α(c) ⊆ ˆS ∩ ↓c, because α(c) ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose that d ∈ ˆS ∩ ↓c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We need to show that d ≤ α(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3, because now α(c) = α ˆS(c) = mub( ˆS ∩ ↓c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Let C be strongly bounded from above, let M be an RC-module, and let S ⊆ C be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The morphism α gives rise to a natural transformation Tα : resα res ˜S M → M, where for any c ∈ C, Tα,c is the morphism M(α(c) ≤ c): (resα res ˜S M)(c) = M(α(c)) → M(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We are now able to prove our main result Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let C be strongly bounded from above, and let M be an RC-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Given a finite subset S ⊆ C, the following conditions are equivalent: 1) For all c ≤ d in C, S ∩ ↓c = S ∩ ↓d ⇒ M(c ≤ d) is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) Tα : resα res ˜S M → M is an isomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 3) α is an encoding of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If min(C) ∈ S, then condition 1) says that M is S-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose first that 1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We can safely assume that S includes the minimum element of C, so that Supp(M) ⊆ ↑S = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This will not affect the sets ˆS or ˜S, nor the functor α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Therefore M is S-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We have proved in [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 25, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='13] that an S-determined module is ˆˆS-presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2 now tells us that M is ˆS-presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' So M ∼= ind ˆS res ˆS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This implies that for c ∈ C, (resα res ˜S M)(c) = M(α(c)) ∼= colim d≤α(c), d∈ ˆS M(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Furthermore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4, we get colim d≤α(c), d∈ ˆS M(d) = colim d≤c, d∈ ˆS M(d) = (ind ˆS res ˆS M)(c) ∼= M(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If 2) holds, we immediately see that the functor α with the R ˜S-module res ˜S M is an encoding of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Finally, suppose that 3) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Assume that c ≤ d and S ∩ ↓c = S ∩ ↓d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We need to show that M(c ≤ d) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since α is an encoding of M, there exists an R ˜S-module N such that resα N ∼= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here resα N(c ≤ d) is the morphism N(α(c) ≤ α(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We note that α(c) = mub(S ∩ ↓c) = mub(S ∩ ↓d) = α(d), 6 EERO HYRY AND MARKUS KLEMETTI so the morphism resα N(c ≤ d) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus M(c ≤ d) is an isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Therefore 1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Adding infinitary points One approach to understand RZn-modules better is to expand the set Zn to include points at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This idea has been utilized by Perling in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Set Z := Z ∪ {−∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' It is easy to see that Z n inherits a poset structure from Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Any RZn-module M can be naturally extended to an RZ n-module M by setting M(c) = lim d≥c, d∈Zn M(d) for all c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' More formally, this is the coinduction of M with respect to the inclusion Zn → Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The functor M �→ M establishes an equivalence of categories between the category RZn-Mod and its essential image in RZ n-Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let S ⊆ Z n be a finite non-empty subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We denote by mlb(S) the (unique) maximal lower bound of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this section, we will define a morphism β “dual” to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The idea is to map an element to the maximal lower bound of the elements of S above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The morphism β will play a crucial role in the proof of our main result, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We restrict ourselves to cartesian subsets of Z n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' subsets of the form S = S1 × · · · × Sn, where S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , Sn are subsets of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this situation, we can calculate α and β coordinatewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We begin with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let pi : Z n → Z be the canonical projection for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=', n}, and let S ⊆ Z n be a finite non-empty subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then 1) mub(S) = (max(p1(S)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , max(pn(S)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) mlb(S) = (min(p1(S)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , min(pn(S))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since both 1) and 2) are proved in the same way, we will only present the proof of 1) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The existence of max(pi(S)) follows from the fact that pi(S) is non-empty, linearly ordered and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Write d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , dn) := mub(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We will show that di = max(pi(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' First, since d is an upper bound of S and the canonical projection pi preserves order, we see that di = pi(d) ≥ max(pi(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Secondly, if max(pi(S)) < di, then d′ := (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , di−1, max(pi(S)), di+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , dn) is an upper bound of S such that d′ < d, contradicting the minimality of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus di = max(pi(S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Let S := S1 × · · · × Sn ⊆ Z n be a cartesian subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We write S = ˜S1 × · · · × ˜ Sn, where �Si = Si ∪ {−∞} ⊆ Z for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that if S is finite, then so is S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let a ≤ b in Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We write a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , an) and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For the closed interval [a, b] = {c ∈ Zn | a ≤ c ≤ b} = [a1, b1] × · · · × [an, bn], FINITE PRESENTATION OF FINITELY DETERMINED MODULES 7 we have [a, b] = � [a1, b1] × · · · × � [an, bn] = {(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) | ai ≤ ci ≤ bi or ci = −∞ (i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n})}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We now have Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let S := S1 × · · · × Sn ⊆ Z n be a finite cartesian subset, and let T ⊆ S be a finite non-empty subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then 1) mub(T ) ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) mlb(T ) ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 3) ˜S = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' To prove 1), let pi be the canonical projection Z n → Z for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1 1), we get that mub(T ) = (max(p1(T )), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , max(pn(T ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus mub(T ) ∈ S, because pi(T ) ⊆ pi(S) = Si for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Next, the proof for 2) is done in the same way as 1), this time using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Finally, for 3), we note that S is finite and cartesian, so 1) implies ˆS = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since S already contains the minimum element of Z n, we get ˜S = ˆS ∪ {(−∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , −∞)} = S ∪ {(−∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , −∞)} = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Let S := S1 ×· · ·×Sn ⊆ Z n be a finite cartesian subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since ˜S = S by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3 3), we have a poset morphism α := αS : Z n → S, where α(c) = mub(S ∩ ↓c) for all c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3 2), we now can define a “dual” poset morphism β := βS : S → S by setting β(c) = mlb(S ∩ ↑c) for all c ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here the set S ∩ ↑c is always non-empty, because S is final in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We can now give coordinatewise formulas for α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We write αi := αSi and βi := βSi for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Z n, we have 1) α(c) = (α1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , αn(cn));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) if c ∈ S, then β(c) = (β1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , βn(cn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' To prove 1), we will first show that pi(S ∩ ↓c) = Si ∩ ↓ci, where pi : Z n → Z is the canonical projection for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since pi(S) = Si and pi(↓c) = ↓ci, we see that pi(S ∩↓c) ⊆ Si ∩↓ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For the other direction, suppose that d ∈ Si ∩ ↓ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then d ≤ ci, so we have an element d′ := (−∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , −∞, d, −∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=', −∞) ∈ S ∩ ↓c 8 EERO HYRY AND MARKUS KLEMETTI such that pi(d′) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Hence pi(S ∩ ↓c) = Si ∩ ↓ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Now, using this result and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1 1), we get α(c) = mub(S ∩ ↓c) = (max(S1 ∩ ↓c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , max(Sn ∩ ↓cn)) = (α1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , αn(cn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For 2), the proof is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We will first show that pi(S ∩ ↑c) = Si ∩ ↑c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' From pi(S) = Si and pi(↑c) = ↑ci, we see that pi(S ∩ ↑c) ⊆ Si ∩ ↑ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Next, suppose that d ∈ Si ∩ ↑ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since c ∈ S, there is an element s := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , sn) ∈ S such that s ≥ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Because d ≥ ci and S is cartesian, we again have an element d′ := (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , si−1, d, si+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , sn) ∈ S ∩ ↑c such that pi(d′) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus pi(S ∩ ↑c) = Si ∩ ↑c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' To finish the proof, we use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1 2): β(c) = mlb(S ∩ ↑c) = (min(S1 ∩ ↑c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , min(Sn ∩ ↑cn)) = (β1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , βn(cn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ We note that α and β ◦ α are “continuous” in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 1) If N is an RS-module, then lim d≥c, d∈Zn N(α(d)) ∼= N(α(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) If Q is an RS-module, then lim d≥c, d∈Zn Q((β ◦ α)(d)) ∼= Q((β ◦ α)(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For 1), suppose that N is an RS-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c′ := (c′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , c′ n) ∈ Z n as follows: For any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}, we set ai = min(Si ∩ Z), if it exists, and c′ i := � max(ci, 0), if Si ∩ Z = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' max(ci, ai − 1), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This guarantees that we always have c ≤ c′ and c′ ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' With the notation from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4, we may write α(c) = (α1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , αn(cn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If Si ∩ Z = ∅, then αi(c′ i) = −∞ = αi(ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Similarly, if c′ i = ai − 1, then αi(c′ i) = −∞ = αi(ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus α(c) = α(c′) in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since α is a poset morphism, we see that for all d ∈ Zn such that c ≤ d ≤ c′, α(c) = α(d) = α(c′), and therefore N(α(c)) = N(α(d)) = N(α(c′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' FINITE PRESENTATION OF FINITELY DETERMINED MODULES 9 Furthermore, because the set {d ∈ Zn | c ≤ d ≤ c′} is an initial subset of the set {d ∈ Zn | c ≤ d}, we have lim d≥c, d∈Zn N(α(d)) ∼= lim c≤d≤c′, d∈Zn N(α(d)) ∼= N(α(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Next, for 2), let Q be an RS-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Now resβ Q is an RS-module, so by 1), we have lim d≥c, d∈Zn(resβ Q)(α(d)) ∼= (resβ Q)(α(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' On the other hand, by definition, for all e ∈ Z n, (resβ Q)(α(e)) = Q(β(α(e))) = Q((β ◦ α)(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This means that we may write the above isomorphism as lim d≥c, d∈Zn Q((β ◦ α)(d)) ∼= Q((β ◦ α)(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let N be an RZ n-module, and let c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then 1) lim d≥c, d∈Zn N(α(d)) ∼= N(α(c));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) lim d≥c, d∈Zn N((β ◦ α)(d)) ∼= N((β ◦ α)(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For 1), we note that resS N is an RS-module, where (resS N)(d) = N(d) for all d ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We may then apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5 1) to get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For 2), we use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5 2) on the RS-module resS N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Finitely determined modules Let M be an RC-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We say that M is pointwise finitely presented if M(c) is finitely presented for all c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Slightly generalizing the definition of Miller in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 25, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5], where R = k is a field, we say that an RZn-module M is finitely determined, if M is pointwise finitely presented, and for some a ≤ b in Zn, the convex projection π: Zn → [a, b] gives M an encoding by the closed interval [a, b] ⊆ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here the convex projection π takes every point in Zn to its closest point in the interval [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , an) and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , bn), we have for any c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Zn, π(c) = (π1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , πn(cn)), where πi(ci) = max(ai, min(ci, bi)) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that a pointwise finitely presented RZn-module M is finitely determined if and only if there exists a closed interval [a, b] ⊆ Zn such that the morphisms M(c ≤ c + ei) (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n) are isomorphisms whenever ci lies outside [ai, bi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be an RZn-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then M is encoded by the closed interval [a, b] with the convex projection π: Zn → [a, b] if and only if M ∼= resπ res[a,b] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Indeed, if M ∼= resπ N for some R[a, b]-module N, then for all c ∈ Zn, we have M(c) ∼= (resπ N)(c) = N(π(c)) = N(π(π(c))) ∼= M(π(c)) because for all c ∈ Zn, π(π(c)) = π(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 10 EERO HYRY AND MARKUS KLEMETTI We would now like to investigate how the notion of finite determinacy relates to our notion of S-determinacy, when S is finite and M is pointwise finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' While the requirement that Supp(M) ⊆ ↑S does not necessarily hold for finitely determined modules, we do have the following: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be an RZn-module, and a, b ∈ Zn such that a ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Set u := (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , 1) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If M is [a + u, b]-determined, then M has an encoding by the closed interval [a, b] with the convex projection π: Zn → [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The converse implication holds if Supp(M) ⊆ ↑a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For the first implication, suppose that M is [a + u, b]-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We write a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , an) and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We note that if ci ≤ ai for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}, then also πi(ci) ≤ ai, so that c, π(c) /∈ Supp(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Otherwise c > a, in which case π(c) ≤ c and [a + u, b] ∩ ↓π(c) = [a + u, b] ∩ ↓c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus M(π(c)) → M(c) is an isomorphism by the definition of [a + u, b]-determined modules, and M ∼= resπ res[a,b] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' To prove the converse, assume that Supp(M) ⊆ ↑a and M has an encoding by the closed interval [a, b] with the encoding convex projection π: Zn → [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose that ci < ai for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' From the condition Supp(M) ⊆ ↑a, we see that M(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since M is finitely determined, we also have M(π(c)) = M(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus M(c) = 0 if ci ≤ ai for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If this is not the case, we have c ≥ a + u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c ≤ d in C such that a + u ≤ c ≤ d and [a + u, b] ∩ ↓ c = [a + u, b] ∩ ↓ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This implies that π(c) = π(d), so M(c ≤ d) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ To proceed, we have to shift our focus to RZ n-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let a ≤ b in Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' With the notation from section 3, we will view the case S = [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In particular, we have α = α[a,b] and β = β[a,b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4 gives us formulas for α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Z n and d := (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , dn) ∈ [a, b], then α(c) = (α1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , αn(cn)) and β(d) = (β1(d1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , βn(dn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here αi := αSi and βi := βSi for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Explicitly, αi(ci) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∞, if ci < ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' ci, if ai ≤ ci ≤ bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' bi, if ci > bi and βi(di) = �ai, if di = −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' di, otherwise for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The next proposition shows us that the composition β ◦ α is an extension of the convex projection π from Zn to Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let π: Zn → [a, b] be the convex projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then for any c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , cn) ∈ Zn, π(c) = (β ◦ α)(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose first that n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Recall that π(c) = max(a, min(c, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Now there are three cases: If c ∈ [a, b], then (β ◦ α)(c) = β(c) = c = π(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If c < a, then (β ◦ α)(c) = β(−∞) = a = π(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If c > b, then (β ◦ α)(c) = β(b) = b = π(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose next that n > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4, we may write α(c) = (α1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , αn(cn)) and β(d) = (β1(d1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , βn(dn)) FINITE PRESENTATION OF FINITELY DETERMINED MODULES 11 for all d ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Similarly, recall that π(c) = (π1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , πn(cn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' It now follows from the case n = 1 that (β ◦ α)(c) = β(α1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , αn(cn)) = ((β1 ◦ α1)(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , (βn ◦ αn)(cn)) = (π1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , πn(cn)) = π(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In an effort to keep the notation simpler, we only defined β for the elements in the image of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Of course, we could have defined β in a fully dual fashion to α, starting from posets that are strongly bounded from below, adding the point ∞ to Z, and defining a set S dually to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This would have resulted in the situation where (β|S ◦ α)(c) = (α|S ◦ β)(c) = π(c) for all c ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In other words, the same result would have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We saw in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1 that if M is encoded by a closed interval [a, b] with the convex projection π: Zn → [a, b], we have M(c) ∼= M(π(c)) for all c ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' On the other hand, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, we have M(α(c)) ∼= M(c) for all c ∈ Z n if M is S-determined and S ⊆ Z n is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In preparation for the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7, we will now show that a similar result applies to β in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Set u := (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , 1) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be an RZn-module, and let c ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 1) If M has an encoding by the closed interval [a, b] with the convex projection π: Zn → [a, b], then M(c) ∼= M(β(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) If M is [a + u, b]-determined, then M(c) ∼= M(β(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' To show 1), suppose that M has an encoding by the closed interval [a, b] with the convex projection π: Zn → [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then, by the definition of M, M(c) = lim d≥c, d∈Zn M(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The encoding gives us M(d) ∼= M(π(d)) for all d ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This implies that M(c) ∼= lim d≥c, d∈Zn M(π(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We may now apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='6 to see that M(c) ∼= M(β(α(c))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that c ∈ [a, b] implies α(c) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus M(c) ∼= M(β(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Next, to prove 2), let M be [a + u, b]-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Since c ≤ β(c), it is then enough to show that [a + u, b]∩↓c = [a + u, b]∩↓β(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We instantly have ↓c ⊆ ↓β(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For the other direction, let d := (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , dn) ∈ [a + u, b] ∩ ↓β(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We want to show that d ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Recall that we may write β(c) = (β1(c1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , βn(cn)), where βi(ci) = �ai, if ci = −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' ci, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 12 EERO HYRY AND MARKUS KLEMETTI for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Suppose that i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If βi(ci) = ci, we have di ≤ βi(ci) = ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Otherwise, if βi(ci) = ai, we must have di = ci = −∞, because di, ci ∈ [ai + 1, bi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We conclude that d ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be a pointwise finitely presented RZn-module and let c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If M is finitely determined with the convex projection π: Zn → [a, b], then from the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, we have M(c) ∼= M((β ◦α)(c)), so that M is pointwise finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We are now ready to state Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be a pointwise finitely presented RZn-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then the following are equivalent: 1) M is finitely determined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) M is S-determined for some finite S ⊆ Z n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 3) M is finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We will first show the equivalence of 1) and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that for any finite subset S ⊆ Z n, we can always find a, b ∈ Zn such that S ⊆ [a + u, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Consider the functor α′ = α[a+u,b] : Z n → [a + u, b], and denote its restriction to Zn by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, M is [a + u, b]-determined if and only if M is encoded by α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' That is, M ∼= resα′ N for some R[a + u, b]-module N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' By restricting to Zn, we see that M ∼= resZn resα′ N = resα N, so α encodes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Conversely, if M is encoded by α, then M has an obvious encoding by α′, because α is a surjection on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Next, we note that the restriction of β to [a + u, b], β : [a + u, b] → [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' is an isomorphism of posets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Therefore β ◦ α is an encoding of M if and only if α is an encoding of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' These conditions are equivalent to M being finitely determined, because β ◦ α = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Namely, for all c ∈ Zn, we have (β ◦ α)(c) = (β ◦ α′)(c), where (β ◦ α′)(c)i = � βi(−∞), if ci = ai, βi(αi(ci)), else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' = �ai, if ci = ai, πi(ci), else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' = π(c)i for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Finally, we observe that the equivalence of 2) and 3) follows from the main result of our previous paper, [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 25, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For Z n-modules, it states that being pointwise finitely presented and S-determined for some finite S ⊆ Z n is equivalent to being finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Also note Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='6, which shows us that M is pointwise finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ We are now able to give a “sharpened” version of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' FINITE PRESENTATION OF FINITELY DETERMINED MODULES 13 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If M is an RZn-module and a, b ∈ Zn such that a ≤ b, then the following are equivalent: 1) M is encoded by the convex projection π: Zn → [a, b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 2) M is [a + u, b]-determined, where u := (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , 1) ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We showed in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7 that 2) implies 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Conversely, suppose that 1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c ≤ d in Z n such that [a + u, b] ∩ ↓c = [a + u, b] ∩ ↓d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Coordinatewise, for i = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , n}, this implies that either ci = di, bi ≤ ci < di or ci < di ≤ ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In any case, (β ◦ α)(c) = (β ◦ α)(d), so that M(c) ∼= M((β ◦ α)(c)) = M((β ◦ α)(d)) ∼= M(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus M(c ≤ d) is an isomorphism, and M is [a + u, b]-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' □ To demonstrate Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='8, it is convenient to take the point of view of topological data analysis, and consider the births and deaths of elements of a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Given an RZn module M, one can track how an element x ∈ M(c), where c ∈ Z n, evolves when mapped with the homomorphisms M(c ≤ c′), (c, c′ ∈ Z n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We say that the element x is born at c if it is not in the image of any morphism M(c′ ≤ c), where c′ < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' On the other hand, the element x dies at c′′ if M(c ≤ c′′)(x) = 0, but M(c ≤ c′)(m) ̸= 0 for all c ≤ c′ < c′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Consider now an RZ2-module M that is finitely determined, and let π: Z2 → [a, b] be the accompanying convex projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that no new elements are born or die in the leftmost edge or the bottom edge of the box [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This follows from the fact that every element on these two edges has already appeared infinite times before, and was born at some infinitary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let us write a = (a1, a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For example, if an element, say x ∈ M((a1, c)), maps to zero on the leftmost edge of [a, b], in M((a1, c + 1)), then x ∈ M((−∞, c)) will also map to zero in M((−∞, c + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus x does not “die” at the point (a1, c + 1), but rather at the infinitary point (−∞, c + 1) ∈ [a + u, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be an RZn-module and c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Consider the natural homo- morphism λM,c : colim d≤c,d∈[a+u,b] M(d) → M(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Following [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 15, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='6], we say that c is a birth if λM,c is a non-epimorphism, and a death if λM,c is a non-monomorpism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Furthermore, suppose that M is [a + u, b]-determined, and the births are “well-behaved” enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' That is, for any birth c, the module M(c)/ Im λM,c is projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The latter of course holds if R is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then, as we discussed in [4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 21, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='27], births and deaths show the positions of the minimal generators and relations of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In the next example, we will demonstrate how, for a finitely determined module M, the extension M has births and deaths at infinitary points that guarantee the existence of a finite presentation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be an RZ2-module that is defined on objects by M(c) = �R, if c ≤ (0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 0, otherwise, 14 EERO HYRY AND MARKUS KLEMETTI for all c ∈ Z2, and where a morphism R → R is always idR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then M is finitely determined with the convex projection π: Z2 → [(0, 0), (1, 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Now, by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='8, M is [(1, 1), (1, 1)]-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here [(1, 1), (1, 1)] is the set {(−∞, −∞), (1, −∞), (−∞, 1), (1, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In particular, we have M((−∞, −∞)) = R, and M((−∞, 1)) = M((1, −∞)) = M((1, 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Furthermore, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7, M is now finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In more concrete terms, we have an exact sequence of RZ 2-modules K → N → M → 0, where N = R[MorZ 2((−∞, −∞), −)] and K = R[MorZ 2((1, −∞), −)] ⊕ R[MorZ 2((−∞, 1), −)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Here (−∞, −∞) is the only birth of M, while (1, −∞) and (−∞, 1) are the deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If k is a field, then it is well known that finitely generated kZn- modules are finitely presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This result, however, does not apply to kZ n- modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' For a counterexample, consider a kZ2-module M, where M((x, y)) = �k, if x + y < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Clearly M is finitely generated with its only birth in (−∞, −∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' It is not finitely presented, since the deaths happen at points (n, −n) for all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Finally, we want to relate Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='7 to the work of Perling ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Recall that a subset L ⊆ Z n is a join-sublattice if mub(S) ∈ L for every finite subset S ⊆ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that this is equivalent to the condition that L = ˆL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Given a join-sublattice L ⊆ Z n, following Perling in [8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 16-19, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='1], we define the zip-functor zipL : RZn-Mod → RL-Mod and the unzip-functor unzipL : RL-Mod → RZ n-Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Contrary to Perling, we do not assume that R is a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The zip-functor maps an RZn-module M to the RL-module resL M, whereas he unzip-functor maps an RL-module N to an RZ n-module unzipL N defined by (unzipL N)(c) = �N(mub(L ∩ ↓c)), if L ∩ ↓c ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 0, otherwise for all c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Note that Supp(unzipL N) ⊆ ↑L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' It turns out that unzipL is essentially the same thing as resα, when L is finite and α := αL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' There is the slight complication that unzipL is defined for RL-modules, while resα is defined for R˜L-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' We may, however, extend an RL-module N to an R˜L-module ˜N by setting ˜N((−∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , −∞)) = 0, FINITE PRESENTATION OF FINITELY DETERMINED MODULES 15 if (−∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' , −∞) /∈ L, and ˜N(c) = N(c), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Having defined the module ˜N in this way, we see that unzipL N ∼= resα ˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Given an RZn-module M, the join-sublattice L is called M-admissible in [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 18, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='4] if the condition M ∼= unzipL zipL M is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' This leads us to the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let M be an RZn-module, and L a finite join-sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Then L is M-admissible if and only if M is L-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Let c ∈ Z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' With the earlier notation, we see that unzipL zipL M = unzipL resL M ∼= resα � resL M, where (resα � resL M)(c) = � (resα res˜L M)(c), if L ∩ ↓c ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Assume first that M ∼= unzipL zipL M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If L ∩ ↓c = ∅, we have M(c) = 0 by the definition of the functor unzipL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' But in this case α(c) ≤ c, so that L ∩ ↓α(c) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Using the definition of unzipL again, we get (resα res˜L M)(c) = M(α(c)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' On the other hand, if there is an element d ∈ L ∩ ↓c, then, by the above formula, M(c) ∼= (resα res˜L M)(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus, M ∼= resα res˜L M and Supp(M) ⊆ ↑L, so M is L-determined by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Conversely, suppose that M is L-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='5, we have M ∼= resα res˜L M and Supp(M) ⊆ ↑L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' The above formula shows us that (unzipL zipL M)(c) = (resα res˜L M)(c) for all c ∈ ↑L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' If c /∈ ↑L, then c /∈ Supp(M), which means that M(c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' In this case, we also have (unzipL zipL M)(c) = 0 by the definition of the functor unzipL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content=' Thus we have an isomorphism M ∼= unzipL zipL M.' metadata={'source': 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+page_content='hyry@tuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='fi Faculty of Information Technology and Communication Sciences, Tampere Univer- sity, Kanslerinrinne 1 (Pinni B), Tampere, 33100, Finland Email address: markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='klemetti@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} +page_content='com Faculty of Information Technology and Communication Sciences, Tampere Univer- sity, Kanslerinrinne 1 (Pinni B), Tampere, 33100, Finland' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQfYR04/content/2301.08538v1.pdf'} diff --git a/d9E3T4oBgHgl3EQfegrm/vector_store/index.faiss b/d9E3T4oBgHgl3EQfegrm/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1e8bc7a5f82a9e1fa4a0d6490a9f6982a5eef2db --- /dev/null +++ b/d9E3T4oBgHgl3EQfegrm/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7b542f37261df60e7df1d8973bffee96609b90e86da33441fc81092802554bc0 +size 19791917 diff --git a/edFQT4oBgHgl3EQfkDbq/vector_store/index.faiss b/edFQT4oBgHgl3EQfkDbq/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..fef990fb5c63d9d7c1c127ae78ffb84c223db3ee --- /dev/null +++ b/edFQT4oBgHgl3EQfkDbq/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5998fbbc8a872bdd884783da105a785e5d2a14bbb9413f455a76c4b1d59a42f3 +size 1966125 diff --git a/ftFJT4oBgHgl3EQfUCze/content/tmp_files/2301.11507v1.pdf.txt b/ftFJT4oBgHgl3EQfUCze/content/tmp_files/2301.11507v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c14e39ce59970ac7c40c54dffc1a30a6eda4322a --- /dev/null +++ b/ftFJT4oBgHgl3EQfUCze/content/tmp_files/2301.11507v1.pdf.txt @@ -0,0 +1,1848 @@ +Semi-Parametric Video-Grounded Text Generation +Sungdong Kim 1 2 Jin-Hwa Kim 1 3 Jiyoung Lee 1 Minjoon Seo 2 +Abstract +Efficient video-language modeling should con- +sider the computational cost because of a large, +sometimes intractable, number of video frames. +Parametric approaches such as the attention mech- +anism may not be ideal since its computational +cost quadratically increases as the video length +increases. Rather, previous studies have relied +on offline feature extraction or frame sampling +to represent the video efficiently, focusing on +cross-modal modeling in short video clips. In +this paper, we propose a semi-parametric video- +grounded text generation model, SeViT, a novel +perspective on scalable video-language modeling +toward long untrimmed videos. Treating a video +as an external data store, SeViT includes a non- +parametric frame retriever to select a few query- +relevant frames from the data store for a given +query and a parametric generator to effectively +aggregate the frames with the query via late fu- +sion methods. Experimental results demonstrate +our method has a significant advantage in longer +videos and causal video understanding. Moreover, +our model achieves the new state of the art on four +video-language datasets, iVQA (+4.8), Next-QA +(+6.9), and Activitynet-QA (+4.8) in accuracy, +and MSRVTT-Caption (+3.6) in CIDEr. +1. Introduction +Recently, there has been the impressive success of vision- +language models (Lu et al., 2019; Radford et al., 2021; +Li et al., 2022a; Alayrac et al., 2022) demonstrating a re- +markable transferability on video-language tasks including +video retrieval, video captioning, and video question answer- +ing (Video QA). Considering video input’s spatio-temporal +aspects, it is often more challenging to process than other +modalities, especially combining video and language modal- +ities. Moreover, modeling video information requires heavy +computations since it comprises lengthy image sequences +(frames). Conventionally, many video-language works have +1NAVER AI Lab 2KAIST AI 3SNU AIIS. Correspondence to: +Minjoon Seo . +Figure 1. Overview of semi-parametric video-grounded text gener- +ation. Treating an untrimmed long video as an external data store, +it first retrieves top-k relevant frames from the data store with +a given input query. Then, a vision-language (VL) transformer +encodes each frame and the input independently and generates +textual output by performing late fusion over top-k frames. +relied on the pre-trained vision or video encoder as an of- +fline feature extractor to represent video densely but effi- +ciently (Sun et al., 2019; Li et al., 2020; Yang et al., 2021). +A line of research has shown the effectiveness of sparse +video representation for video-language modeling (Lei et al., +2021). The sparse video representation approximates a +video with sparsely sampled frames instead of all frames +while allowing gradient updates of the visual encoder with +computationally feasible implementation. Lei et al. (2021) +argue that randomly selected sparse frames work well on var- +ious video-language tasks even with very few frames, e.g., +1-20 frames for 3-180 seconds clips. Recent video-language +studies outperform the performance of the models trained +in the single task by massive video-language pre-training, +employing the sparse frame paradigm (e.g., uniform sam- +pling) (Zellers et al., 2021; Wang et al., 2022a; Zellers et al., +2022; Yang et al., 2022). +However, these studies overlook the limitations of the sparse +video representation based on naive frame sampling. The +arXiv:2301.11507v1 [cs.CV] 27 Jan 2023 + +Input +Search +(MIPS) +Non-Parametric +Frame Retriever +Top-krelevantframes +Untrimmed +Long Video +External Data Store +Parametric +VL Transformer +OutputSemi-Parametric Video-Grounded Text Generation +pre-trained models have been tested on only benchmarks +with short video clips that are usually less than a minute. +We are curious whether the benefits of the sparse frame +sampling are still valid if the length of the source video +gets longer. In particular, we hypothesize the model relying +on a few sampled frames might fail on long untrimmed +videos since the scene changes frequently and the frame +length affects the size of the population. It requires more +frames to be sampled to retain the performance, resulting +in an increase in the computational cost, i.e., an efficiency- +accuracy trade-off. +On the other hand, recent semi-parametric NLP models +show success on knowledge-intensive tasks having a similar +challenge regarding large search space of external knowl- +edge (Lewis et al., 2020; Izacard & Grave, 2021). The +semi-parametric models often consist of a non-parametric +retriever and a parametric generator. The non-parametric +retrieval drastically reduces the search space of large knowl- +edge sources (millions of text such as Wikipedia) to a man- +ageable size, e.g., less than 100, allowing the parametric +model to ground relevant knowledge for a given query ef- +fectively. Also, it provides controllability of the external +knowledge and explainability over model decisions with +their provenance. Motivated by their success, we explore +semi-parametric video-grounded text generation as depicted +in Figure 1, another way for the scalable sparse video repre- +sentation toward long videos with minutes and even hours. +In this paper, we propose the Semi-parametric Video- +grounded Text generation model (SeViT) to take benefits +from both efficiency of the sparse frame paradigm and scal- +ability over long-form videos. SeViT consists of a non- +parametric frame retriever and a parametric video-grounded +text generator. In particular, we treat a video as an external +data store and perform cross-modal retrieval to get top-k +query-relevant frames from the data store with a given query. +The video-grounded text generator independently encodes +each frame with the query. Then, late fusion methods are +followed to produce the final output by aggregating the sep- +arately encoded query-aware frames, e.g., marginalization +in the final decoding step or cross-attention in the decoder +layer (Lewis et al., 2020; Izacard & Grave, 2021). +In our experiments, SeViT achieves competitive or even bet- +ter performances on five Video QA (Xu et al., 2017; Yu et al., +2019; Yang et al., 2021; Xiao et al., 2021) and two video +captioning (Chen & Dolan, 2011; Xu et al., 2016) tasks +compared to previous baseline models, which are massively +pre-trained on video-text pairs, without any video-language +pre-training. Our analysis demonstrates that SeViT has a +significant advantage on the longer videos and questions +requiring causal video understanding. Especially, our SeViT +achieves new state-of-the-art performances on three Video +QA benchmarks, iVQA, Next-QA, and ActivityQA, which +have relatively long source videos, by improving 4.8-6.9% +point of accuracy, and one video captioning, MSRVTT- +Caption by improving 3.6% point of CIDEr. +Our contributions are three folds: +• To our best knowledge, we propose the semi- +parametric architecture in the video-language domain, +SeViT, by treating a video as an external data store for +the first time. +• We demonstrate that SeViT based on retrieval- +augmented generation shows strong performance in +long videos and causal video understanding compared +to its baseline relying on frame sampling. +• SeViT achieves the new state of the art on three Video +QA with longer videos, iVQA, Next-QA, Activitynet- +QA, and one video captioning, MSRVTT-Caption with- +out any video-language pre-training. +2. Related Work +2.1. Video-Language Models +Previous video-language models (Sun et al., 2019; Li et al., +2020; Yang et al., 2021) often rely on offline feature ex- +traction leveraging pre-trained 2D/3D vision encoders such +as ResNet (He et al., 2016), S3D (Xie et al., 2018) and +SlowFast (Feichtenhofer et al., 2019) to efficiently represent +video frames, while adopting pre-trained language models +like BERT or RoBERTa (Devlin et al., 2019; Liu et al., +2019) for the textual representations of subtitles or captions. +Recently, some studies adopt end-to-end trainable video- +specific transformer (Liu et al., 2022; Arnab et al., 2021) for +video captioning tasks (Lin et al., 2022; Seo et al., 2022). +Contrary to the models relying on the feature extraction +for densely sampled frames, Lei et al. (2021) propose Clip- +BERT representing a video with sparsely sampled frames. +It allows end-to-end training of the pre-trained vision and +text encoders, leading to comprehensive performances on +video-language downstream tasks, i.e., text-to-video re- +trieval and Video QA. Recent video-language studies use +a few uniformly sampled frames per video to pre-training +video-language models (Zellers et al., 2021; Wang et al., +2022a; Zellers et al., 2022; Yang et al., 2022). They lever- +age millions of video-text pairs utilizing automatically gen- +erated subtitles via automatic speech recognition API for +their pre-training procedure (Miech et al., 2019; Bain et al., +2021). The massive pre-training on the large video-text +pairs boosts the performance of downstream video-language +tasks, achieving state-of-the-art. Our approach shares the +sparse frame strategy, but is more scalable toward long +videos. Also, we focus on fine-tuning our semi-parametric +model rather than video-language pre-training. + +Semi-Parametric Video-Grounded Text Generation +Figure 2. Detailed illustration for mechanism of SeViT. 1) It first retrieves top-k query-relevant frames from a video via maximum inner +product search (MIPS) between a query vector and (pre-computed) |V | frame vectors, where k ≪ |V |. 2) Each frame is encoded with +the query q independently by the encoder of VGT-Generator. It produces k query-aware representations. 3) We explore two late fusion +methods, 3-(a) Marginalization (Lewis et al., 2020) and 3-(b) Fusion-in-Decoder (Izacard & Grave, 2021) to produce the final output ˆa by +aggregating the k query-aware frames in the decoder. +2.2. Semi-Parametric Language Models +Semi-parametric language models show impressive success +on many knowledge-intensive NLP tasks such as open- +domain question answering and fact varification (Guu et al., +2020; Lewis et al., 2020; Izacard & Grave, 2021; Izacard +et al., 2022), or language modeling (Khandelwal et al., 2019; +Borgeaud et al., 2021). The semi-parametric model often +consists of a non-parametric module, i.e., a retriever, and +a parametric generator. This approach assumes large exter- +nal knowledge such as Wikipedia or other large text cor- +pora. The non-parametric retriever returns top-k relevant +knowledge for a given input from the large data store. The +retrieval is often based on a maximum inner product search +(MIPS) within pre-computed vectors of the data store (John- +son et al., 2019). Then, the parametric generator effectively +aggregates the knowledge with the given input. The non- +parametric module has several useful properties like con- +trollability, explainability, and debuggability by providing +the origin of model decisions. Meanwhile, the parametric +model provides better empirical performances than the non- +parametric model. Combining the best of two worlds, a +semi-parametric architecture is limitedly adopted for pro- +tein structure prediction (Jumper et al., 2021), image gen- +eration (Blattmann et al., 2022), and image-text QA (Chen +et al., 2022; Lin & Byrne, 2022). Inspired by these studies, +we adapt the retrieval-augmented generation framework to +the video-language domain for the first time. +2.3. Informative Frame Selection +Focusing on the fact that a video contains a lot of redun- +dant and worthless frames, many works tried to select in- +formative frames from the video (Chen et al., 2018; Yu +et al., 2019). Chen et al. (2018) introduce the informative +frame selection with the reinforcement learning algorithm +for video captioning task. Yu et al. (2019) show informa- +tive frame selection leveraging off-the-shelf action proposal +network is effective for Video QA with long untrimmed +videos. Dense video captioning (Krishna et al., 2017; Zhou +et al., 2018) also contains the frame proposal procedure, +but many works evaluate their models with ground-truth +proposals (Seo et al., 2022). On the other hand, Lei et al. +(2018) introduce TVQA requiring temporal localization for +answering questions in the TV show domain. However, it +often needs relevant segment selection with its start and end +positions requiring more consideration over corresponding +subtitles, i.e., multi-channel tasks. Moreover, the questions +containing “before” and “after” are difficult to find based +on the given query because the segment that can infer the +answer and the segment that corresponds to the query are +often different. We hope our work would be extended with +better frame segment selection in future works. +3. Method +In this section, we introduce SeViT, a Semi-parametric +Video-grounded Text generator. As illustrated in Figure 2, +it includes informative frame selection leveraging cross- +modal retrieval and effective late fusion methods such as + +1. Retrieving Top-k query-relevant frames +2. Independentlyencoding each frame and query q +3.Performing late fusion over top-k query-aware frames +Query +Decoder +[ +Marginalization +Encoder +b +Encoder +Decoder +Decoder +Encoder +b +Decoder +Encoder +Fusion-in-Decoder +Frame +Encoder +b +Encoder +Decoder +Frame Retriever +VGT-GeneratorSemi-Parametric Video-Grounded Text Generation +marginalization and fusion-in-decoder for video-grounded +text generation. We start by defining task formulation and +then explain each method and training details. +3.1. Overview: Video-Grounded Text Generation +Let V = {f1, f2, ..., f|V |} be a video clip consisting of +|V | number of sequential frames and q be a textual query. +Video-grounded text generation is a task that produces the +textual output ˆa conditioned on V and q, i.e., p(a | V, q). +Video captioning and video question answering (Video QA) +are popular examples of video-grounded text generation. +Basically, we inherit the sparse frame paradigm (Lei et al., +2021) representing a video with sparsely selected frames +Vk ⊂ V to approximate a video V where |Vk| = k and +k ≪ |V |. However, in contrast to previous approaches that +uniformly select Vk, we propose a semi-parametric model +which dynamically retrieves the k query-relevant frames +using a non-parametric retriever and aggregates them with a +parametric generator. +3.2. Frame Retriever +Conventionally, many studies perform random uniform sam- +pling to choose Vk (Lei et al., 2021; Zellers et al., 2021; +Wang et al., 2022a; Yang et al., 2022). In other words, the +random sampling selects k frames from V regardless of q. +Contrary to the prior studies, we select the relevant frames +conditioned on q by introducing a frame retriever. The frame +retriever η takes V and q, and returns a subset Vk ⊂ V , +modeled as: pη(Vk | V, q). In particular, the frame retriever +consists of two separated query and frame transformer en- +coders, EQ and EF, respectively. Each encoder takes q and +f to represent embedding vectors, respectively, where fi is +the i-th frame in V . Then, the cosine similarity between the +two vectors is used for the relevance score between q and fi +as follows: +sim(q, fi) = +EQ(q)T EF(fi) +∥EQ(q)∥2∥EF(fi)∥2 +(1) +where ∥·||2 denotes the l-2 normalization. Frame retriever re- +turns the top-k frames based on the relevance scores among +q and all frames in V as follows: +Vk ← argsort +fi∈V +(sim(q, fi))[: k]. +(2) +Also, we compute the relative importance of each selected +frame fj ∈ Vk by performing softmax over the cosine +similarities (Equation 1) where τ is a temperature hyper- +parameter. The frame score is computed as follows: +pη(fj | q) = +esim(fj,q)/τ +� +k esim(fk,q)/τ +(3) +3.3. Video-Grounded Text Generator +A video-grounded text (VGT) generator θ takes Vk and +q, and it outputs a. For θ, we leverage the transformer +encoder-decoder architecture taking both image and text +together to generate textual output (Vaswani et al., 2017; +Wang et al., 2021; 2022b). Specifically, it first embeds +each frame fj ∈ Vk and a text query q with convolution +blocks such as ResNet (He et al., 2016) and embedding +matrix lookup corresponding to subwords from byte-pair +encoding (Sennrich et al., 2015), respectively. Then, the +frame patches and subword tokens vectors are combined and +fed into the multi-modal transformer encoder to produce +k query-aware frame representations. Beyond the single +frame and query interaction, we investigate two effective +late fusion methods, Marginalization (Lewis et al., 2020) +and Fusion-in-Decoder (Izacard & Grave, 2021), to aggre- +gate the independently encoded k query-aware frames for +generating target text a in the decoder. +Marginalization (MAR) +It integrates the k query-aware +frames by marginalization (Lewis et al., 2020). First, the +decoder also produces independent k predictions. Then, it +aggregates the k predictions by marginalizing out weighting +by the frame score pη(fj | q) resulting in the output a = +{w1, w2, ..., wN}, where the w is a subword token of a. +p(a | V, q) = +N +� +i +� +fj∈Vk +pη(fj | q)pθ(wi | q, fj, w1:i−1) +(4) +The marginalization procedure allows joint optimization of +the cross-modal retriever and generator. In other words, +it enables gradient updates of encoders in a cross-modal +retriever to select query-relevant frames with Equation 4, +while not requiring explicit supervision for ground-truth +query-relevant frame pairs. +Fusion-in-Decoder +(FiD) +Fusion-in-Decoder +relies +purely on cross-attention between the hidden states of the +encoder and decoder for the fusion (Izacard & Grave, 2021). +Like the marginalization, it encodes q with each fj ∈ Vk +independently. However, it aggregates the encoder outputs +jointly in the decoder with cross-attention as illustrated in +Figure 2. Specifically, the encoder produces the hidden +states H ∈ Rk×L×d, where the L is the length of the +combined frame and query outputs, and the d is a hidden +dimension. The k hidden outputs are concatenated together +as H ∈ Rk·L×d, as a single sequence before being fed +into the decoder. Finally, the decoder can consider the +k query-aware frames at the same time for target text +generation. +p(a | V, q) = +N +� +i +pθ(wi | q, Vk, w1:i−1) +(5) + +Semi-Parametric Video-Grounded Text Generation +3.4. Training +VGT-generator is trained by minimizing the negative log- +likelihood of p(a | V, q) with either Equation 4 or 5. For +efficient implementation, we pre-compute the frame vec- +tors in advance for all training videos of the target dataset +using the frame encoder of the frame retriever. Then, an +efficient search algorithm, i.e., Maximum Inner Product +Search (MIPS), becomes convenient especially when the +length of the source video gets longer. We further describe +some training techniques considering the frame retriever. +Query-side Fine-Tuning +With the objective of Marginal- +ization (Equation 4), we can jointly optimize the frame +retriever and VGT-generator. +However, re-computation +of frame vectors is required for all videos when we up- +date the frame encoder of the frame retriever. +Lewis +et al. (2020); Izacard et al. (2022) report that the updat- +ing context encoder does not show significant advantages +for knowledge-intensive NLP tasks despite the heavy com- +putation. Thus, we keep the frame encoder EF fixed during +the training while only updating the query encoder EQ for +efficiency (Lewis et al., 2020; Izacard et al., 2022). +Retriever Warm-up for FiD +On the other hand, joint +training of the frame retriever with the objective of FiD +is not straightforward. Even though Izacard et al. (2022) +propose various methods for joint training of retriever and +generator in FiD fusion scheme, it does not work well for +ours in our preliminary experiments. Thus, we initialize +the frame retriever with a fine-tuned retriever by Marginal- +ization. Then, we fix the frame retriever during training +VGT-generator with the FiD manner. It is a similar approach +to FiD-RAG in Shuster et al. (2021). +Top-k Annealing +We promote diverse top-k frame se- +lection with the fixed retriever in FiD training. Basically, +we choose a frame from V in order of high relevance +score (Equation 2). However, VGT-generator might show +a lower generalization ability if trained with the same k +frames from the fixed retriever for every training instance. +Thus, we set a window size u ∈ R1 and prevent sub- +set {fi−u, fi−u+1, ..., fi, ..., fi+u−1, fi+u} from being se- +lected once fi is selected as one of top-k frames. We gradu- +ally decrease the u → 0 at every training epoch, resulting +in diverse top-k frames. +4. Experiments +In this section, we demonstrate the effectiveness of SeViT +compared to its baselines on eight video-language datasets. +We denote our models as SeViTMAR and SeViTFiD according +to their training objectives, Marginalization and Fusion-in- +Decoder, respectively. +Table 1. Comparison between two fusion methods, Marginaliza- +tion (MAR) and Fusion-in-Decoder (FiD) on Video QA and cap- +tioning tasks based on our baseline with uniform frame sampling, +SeViT⊗ explained in Section 4.1, to identify their differences apart +from frame retrieval. We report top-1 accuracy for Video QA and +CIDEr for video captioning. +Dataset +# Frame +train/test +MAR +FiD +Video QA +TGIF-Action +3/6 +94.9 +94.8 +TGIF-Transition +3/6 +98.2 +98.0 +TGIF-Frame +3/6 +70.6 +71.1 +MSVD-QA +5/10 +49.3 +49.3 +MSRVTT-QA +5/10 +41.9 +42.3 +iVQA +5/10 +35.3 +36.4 +Next-QA +5/10 +54.4 +54.6 +Activitynet-QA +5/10 +46.3 +47.1 +Video Captioning +MSVD-Caption +5/10 +127.4 +134.9 +MSRVTT-Caption +5/10 +58.6 +61.8 +4.1. Main Baseline: SeViT with Frame Sampling +Although there are several baseline models, we would like +to strictly compare the effect of employing the frame re- +trieval while controlling other factors such as model size, +pre-training steps, and other fusion methods. To this end, we +introduce a strong baseline utilizing uniform frame sampling +instead of performing frame retrieval to choose k frames. +We refer to the baseline as SeViT-with-frame-sampling, +denoting it simply SeViT⊗. When we train the SeViT⊗ +with the Marginalization, we use a uniform prior 1/k for +the frame score per frame instead of using Equation 3 for +the late fusion (Equation 4). We describe details of other +baselines in our experiments in Appendix C. +4.2. Dataset +We evaluate our models on six Video QA datasets, TGIF- +QA (Jang et al., 2017), MSVD-QA (Xu et al., 2017), +MSRVTT-QA (Xu et al., 2017), iVQA (Yang et al., 2021), +Next-QA (Xiao et al., 2021), and Activitynet-QA (Yu +et al., 2019), and two video captioning datasets, MSVD- +Caption (Chen & Dolan, 2011) and MSRVTT-Caption (Xu +et al., 2016). We mainly report top-1 accuracy for the Video +QA and CIDEr (Vedantam et al., 2015) for the video cap- +tioning. More details including statistics are in Appendix A. +4.3. Implementation Details +We use pre-trained CLIP-base/16 (Radford et al., 2021) for +our frame retriever and pre-trained OFA-Base (Wang et al., +2022b) for the VGT-generator. CLIP is a pre-trained bi- +encoder on large image-text pairs. OFA is a pre-trained +vision-language transformer on multi-tasks consisting of + +Semi-Parametric Video-Grounded Text Generation +Table 2. We compare our SeViT leveraging frame retriever with its +counterpart baseline, SeViT⊗ relying on uniform frame sampling +instead of frame retrieval, on Video QA datasets. +Dataset +(Avg. video length) +Frame +Retrieval +MAR +FiD +MSVD-QA (10s) + +49.3 +49.3 + +49.5 +49.7 +MSRVTT-QA (15s) + +41.9 +42.3 + +41.7 +42.1 +iVQA (18s) + +35.3 +36.4 + +36.4 +36.9 +Next-QA (44s) + +54.4 +54.6 + +54.8 +55.2 +Activitynet-QA (180s) + +46.3 +47.1 + +47.2 +47.6 +Table 3. Evaluation breakdown on Next-QA and Activitynet-QA +by their source video length. We split the test set of Next-QA and +Activitynet-QA into long and short subsets according to whether +the source video length is longer than 60 seconds. +Model +Next-QA +Activitynet-QA +Short +Long +All +Short +Long +All +SeViT⊗ +MAR +54.6 +53.4 +54.4 +47.3 +45.9 +46.3 +SeViTMAR +55.1 +53.9 +54.8 +47.4 +47.1 +47.2 +SeViT⊗ +FiD +54.9 +53.4 +54.6 +47.9 +46.6 +47.1 +SeViTFiD +54.8 +56.6 +55.2 +47.5 +47.6 +47.6 +image-text, text-only, and image-only tasks, by unifying +the input and output protocol. As described in Section 3.4, +we pre-compute frame vectors of all videos in the target +dataset in advance to perform an efficient search, MIPS. +The temperature τ is empirically set to 1. First, we train +SeViTMAR in the Marginalization, i.e., joint optimization of +the retriever and generator on the target dataset. Then, the +fine-tuned retriever is reused for training SeViTFiD on the +same dataset as described in Section 3.4. Also, we set k +to 5 for training and 10 at test time for all datasets except +for TGIF-QAs where we set k to 3 and 6 for the training +and test. For multiple-choice QA, TGIF-Action, TGIF- +Transition, and Next-QA, we concatenate answer options +and query together introducing a separation token. For video +captioning tasks, we use a null query, “What does the image +describe?”, used for image captioning tasks by Wang et al. +(2022b). All models are trained with {1e-5, 3e-5} learning +rate and {16, 32} batch size for 5 epochs on 1-2 NVIDIA +A100 GPUs. We use mainly PyTorch and Huggingface’s +Transformers library for our implementation (Paszke et al., +2019; Wolf et al., 2020). Please see Appendix B for more +details. +Table 4. Evalution results on Next-QA validation set. In addition +to the original validation set, we include a hard subset requiring +video-level understanding identified by ATP (Buch et al., 2022). +Model +Causal +Original / Hard +Temporal +Original / Hard +Descriptive +All +HGA +46.3 / 43.3 +50.7 / 45.3 +59.3 +49.7 +HGQA +48.5 / +- +51.2 / +- +61.7 +51.4 +APT +48.3 / 19.6 +46.7 / 22.6 +58.9 +49.2 +Temp[APT] +48.6 / 38.4 +49.3 / 36.5 +65.0 +51.5 ++ APT +53.1 / +- +50.2 / +- +66.8 +54.3 +VGT +52.3 / +- +55.1 / +- +64.1 +55.0 +SeViT⊗ +MAR +52.3 / 41.9 +52.3 / 44.7 +71.2 +55.2 +SeViTMAR +53.5 / 43.2 +54.0 / 46.3 +69.2 +56.1 +SeViT⊗ +FiD +53.0 / 42.7 +54.1 / 46.4 +71.9 +56.3 +SeViTFiD +54.0 / 43.3 +54.1 / 46.5 +71.3 +56.7 +4.4. Comparison between Late Fusion Methods +Before we discuss the benefits of frame retrieval, we com- +pare our two late fusion methods based on our baseline +model, SeViT⊗. Table 1 shows the results on ten down- +stream datasets. Both methods show comparable perfor- +mance to each other, but FiD performs slightly better than +MAR in most Video QA datasets, especially in TGIF-Frame, +iVQA, and Activitynet-QA. Those datasets contain descrip- +tive QA pairs. We find that the late fusion methods perform +surprisingly well on the datasets requiring temporal rea- +soning, e.g., TGIF-Action, TGIF-Transition, and Next-QA, +even though they do not consider the temporal order among +frames explicitly. Also, FiD shows better performances than +MAR in the two video captioning tasks indicating FiD is +good at generating longer text. +4.5. Benefits from Frame Retrieval +Table 2 shows the effect of frame retrieval on the Video +QA datasets 1. The frame retrieval consistently improves +the performances of all datasets except for MSRVTT-QA, +regardless of video length. Notably, it improves the per- +formances of longer Video QA datasets, iVQA, Next-QA, +and Activitynet-QA. We find that gains are slightly larger +in MAR fusion improving the 1.0 and 0.9 accuracies of +iVQA and Activitynet-QA, respectively. We presume the +joint training of frame retrieval boosts the gain. In Table 12 +of Appendix D, we find that frame retrieval consistently +improves performances of video captioning as well, even +though the null query is used for the retrieval. We think +the null query effectively filters out uninformative frames +resulting in performance gains. +1We exclude TGIF-QAs since their video length (3s) makes +our frame retrieval meaningless. + +Semi-Parametric Video-Grounded Text Generation +Figure 3. Breakdown results on Activitynet-QA (Yu et al., 2019) by (a) source video length and (b) the number of frames at the test time. +Table 5. Ablation study on two long Video QA datasets, Next-QA +and Activitynet-QA. +Model +Next-QA +Activitynet-QA +SeViTMAR +54.8 +47.2 +w/o. QS fine-tuning +54.2 +46.0 +SeViTFiD +55.2 +47.6 +w/o. Retriever warm-up +54.4 +47.0 +w/o. Top-k annealing +54.3 +46.7 +SeViTFiD +w. OFA-Medium (93M) +51.1 +44.8 +w. OFA-Base (182M) +55.2 +47.6 +w. OFA-Large (472M) +60.6 +48.9 +Results on Long Video Subset +In Table 3, we further +break down the evaluation results according to source video +length to identify the benefits of frame retrieval in longer +videos. Specifically, we divide the original test set of Next- +QA and Activitynet-QA into long and short sub-splits ac- +cording to whether the source video length is longer than +60 seconds. We can find that most improvements by frame +retrieval are from the long videos in both datasets. Espe- +cially, SeViTFiD improves 3.2% point accuracy of long video +subset in Next-QA by employing frame retrieval. +Results by Question Types +Table 4 shows that the perfor- +mance gains by the frame retrieval in Next-QA are related to +causal and temporal QA types for both fusion methods. In +contrast, the performance of the descriptive type is slightly +degraded by the frame retrieval. It is notable that the im- +provements are significant in SeViTMAR for both causal and +temporal types while the improvement is limited to causal +type in SeViTFiD. It reminds us of the importance of joint +retriever training, again. However, SeViTFiD consistently +performs better in all three types compared to SeViTMAR. +Moreover, it outperforms previous best-performing models +utilizing sophisticated graph representation, HGA (Jiang & +Han, 2020), HGQA (Xiao et al., 2022a), and VGT (Xiao +et al., 2022b), especially in causal and descriptive types. +More results by question types are in Table 10 and 11 of +Appendix D. +Analysis on Untrimmed Long Videos +Finally, we fur- +ther analyze the advantages of frame retrieval on the longest +video benchmark, Activitynet-QA. Figure 3 illustrates the +advantages in two folds. First, we divide the test set into sub- +sets according to more fine-grained video lengths as shown +in Figure 3 (a). As hypothesized, there is a clear tendency +for significant performance gaps according to the usage of +frame retrieval to become more pronounced in longer videos. +Specifically, SeViT⊗ +MAR and SeViT⊗ +FiD both drop their perfor- +mances significantly with videos longer than 180 seconds. +However, both SeViTMAR and SeViTFiD successfully retain +their performances with the longer videos. Second, we inves- +tigate the sample efficiency in terms of the number of frames +at the inference time. We believe that if the frame retriever +selects informative frames well, the model works better with +fewer frames than non-informative frames from the uniform +frame sampler. Figure 3 (b) shows such a tendency as we +have hypothesized. Even though the performances decrease +gradually with fewer frames, the performance gaps between +SeViT and SeViT⊗ also becomes significant. It implies the +frames obtained by frame retriever are more informative +than the frames by random uniform sampling. We also find +the strength of SeViT in long videos by a qualitative analysis +as shown in Figure 4 of Appendix E. +4.6. Ablation Study +Table 5 shows ablation results on two long Video QA +datasets, Next-QA and Activitynet-QA. Once again, we +can observe the importance of frame retrieval from this +study. If we fix the frame retriever during training instead + +47 +48 +46 +45 +44 +46 +A +43 +SeViTiD +45 +42 +SeViTMAR +SeViTFiD +44 +10-30 +30-60 +60-120 +120-180 +>180 +3 +10 +20 +5 +(a) Video Length (Sec.) +(b) # Frames at Inference timeSemi-Parametric Video-Grounded Text Generation +Table 6. Comparison with baselines including state-of-the-art models on five Video QA datasets. † indicates our VGT-generator is +initialized with OFA-Large (Wang et al., 2022b). ∗ indicates the score is obtained from Xiao et al. (2022b). Bold indicates the best score +and underline indicates the second best score. +Model +Video-Text +Pre-training +MSVD-QA +(10s) +MSRVTT-QA +(15s) +iVQA +(18s) +Next-QA +(44s) +Activitynet-QA +(180s) +JustAsk + +47.5 +41.8 +35.4 +50.8∗ +39.0 +MERLOT + +- +43.1 +- +- +41.4 +All-in-One + +48.3 +46.8 +- +- +- +FrozenBiLM + +54.4 +47.0 +39.7 +- +43.2 +LAVENDER + +56.6 +45.0 +- +- +- +ClipBERT + +- +37.4 +- +- +- +SINGULARITY + +- +43.9 +- +- +44.1 +IGV + +40.8 +38.3 +- +51.3 +- +HQGA + +41.2 +38.6 +- +51.8 +- +VGT + +- +39.7 +- +53.7 +- +Ours +SeViTMAR + +49.5 +42.3 +36.4 +54.8 +47.2 +SeViTFiD + +49.7 +42.1 +36.9 +55.2 +47.6 +SeViT† +FiD + +52.6 +43.8 +44.5 +60.6 +48.9 +of performing query-side (QS) fine-tuning, the final per- +formance of SeViTMAR drops 0.6 and 1.2 of accuracy in +Next-QA and ActivitynetQA, respectively. Similarly, using +the warmed-up frame retriever by SeViTMAR boosts the final +performance of SeViTFiD in both datasets. We find diverse +retrieval by top-k annealing also contributes to the final +performance. Furthermore, we also find the larger back- +bone model for the VGT-generator significantly improves +performance. +4.7. Comparison with State-of-the-arts +We also compare ours with previous state-of-the-art models +as shown in Table 6. Notably, our models show competitive +performances on the short video-based Video QA datasets +compared to baseline models pre-trained on large video- +text pairs, JustAsk (Yang et al., 2021), MERLOT (Zellers +et al., 2021), All-in-One (Wang et al., 2022a), Frozen- +BiLM (Yang et al., 2022) and LAVENDER (Li et al., 2022b), +even without any video-text pre-training. Also, our model +outperforms other baselines utilizing graph representation, +HQGA (Xiao et al., 2022a), IGV (Li et al., 2022c), and +VGT (Xiao et al., 2022b), and baselines pre-trained on +image-text pairs, ClipBERT (Lei et al., 2021) and SINGU- +LARITY (Lei et al., 2022). Moreover, our models outper- +form in relatively longer Video QA datasets, Next-QA and +Activitynet-QA. Especially, our FiD-based model achieves +new state-of-the-art performances on iVQA, Next-QA, and +Activitynet-QA, when using a large-sized backbone, i.e., +OFA-Large, for our VGT-generator. Moreover, in Table 12 +of Appendix D, our model shows competitive performances +on the video captioning dataset compared to end-to-end +video transformer-based baselines, SwinBERT (Lin et al., +2022), MV-GPT (Seo et al., 2022), and LAVENDER (Li +et al., 2022b). Notably, SeViTFiD based on OFA-Large +achieves a new state-of-the-art performance in terms of +CIDEr (Vedantam et al., 2015) on the MSRVTT-Caption +dataset even without video-text pre-training. +5. Conclusion +In this work, we present SeViT for scalable video repre- +sentation toward untrimmed long videos. In particular, we +regard a video as an external data store and leverage the +non-parametric retriever to get relevant frames. 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MSVD-QA and MSRVTT- +QA are also widely used open-ended Video QA datasets +containing short video clips of 10-15 seconds on average +and auto-constructed synthetic questions (Xu et al., 2017). +iVQA (Yang et al., 2021) is an open-ended Video QA dataset +with video clips of 18 seconds on average and human an- +notations of 5 reference answers per question. We fol- +low the weighted accuracy evaluation setup of Yang et al. +(2021). Recently, Xiao et al. (2021) introduce Next-QA +to evaluate video understanding focusing on causal and +temporal aspects of daily activity contents (Xiao et al., +2021). It is a multiple-choice QA dataset including rela- +tively longer videos with 44 seconds on average. We also +report validation results on hard sub-split of Next-QA re- +quiring video-level understanding identified by Buch et al. +(2022). Activitynet-QA contains human-generated QA pairs +based on the longest untrimmed videos with 180 seconds +on average (Yu et al., 2019). MSVD-Caption and MSRVTT- +Caption datasets provide 40 and 20 ground-truth captions +based on 10-15 seconds video clips, respectively (Chen & +Dolan, 2011; Xu et al., 2016). For all datasets, we compress +videos with {0.5, 1} FPS to increase processing efficiency. +The statistics of datasets are shown in Table 7. +Table 7. Dataset statistics used in our experiments. +Dataset +Video +QA pairs +Train +Val +Test +Train +Val +Test +TGIF-QA +62,841 +- +9,575 +139,395 +- +25,751 +MSVD-QA +1,200 +250 +520 +30,933 +6,415 +13,157 +MSRVTT-QA +6,513 +497 +2,990 +158,581 +12,278 +72,821 +iVQA +5,994 +2,000 +2,000 +5,994 +2,000 +2,000 +Next-QA +3,870 +570 +5,440 +34,132 +4,996 +8,564 +Activitynet-QA +3,200 +180 +800 +32,000 +18,000 +8,000 +MSVD-Cap. +1,200 +100 +670 +- +- +- +MSRVTT-Cap. +6,513 +497 +2,990 +- +- +- +B. More Implementation Details +We set the maximum sequence length of the query to 32 +for open-ended Video QA and video captioning tasks, and +64 for multiple-choice Video QA datasets. Also, we set +the beam size to 6 and no-repeat-ngram-size to 3 for the +generation. All hyper-parameter setups are in Table 8 and +Table 9. For all experiments, we choose the best-performing +checkpoint based on the validation score. +Table 8. Common hyper-parameter setups in our experiments. +Hyper-parameter +Setup +learning rate (lr) +3e-5 +# epoch +5 +lr scheduling +linear +warmup ratio +0.01 +weight decay +0.01 +label smoothing +0.1 +Table 9. Task-specific hyper-parameter setups in our experiments. +Dataset +Batch Size +Max input len. +Max output len. +TGIF-Action +16 +64 +10 +TGIF-Transition +16 +64 +10 +TGIF-Frame +16 +32 +10 +MSVD-QA +16 +32 +10 +MSRVTT-QA +32 +32 +10 +iVQA +16 +32 +10 +Next-QA +16 +64 +16 +Activitynet-QA +32 +32 +10 +MSVD-Caption +16 +10 +32 +MSRVTT-Caption +32 +10 +32 +C. Other Baselines +Video QA +Lei et al. (2021) propose ClipBERT, an end- +to-end video-language model with sparse frame sampling. +Contrary to previous approaches relying on pre-extracted +dense video features, it leverages sparsely sampled frames +as a video representation allowing gradient updates. Also, +Lei et al. (2021) conduct image-text pre-training based on +BERT (Devlin et al., 2019) and ResNet-50 (He et al., 2016) +backbones. It shows the benefits of sparse sampling in many +video-language downstream tasks. On the other hand, Lei +et al. (2022) explain single frame bias in video-language +tasks by introducing SINGULARITY model. The authors +argue that some video-language tasks do not require rea- +soning over multiple frames. Rather, they claim that it is +enough for the tasks to perform the proper fusion method +over frames at test time, after image-text pre-training and +fine-tuning. +Yang et al. (2021) introduce synthetic Video QA dataset, +HowToVQA69M, by generating question-answer pairs con- +ditioned on only transcripts. They show the model pre- +trained on the synthetic dataset works well on various Video +QA datasets. The resulting model is VQA-T, which we refer +to JustAsk. The model utilizes offline features from Dis- +tillBERT (Sanh et al., 2019) and S3D (Xie et al., 2018) for +text and video representation, respectively. It is optimized +by contrastive loss, between the video-question pair and +the correct answer, and masked language modeling (MLM) +loss. MERLOT (Zellers et al., 2021) is another video- +language model pre-trained on the YT-Temporal-180M, + +Semi-Parametric Video-Grounded Text Generation +the curated video-transcript dataset from 6M unlabelled +YouTube videos. They employ a vision transformer on the +top of ResNet-50 (He et al., 2016) to represent video with a +few sampled frames. It is trained by optimizing contrastive +frame-text matching, masked language modeling (MLM), +and temporal reordering objectives. They demonstrate the +efficacy of the large pre-training on various visual reasoning +and Video QA tasks. Similarly, All-in-One adopts MLM +and the frame-text matching loss for its pre-training objec- +tives (Wang et al., 2022b). Specifically, it unifies the input +representation that embeds raw pixels and text jointly. More- +over, it also relies on sparsely sampled frames, e.g., 3 or 9 +frames. FrozenBiLM is devised for zero-shot Video QA by +Yang et al. (2021). They conduct parameter-efficient tuning +by introducing a few trainable parameters in frozen pre- +trained bidirectional encoder (He et al., 2021). It is further +pre-trained on 10M video-text pairs with MLM objective. +The model achieves state-of-the-art performances on many +(zero-shot) Video QA benchmarks. +On the other hand, HGA (Jiang & Han, 2020) performs +cross-modal reasoning over heterogeneous graphs among +the video frames and question words for reach modality +interaction. Xiao et al. (2021) test the model in their bench- +mark, Next-QA, with BERT (Devlin et al., 2019) for the +text representations. HQGA (Xiao et al., 2022a) represents +a video as a hierarchical semantics consisting of multiple +granularities, e.g., entity, frame, clip, and video, with textual +cues. It shows good performance on the Video QA such as +Next-QA. Similarly, VGT (Xiao et al., 2022b) also utilizes +the video graph representation explicitly capturing objects in +the frames, their relationships, and spatio-temporal dynam- +ics. Li et al. (2022c) introduce IGV with a training frame- +work to prevent Video QA models from exploiting spurious +correlations. In particular, they split a video into causal and +complement parts with respect to the given question. Then, +the Video QA models are trained to exploit causal frames +while not grounding on the complement frames which do +not contain critical clues. +Video Captioning +ORG-TRL (Zhang et al., 2020) em- +ploys graph convolutional networks to model objects in the +video and conducts teacher-recommended learning leverag- +ing pre-trained language model (Devlin et al., 2019). Tang +et al. (2021) introduce DECEMBERT pre-trained by noisy +video-transcript pairs along with corresponding dense cap- +tions from the off-the-shelf dense captioning model. They +pre-train the model with MLM and video-text matching +loss. Moreover, they introduce constrained attention loss to +mitigate misalignment errors of noisy transcripts. Lin et al. +(2022) propose a fully end-to-end trainable transformer, +SwinBERT on the top of VidSwin Transformer (Liu et al., +2022) for video captioning. To handle long video input effi- +ciently, Lin et al. (2022) introduce learnable sparse attention +masks that reduce redundant video inputs. LAVENDER +also adopts the VidSwin as a backbone and unifies the for- +mulation of pre-training and fine-tuning with MLM (Li +et al., 2022b). Seo et al. (2022) propose MV-GPT which is +another end-to-end video-language model for video caption- +ing. They employ the video-specific transformer backbone, +ViViT (Arnab et al., 2021) to represent a video. They also +introduce a bidirectional pre-training objective predicting +either present or future subtitles from masked text input and +corresponding frames. +D. More Experimental Results +We further report breakdown results by question types on +Activitynet-QA test set in Table 10 and Next-QA validation +set in Table 11. Also, this section includes the comparison +with state-of-the-art models of video captioning in Table 12. +E. Qualitative Examples +In Figure 4, we show qualitative examples comparing +SeViTFiD and SeViT⊗ +FiD on the 10 minute-long videos in +Activitynet-QA. We also include qualitative examples of +video captioning in Figure 5. + +Semi-Parametric Video-Grounded Text Generation +Table 10. Breakdown results on Activitynet-QA test set by fine-grained question types. +Model +Motion +Spatial +Temporal +Yes/No +Color +Object +Location +Number +Other +All +JustAsk +28.0 +17.5 +4.9 +66.3 +34.3 +26.7 +35.8 +50.2 +36.8 +39.0 +MERLOT +33.9 +18.1 +4.0 +72.5 +36.2 +24.5 +36.5 +51.7 +37.8 +41.4 +SeViTMAR +31.8 +24.6 +3.5 +78.6 +64.0 +36.2 +40.9 +56.9 +39.1 +47.2 +SeViTFiD +30.6 +24.8 +4.4 +78.6 +65.4 +32.1 +40.9 +59.4 +40.3 +47.6 +Table 11. Breakdown results on Next-QA validation set by fine-grained question types. +Model +Causal +Temporal +Descriptive +All +Why +How +All +Bef&Aft +Present +All +Count +Location +Other +All +HGA +47.0 +44.2 +46.3 +49.5 +52.5 +50.7 +44.1 +72.5 +55.4 +59.3 +49.7 +SeViTMAR +54.2 +51.8 +53.5 +51.0 +58.4 +54.0 +55.4 +81.7 +65.2 +69.2 +56.1 +SeViTFiD +55.5 +49.9 +54.0 +50.2 +59.7 +54.1 +56.5 +82.4 +69.2 +71.3 +56.7 +Table 12. Comparison with baseline models including state-of-the-art models on two video captioning datasets. † indicates SeViT is +initialized with OFA-Large (Wang et al., 2022b) backbone for VGT-generator. BLEU-4, METEOR, ROUGE-L, CIDEr are reported (Pap- +ineni et al., 2002; Banerjee & Lavie, 2005; Lin, 2004; Vedantam et al., 2015). Bold indicates the best score and underline indicates the +second best score. +Model +Video-Text +Pre-training +MSRVTT +MSVD +BLEU +METEOR +ROUGE +CIDEr +BLEU +METEOR +ROUGE +CIDEr +DECEMBERT + +45.2 +39.7 +64.7 +52.3 +- +- +- +- +MV-GPT + +48.9 +38.7 +64.0 +60.0 +- +- +- +- +LAVENDER + +- +- +- +60.1 +- +- +- +150.7 +ORG-TRL + +43.6 +28.8 +62.1 +50.9 +54.3 +36.4 +73.9 +95.2 +SwinBERT + +41.9 +29.9 +62.1 +53.8 +58.2 +41.3 +77.5 +120.6 +Ours +SeViT⊗ +MAR + +44.9 +30.7 +63.0 +58.6 +64.0 +42.9 +79.5 +127.4 +SeViTMAR + +44.9 +31.0 +63.0 +57.6 +64.6 +42.9 +80.3 +136.1 +SeViT⊗ +FiD + +46.2 +31.4 +64.3 +61.8 +65.7 +43.3 +79.8 +134.9 +SeViTFiD + +47.1 +31.6 +64.4 +62.2 +66.9 +43.7 +80.9 +135.5 +SeViT† +FiD + +48.2 +31.7 +64.9 +63.7 +69.1 +46.3 +83.0 +148.1 + +Semi-Parametric Video-Grounded Text Generation +Figure 4. We show efficacy of SeViT by qualitative examples. To this end, we compare SeViTFiD and SeViT⊗ +FiD in videos longer than 10 +minutes from Activitynet-QA. The results show that a baseline based on frame sampling, SeViT⊗ +FiD, fails to find relevant frames from the +long videos while SeViTFiD successfully finds query-relevant frames. + +Q:Does themanwearglasses? +SeViTFiD +A: Yes +SeViT +A: No +Q: Does she wear watch? +SeViTFiD +A: Yes +XSemi-Parametric Video-Grounded Text Generation +Figure 5. Qualitative examples of video captioning. We compare SeViTFiD and SeViT⊗ +FiD on videos in MSRVTT-Caption (top) and +MSVD-Caption (bottom). Especially, the top example shows the case that frames by SeViT⊗ +FiD, i.e., random frame sampling, include +uninformative frames resulting in performance degradation. + +GT:a woman demonstratingthefunctions ofababy stroller +SeViTFiD +A woman is showing howto fold a strolle +X +X +Available Accessory Weather Shield +Available Accessory - Weather Shield +Available AccessoryWeather Shield +Awoman ispushingastroller +GT: a cat is playing with a turtle +SeViTFiD +A cat isplaying with a turtle +A cat is playing with a mouse X \ No newline at end of file diff --git a/ftFJT4oBgHgl3EQfUCze/content/tmp_files/load_file.txt b/ftFJT4oBgHgl3EQfUCze/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae017f15b01ddf8786ffb49a2a92575818126999 --- /dev/null +++ b/ftFJT4oBgHgl3EQfUCze/content/tmp_files/load_file.txt @@ -0,0 +1,1534 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf,len=1533 +page_content='Semi-Parametric Video-Grounded Text Generation Sungdong Kim 1 2 Jin-Hwa Kim 1 3 Jiyoung Lee 1 Minjoon Seo 2 Abstract Efficient video-language modeling should con- sider the computational cost because of a large, sometimes intractable, number of video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Parametric approaches such as the attention mech- anism may not be ideal since its computational cost quadratically increases as the video length increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Rather, previous studies have relied on offline feature extraction or frame sampling to represent the video efficiently, focusing on cross-modal modeling in short video clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' In this paper, we propose a semi-parametric video- grounded text generation model, SeViT, a novel perspective on scalable video-language modeling toward long untrimmed videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Treating a video as an external data store, SeViT includes a non- parametric frame retriever to select a few query- relevant frames from the data store for a given query and a parametric generator to effectively aggregate the frames with the query via late fu- sion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Experimental results demonstrate our method has a significant advantage in longer videos and causal video understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Moreover, our model achieves the new state of the art on four video-language datasets, iVQA (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content='8), Next-QA (+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content='9), and Activitynet-QA (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content='8) in accuracy, and MSRVTT-Caption (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content='6) in CIDEr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Introduction Recently, there has been the impressive success of vision- language models (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Alayrac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=', 2022) demonstrating a re- markable transferability on video-language tasks including video retrieval, video captioning, and video question answer- ing (Video QA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Considering video input’s spatio-temporal aspects, it is often more challenging to process than other modalities, especially combining video and language modal- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Moreover, modeling video information requires heavy computations since it comprises lengthy image sequences (frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Conventionally, many video-language works have 1NAVER AI Lab 2KAIST AI 3SNU AIIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftFJT4oBgHgl3EQfUCze/content/2301.11507v1.pdf'} +page_content=' Correspondence to: Minjoon Seo 0 and ϵk + uk > 0 +1, ϵk < 0 and ϵk + uk > 0 +2, ϵk < 0 and ϵk + uk < 0 +(7) +and those regions are distinguished by the surfaces de- +fined by +ϵk = 0, +(8) +ϵk + uk = 0, +(9) +uk = 0. +(10) +As pointed out in [4], we have pseudo-Fermi surfaces +across which occupation numbers change by 2 (∆nk = 2) +where there is a single-particle energy gap of ∣uk∣/2 +(i.e., pseudogap), and Fermi-arcs across which occupa- +tion numbers change by 1 (∆nk = 1) with no such gap. +In the region with nk = 1, each state can be occupied +by either spin-up or spin-down fermions, resulting in a +massive degeneracy. In order to remove this degeneracy, +we can introduce an infinitesimal Zeeman splitting ∆Z +between the spin-up and spin-down fermions so that the +nk = 1 regions are occupied by the spin-down fermions +only in the ground state. The Fermi arcs are then the +Fermi surfaces for spin-up and spin-down fermions re- +spectively, albeit they are not closed (hence arcs). The +single-particle Green’s function of H0, again the same as +the exact Green’s function of HHKY, is +G0 +σ(ωk) =(1 − nkσ)(1 − nk,−σ) +ω + iη − ̵h−1ϵk ++ +(1 − nkσ)nk,−σ +ω + iη − ̵h−1 (ϵk + uk) ++ nkσ(1 − nk,−σ) +ω − iη − ̵h−1ϵk ++ +nkσnk,−σ +ω − iη − ̵h−1 (ϵk + uk) += +(1 − nkσ) +ω + iη − ̵h−1Ekσ ++ +nkσ +ω − iη − ̵h−1Ek,−σ +, +(11) +where η is an infinitesimal positive. +In addition to the Hamiltonian in Eq. (3), we consider +a perturbing Hubbard interaction: +HHubbard =V ∑ +i +ˆni↑ˆni↓ +(12) += V +N ∑ +kk′q +c† +k+q↑c† +k′−q↓ck′↓ck↑, +(13) +where V is the interaction strength, i is site index and +N is system size. Hence, the full Hamiltonian we would +like to consider is the sum of the HKY Hamiltonian and +the Hubbard interaction, +H = HHKY + HHubbard = H0 + H′, +(14) +where we treat H1, H2, and HHubbard perturbatively +such that +H′ = H1 + H2 + HHubbard. +(15) +(a) +(b) +(c) +Figure 1: Vertices given by the interaction +resulting from (a) H1, (b) H2, and (c) HHubbard. +Figure 2: An example of diagrams that are not +part of the particle-particle or particle-hole ladder +of HKY interaction (H2). All such diagrams +vanish. +III. +FEYNMAN RULES AND LADDER SUMS +The Feynman rules can be established following stan- +dard text books [7]. We introduce a solid line to denote +the unperturbed Green’s function G0 +σ(ω,k) in Eq. (11), +a cross symbol to denote the interaction given by H1 in +Eq. (5), a dashed line to denote the one given by H2 +in Eq. (6), and a wavy line to denote the Hubbard in- +teraction in Eq. (12). Hence, it is necessary to consider +those three kinds of vertices in Fig. 1. In terms of these +diagrammatic components, we find the diagrams in Fig. +2 cancel each other, where the second one is the Hartree +diagram in terms of the H2 interaction. Note this cancel- +lation occurs not only when they stand alone in the first +order diagram illustrated here, but also when they are +embedded in higher order diagrams. This cancellation, +guaranteed by the self-consistent Hartree condition, is +a significant simplification, as we can now drop all di- +agrams that involve the cross symbol given by Fig. 1a +and/or a Hartree bubble. Other than this simplification, +the Feynman rules are the same as the usual ones [7]. + +wko +wko^w+sk +Aw'-sk↓ +uk +wk ↑ +.0 +w'k↓b.b +wk ↑ +w'k' ↓wko. +wko +wk.-o ++ += 0 +wkoL +wko3 +Figure 3: Particle-particle ladder diagrams. +In addition to Fig. 2, another major simplification is all other contributions from H2 interaction can be organized +as particle-particle and particle-hole ladder diagrams illustrated in Fig. 3 and Fig. 4. +The first line of the equation +in Fig. 4 includes all particle-particle crossing diagrams, which are equivalent to the particle-hole ladder diagrams +on the following line. Fig. 3 and 4 are the only nonzero contributions given by the H2 term. This is because H2 only +gives rise to forward scattering, and cannot create particle-hole pairs. Also for this reason these ladder diagrams can +be summed up easily because they form geometric series. To see this we inspect the corresponding Bethe-Salpeter +equations [7] +Γpp(p0 +1 + p0 +2,p) = u(p) + u(p)Γ(p0 +1 + p0 +2,p)∫ +dq0 +2π G0 +σ (p0 +1 + p0 +2 +2 ++ q0,p)G0 +−σ (p0 +1 + p0 +2 +2 +− q0,p), +(16) +and +Γph(p0 +1 − p0 +4,p) = u(p) + u(p)Γ(p0 +1 − p0 +4,p)∫ +dq0 +2π G0 +σ (p0 +1 − p0 +4 +2 ++ q0,p)G0 +−σ (q0 − p0 +1 − p0 +4 +2 +,p), +(17) +where p0 +1, p0 +2 and p0 +4 are the frequencies carried by the external propagators. Due to the forward-scattering nature +there is no momentum integral, as a result Γ does not enter the integrals on the RHS, allowing the integrals to be +carried out explicitly, yielding +Γpp(p0 +1 − p0 +2,p) = u(p)(1 − npσ)(1 − np,−σ) +1 − +u(p) +̵h(p0 +1+p0 +2)−(Epσ+Ep,−σ)+iη ++ +u(p)npσnp,−σ +1 + +u(p) +̵h(p0 +1+p0 +2)−(Epσ+Ep,−σ)−iη +, +(18) +Γph(p0 +1 − p0 +4,p) = +u(p)(1 − npσ)np,−σ +1 − +u(p) +̵h(p0 +1−p0 +4)−(Epσ−Ep,−σ)+iη ++ +u(p)npσ(1 − np,−σ) +1 + +u(p) +̵h(p0 +1−p0 +4)−(Epσ−Ep,−σ)−iη +. +(19) +All other diagrams (which inevitably mix particle-particle and particle-hole ladders) vanish, an example of which is +shown in Fig. 5. This is due to the restrictions on the occupations in Eq. (18) and Eq. (19), where we only have +the combinations of npσnp,−σ, (1 − npσ)(1 − np,−σ) for the former, and (1 − npσ)np,−σ, npσ(1 − np,−σ) for the latter. +IV. +THE SELF-ENERGY DIAGRAMS +In this section we study the electron self-energy +Σσ(ω,k), especially its imaginary part, which tells us +the decay rate and the broadening of the electron spec- + +TpP +p3 +p2p +dm +↑ dyd4 +Figure 4: Particle-hole ladder diagrams. +Figure 5: An example of diagrams that are not +part of the particle-particle or particle-hole ladder +of HKY interaction (H2). All such diagrams +vanish. +tral function measured in the angle-resolved photoemis- +sion spectroscopy (ARPES): +1 +τ = ImΣσ(ω,k). +(20) +We evaluate the self-energy diagrams to the 2nd order +of Hubbard interaction (V 2), which is the lowest order +that gives rise to an imaginary part. We will, however, +include all contributions from HKY interaction, using +the ladder sums performed in the previous section. +The simplest diagram is the one with Hubbard inter- +action only (Fig. 6a). Its imaginary part is +Im Σ6a +σ (ω,k) =V 2 ∫ +d2k′ +(2π)2 +d2q +(2π)2 δ(ω + Ek′,−σ + Ek′−q,−σ − Ek+q,σ) +× [(1 − nkσ)nk+q,σ(1 − nk′,−σ)nk′−q,−σ − nkσ(1 − nk+q,σ)nk′,−σ(1 − nk′−q,−σ)], +(21) + +p2p +Iph = +pgp 个 +p4p +pp +dn +dn +p2p↓5 +which yields the familiar Fermi liquid result near the +(pseudo) Fermi surface: +ImΣ6a +σ (ω,k) ≈ −D3V 2(̵hω)2, +(22) +where D is the density of states at the non-interacting +Fermi level. We note however this results in much more +broadening in the pseudogap region as the quasiparti- +cle energy ̵hω ∼ ∣u∣ is bounded below by the size of the +pseudogap, compared to that near the Fermi arcs where +̵hω can be arbitrary small. This is consistent with the +cuprate phenomenology. +We now turn to the diagrams that involve HKY interaction (H2). Those diagrams are in Fig. 6. Here we present +the calculation on the self-energy diagram given by Fig. 6b as an example. A full calculation on each diagram is +presented in Appendix A. +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +Figure 6: Self-energy Feynman diagrams to the second order in Hubbard interaction (V 2). (a) is the self-energy +diagrams in the second order with Hubbard interaction only. (b)-(i) are the self-energy diagrams with both Hubbard and +the HKY (H2) interactions. + +6 +After performing the frequency integrals, the corresponding imaginary part of Fig. 6b takes the form +Im Σ6b +σ (ω,k) =i6πV 2 ∫ +d2q +(2π)2 +× {[δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ) − δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ − uq](1 − nqσ)(1 − nq,−σ)n−k+2q,−σ +−[δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ) − δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ + uq)]nqσnq,−σ(1 − n−k+2q,−σ)}, +(23) +Note compared to Eq. (21), we have one fewer momentum integral to perform, despite the extra loop. This is due +to the fact HKY interaction forces the propagators coupled by it to have the same momentum. This simplification +changes the phase space constraints significantly and enhances Im Σ, as we demonstrate below. +To bring Eq. (23) to a form closer to Eq. (21), letting −k + 2q → q′, we treat q′ as an additional integration +variable, compensated by an additional delta function. Rewrite the integrals in terms of the polar coordinates such +that (qx,qy) = (q cosφ,q sinφ) and (q′ +x,q′ +y) = (q′ cosφ′,q′ sinφ′). Eq. (23) becomes +Im Σ6b +σ (ω,k) =i6πV 2 +1 +(2π)2 ∫ qdqq′dq′dφdφ′ 1 +q′ δ(q′ − ∣ − k + 2q∣)δ(φ′ − φ−k+2q) +× {[δ(ω + Eq′,−σ − 2ϵq) − δ(ω + Eq′,−σ − 2ϵq − uq)]θ(ϵq)θ(−Eq′,−σ) +−[δ(ω + Eq′,−σ − 2ϵq − 2uq) − δ(ω + Eq′,−σ − 2ϵq − 2uq + uq)]θ(−ϵq − uq)θ(Eq′,−σ)}. +(24) +We transform the integral from ∫ qdqq′dq′ to ∫ dϵdE′J(ϵ,E′,φ,φ′) and ∫ dEdE′dφdφ′, where ϵ = ϵq, E = ϵq + uq, +and E′ = Eq′,−σ, and the Jacobian +J(ϵ,E′,φ,φ′) = q ∂q +∂ϵ q′ ∂q′ +∂E′ = D(ϵ,φ)D(E′,φ′), +(25) +J(E,E′,φ,φ′) = q ∂q +∂E q′ ∂q′ +∂E′ = D(E,φ)D(E′,φ′), +(26) +with the angle-dependent density of states +D(ϵ,φ) = q ∂q +∂ϵ , +(27) +D(E,φ) = q ∂q +∂E , +(28) +D(E′,φ′) = q′ ∂q′ +∂E′ . +(29) +In a two-dimensional system it is a good approximation to treat them as a constant in terms of density of states D: +D(ϵ,φ)D(E′,φ′) ≈ D(E,φ)D(E′,φ′) ≈ (2πD)2 . +(30) +Eq. (24) then becomes +Im Σ6b +σ (ω,k) +≈i6πD2 {∫ dϵdE′dφdφ′ 1 +q′2 δ (1 − ∣ − k + 2q∣ +q′ +)δ(φ′ − φ−k+2q)[δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uq)]θ(ϵ)θ(−E′) +−∫ dEdE′dφdφ′ 1 +q′2 δ (1 − ∣ − k + 2q∣ +q′ +)δ(φ′ − φ−k+2q)[δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uq)]θ(−E)θ(E′)}, +(31) +where we have four integrals and three delta functions, and we let δ(q′ − ∣ − k + 2q∣) → +1 +q′ δ (1 − ∣−k+2q∣ +q′ +). We then +consider the angle integral on φ′. In order to express ∣ − k + 2q∣, q, q′ in terms of the integral variables and their +corresponding angle dependence, we solve the equation E(k,φk) = E so that k = f(E,φ). Rewriting uq = uqφ, the + +7 +approximate Eq. (31) becomes +Im Σ6b +σ (ω,k) ≈i6πD2 ∫ dφ′δ(φ′ − φ−k+2q) +× {∫ dϵdE′dφ +1 +f 2(E′,φ′)δ [1 − f(ϵ,φ−k+2q) +f(E′,φ′) ][δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf(ϵ,φ),φ)]θ(ϵ)θ(−E′) +−∫ dEdE′dφ +1 +f 2(E′,φ′)δ [1 − f(E,φ−k+2q) +f(E′,φ′) +][δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf(E,φ),φ)]θ(−E)θ(E′)}. +(32) +Performing the angle integral over φ′ yields +Im Σ6b +σ (ω,k) ≈ +i6πD2 {∫ dϵdE′dφ +1 +f 2(E′,φ−k+2q)δ [1 − f(ϵ,φ−k+2q) +f(E′,φ−k+2q)][δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf(ϵ,φ),φ)]θ(ϵ)θ(−E′) +−∫ dEdE′dφ +1 +f 2(E′,φ−k+2q)δ [1 − f(E,φ−k+2q) +f(E′,φ−k+2q)][δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf(E,φ),φ)]θ(−E)θ(E′)}. +(33) +We notice that φ−k+2q can by further replaced by another function g in terms of k, ϵ, E and φ. (33) becomes +Im Σ6b +σ (ω,k) ≈ +i6πD2 {∫ dϵdE′dφ +1 +f 2[E′,g(k,ϵ,φ)]δ {1 − f[ϵ,g(k,ϵ,φ)] +f[E′,g(k,ϵ,φ)]}[δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf(ϵ,φ),φ)]θ(ϵ)θ(−E′) +−∫ dEdE′dφ +1 +f 2[E′,g(k,E,φ)]δ {1 − f[E,g(k,E,φ)] +f[E′,g(k,E,φ)]}[δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf(E,φ),φ)]θ(−E)θ(E′)}. +(34) +We first perform the integral over φ. The remaining delta functions give us φ’s dependence on k, ϵ, E and E′, and +we denote it as a function h. Hence, (34) becomes +Im Σ6b +σ (ω,k) ≈ +i6πD2 {∫ dϵdE′ +1 +f 2{E′,g[k,ϵ,h(k,ϵ,E′)]} [δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′))]θ(ϵ)θ(−E′) +−∫ dEdE′ +1 +f 2{E′,g[k,E,h(k,E,E′)]} [δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf[E,h(k,E,E′)],h(k,E,E′))]θ(−E)θ(E′)}. +(35) +While we do not know the exact form of the terms 1/f 2, they give us quantities of order 1/(Fermi momentum)2, +which is of order O(1) for generic lattice filling. Its dependence on ϵ, E, E′ and k is unimportant due to the phase +space constraints, as will become clear soon. We are then left with integrals with energy variables ϵ, E and E′: +Im Σ6b +σ (ω,k) ≈i6πV 2D2 {∫ dϵdE′ [δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ − uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′))]θ(ϵ)θ(−E′) +−∫ dEdE′ [δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf[E,h(k,E,E′)],h(k,E,E′))]θ(−E)θ(E′)}, +(36) +We can then carry out the integral by assuming uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′) ≈ uf[E,h(k,E,E′)],h(k,E,E′) ≈ ∣u∣, where ∣u∣ is +some constant average over uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′) or uf[E,h(k,E,E′)],h(k,E,E′). The integrals over E′ give +Im Σ6b +σ (ω,k) ≈ i6πV 2D2 {∫ dϵθ(ϵ)[θ(−2ϵ + ω) − θ(−2ϵ + ω − ∣u∣)] − ∫ dEθ(E)[θ(−2E + ω) − θ(−2E + ω − ∣u∣)]}. +(37) + +8 +The integrals on ϵ, E give +Im Σ6b +σ (ω,k) ≈i6πV 2 [(ω +2 − ω − ∣u∣ +2 +) − (ω +2 − ω + ∣u∣ +2 +)] +(38) +=i6πV 2D2 [∣u∣ +2 − (−∣u∣ +2 )] +(39) +=i6πV 2D2∣u∣. +(40) +Here we note that the linearity in ∣u∣ comes from the energy integral per the phase space restrictions given by the +step functions θ(−2ϵ + ω), θ(−2ϵ + ω − uq), θ(E), and θ(−2E + ω + uq). The final result would not be exactly linear +in ∣u∣. Hence, we append a function f(k) varying in k: +Im Σ6b +σ (ω,k) ≈ − πV 2D2∣u∣f(k), +(41) +where f(k) is a dimensionless quantity of order O(1). Letting ω → Ekσ, we obtain the self-energy results from other +diagrams: +Im Σ6f +σ (k) ≈ −π +2 V 2D2uk(nkσ − nk,−σ), +(42) +Im Σ6g +σ (k) ≈ ImΣ6h +σ (k) ≈ Im Σ6i +σ (k) ≈ −πV 2D2uk(nkσ − nk,−σ)2. +(43) +The rest imaginary parts resulting from Fig. 6c, 6d, and 6e share the same form but appear more complicated as +we presented later in the expressions (A38) and (A63). However, since we are only interested in the cases that occur +near Fermi-arcs and pseudo-Fermi surfaces, further simplifications can be done so that overall these imaginary parts +results in zero or a quantity linear in uk. +V. +SUMMARY AND DISCUSSIONS +In this paper we studied the model introduced in Ref. +[4] (referred to as HKY model) which gives rise to Fermi +arcs and pseudogap, perturbed by Hubbard interaction. +We found the combination of Hubbard and HKY inter- +actions gives rise to a non-zero imaginary part to the +electron self-energy in the low-energy limit. The origin +of such non-Fermi liquid behavior lies in the singular na- +ture of HKY interaction, which has infinite range in real +space. +While our work was motivated by the cuprates, and +gives rise to results that are qualitatively consistent +with its phenomenology, the specific (and certainly over- +simplified) model we studied should not be taken as a +realistic description of the physics of cuprates. Its value +lies, instead, in its simplicity, which demonstrates not +only the possibility of Fermi arcs and pseudogap, but +also that they go hand-in-hand with each other and with +the observed non-Fermi liquid behavior. In fact recent +years have witnessed increasing activities in research us- +ing models that are extensions of the HK model [8–11] +aimed at understanding cuprate phenomenology, includ- +ing superconductivity itself. It is our hope that our work +provides a starting point to build more realistic models +for cuprates and other strongly correlated electron sys- +tems. +ACKNOWLEDGMENTS +This work was supported by the National Science +Foundation Grant No. DMR-1932796, and performed at +the National High Magnetic Field Laboratory, which is +supported by National Science Foundation Cooperative +Agreement No. DMR-1644779, and the State of Florida. +[1] For a recent review, see, e.g., Cyril Proust and Louis +Taillefer, The Remarkable Underlying Ground States of +Cuprate Superconductors, Annual Review of Condensed +Matter Physics Vol. 10:409-429 (2019). +[2] See, e.g., Teppei Yoshida, Makoto Hashimoto, Inna +M. Vishik, Zhi-Xun Shen, Atsushi Fujimori, Pseudo- +gap, Superconducting Gap, and Fermi Arc in High- +Tc Cuprates Revealed by Angle-Resolved Photoemission +Spectroscopy, J. Phys. Soc. Jpn. 81, 011006 (2012). + +9 +[3] See, e.g., Steven M. Girvin and Kun Yang, Modern +Condensed Matter Physics, ISBN: 9781107137394, Cam- +bridge University Press, Cambridge (March 2019). +[4] Kun Yang, Exactly solvable model of Fermi arcs and +pseudogap, Phys. Rev. B 103, 024529 (2021). +[5] Yasuhiro Hatsugai, and Mahito Kohmoto, Exactly Solv- +able Model of Correlated Lattice Electrons in Any Di- +mensions, Journal of the Physical Society of Japan 61, +2056 (1992). +[6] Ganapathy Baskaran, An Exactly Solvable Fermion +Model: Spinons, Holons and a non-Fermi Liquid Phase, +Modern Physics Letters B 5, 643 (1991). +[7] Alexander L. Fetter, and John Dirk Walecka, Quantum +theory of many-particle systems (Courier Corporation, +2012). +[8] Philip W. Phillips, Luke Yeo, and Edwin W. Huang. +Exact theory for superconductivity in a doped Mott in- +sulator. Nature Physics 16, 12: 1175-1180 (2020). +[9] Ryan D. Nesselrodt and James K. Freericks. Exact so- +lution of two simple non-equilibrium electron-phonon +and electron-electron coupled systems: The atomic limit +of the Holstein-Hubbard model and the generalized +Hatsugai-Komoto model. Physical Review B 104, 15: +155104 (2021). +[10] Yu +Li, +Vivek +Mishra, +Yi +Zhou, +and +Fu-Chun +Zhang. Two-stage superconductivity in the Hatsugai- +Kohomoto-BCS model, New Journal of Physics (2022). +[11] Yin Zhong. Solvable periodic Anderson model with +infinite-range Hatsugai-Kohmoto interaction: Ground- +states and beyond. Physical Review B 106, 15: 155119 +(2022). +Appendix A: Calculations on the self-energy diagrams +Here in the appendix we present more calculation details for each self energy diagram in the section . +Fig. 6b +Fig. 6f +The self-energy terms given by Fig. 6b and Fig. 6b are +Σ6b +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 +ds0 +(2π) +dq0 +(2π) +dk′0 +(2π)G0 +σ(q0q)G0 +σ(s0q)G0 +−σ(k′0q) +× G0 +−σ(k′0 + q0 − s0,q)G0 +−σ(q0 + k′0 − ω,2q − k)Γpp(q0 + k′0,q). +(A1) + +k-g +0. +b.b +-w,2-k +0 +0 +w-s°,k-qK +b +w- +k +.010 +Once the Green’s functions are plugged in, it becomes +Σ6b +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 +ds0 +(2π) +dq0 +(2π) +dk′0 +(2π) [ +(1 − nqσ)(1 − nq,−σ)n−k+2q,−σ +(q0 − Eqσ + iη)(s0 − Eqσ + iη)(k′0 − Eq,−σ + iη) +× +1 +(k′0 + q0 − s0 − Eq,−σ + iη)(q0 + k′0 − ω − E2q−k,−σ − iη)( 1 +uq − +1 +q0+k′0−Eqσ−Eq,−σ+iη) ++ +nqσnq,−σ(1 − n−k+2q,−σ) +(q0 − Eqσ − iη)(s0 − Eqσ − iη)(k′0 − Eq,−σ − iη) +× +1 +(k′0 + q0 − s0 − Eq,−σ − iη)(q0 + k′0 − ω − E2q−k,−σ + iη)( 1 +uq + +1 +q0+k′0−Eqσ−Eq,−σ−iη) +⎤⎥⎥⎥⎥⎥⎦ +. +(A2) +The frequency integral gives +Σ6b +σ (ω,k) =i6V 2 ∫ +d2q +(2π)2 +× [nqσnq,−σ(1 − n−k+2q,−σ)( +1 +ω + E−k+2q,−σ − Eqσ − Eq,−σ − iη − +1 +ω + E−k+2q,−σ − Eqσ − Eq,−σ + uq − iη ) ++(1 − nqσ)(1 − nq,−σ)n−k+2q,−σ ( +1 +ω + E−k+2q,−σ − Eqσ − Eq,−σ + iη − +1 +ω + E−k+2q,−σ − Eqσ − Eq,−σ − uq + iη )] +(A3) +The corresponding imaginary part and a sample calculation on Fig. 6b is shown in Section IV. The final result is +Im Σ6b +σ (ω,k) ≈ − πV 2D2∣u∣f(k), +(A4) +where f(k) is a dimensionless quantity of order O(1). +Next we would like to present the calculation on Fig. 6f. +Σ6f +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 +ds0 +(2π) +dq0 +(2π) +dk′0 +(2π)G0 +σ(q0q)G0 +σ(s0q)G0 +−σ(k′0q) +× G0 +−σ(k′0 − q0 + s0,q)G0 +−σ(−q0 + k′0 + ω,k)Γph(q0 − k′0,q). +(A5) +Σ6f +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 +ds0 +(2π) +dq0 +(2π) +dk′0 +(2π) [ +(1 − nqσ)nq,−σ(1 − n−k+2q,−σ) +(q0 − Eqσ + iη)(s0 − Eqσ + iη)(k′0 − Eq,−σ − iη) +× +1 +(k′0 − q0 + s0 − Eq,−σ − iη)(−q0 + k′0 + ω − Ek,−σ + iη)( 1 +uq − +1 +q0−k′0−Eqσ+Eq,−σ+iη) ++ +nqσ(1 − nq,−σ)n−k+2q,−σ +(q0 − Eqσ − iη)(s0 − Eqσ − iη)(k′0 − Eq,−σ + iη) +× +1 +(k′0 − q0 + s0 − Eq,−σ + iη)(−q0 + k′0 + ω − Ek,−σ − iη)( 1 +uq + +1 +q0−k′0−Eqσ+Eq,−σ−iη) +⎤⎥⎥⎥⎥⎥⎦ +. +(A6) + +11 +The frequency integral gives +Σ6f +σ (ω,k) =i6V 2 ∫ +d2q +(2π)2 +× [(1 − nqσ)nq,−σ(1 − nk,−σ)( +1 +ω − Ek,−σ − Eqσ + Eq,−σ + iη − +1 +ω − Ek,−σ − Eqσ + Eq,−σ − uq + iη ) ++nqσ(1 − nq,−σ)nk,−σ ( +1 +ω − E−k,−σ − Eqσ + Eq,−σ − iη − +1 +ω + Ek,−σ − Eqσ + Eq,−σ + uq − iη )] +(A7) +The corresponding imaginary part is +Im Σ6f +σ (ω,k) =i6πV 2 ∫ +d2q +(2π)2 {[−δ(ω − Ek,−σ − Eqσ + Eq,−σ) + δ(ω − Ek,−σ − Eqσ + Eq,−σ − uq)](1 − nqσ)nq,−σ(1 − nk,−σ) ++[δ(ω − Ek,−σ − Eqσ + Eq,−σ) − δ(ω − Ek,−σ − Eqσ + Eq,−σ + uq)]nqσ(1 − nq,−σ)nk,−σ} +=πV 2 ∫ +d2q +(2π)2 {[−δ(ω − Ek,−σ − ϵq − uq + ϵq) + δ(ω − Ek,−σ − ϵq − uq + ϵq − uq)]θ(ϵq + uq)θ(−ϵq)(1 − nk,−σ) ++[δ(ω − Ek,−σ − ϵq + ϵq + uq) − δ(ω − Ek,−σ − ϵq + uq + ϵq + uq)]θ(−ϵq)θ(ϵq + uq)nk,−σ}. +(A8) +After simplification we have +Im Σ6f +σ (ω,k) =i3πV 2 ∫ +d2q +(2π)2 {[−δ(ω − Ek,−σ − uq) + δ(ω − Ek,−σ − 2uq)]θ(ϵq + uq)θ(−ϵq)(1 − nk,−σ) +−[δ(ω − Ek,−σ + uq) − δ(ω − Ek,−σ + 2uq)]θ(−ϵq)θ(ϵq + uq)nk,−σ}. +(A9) +We let Eq′ = ϵq + uq such that Eq′φ′ = ϵqφ + uqφ in polar coordinates. We solve this equation in favor of q′ and φ′: +q′ = f(ϵqφ + uqφ,φ′), +(A10) +φ′ = g(ϵqφ + uqφ,q′) = g[ϵqφ + uqφ,f(ϵqφ + uqφ,φ′)]. +(A11) +Eq. (A9) becomes +Im Σ6f +σ (ω,k) =i6 +π +(2π)2 V 2 ∫ qdqq′dq′dφdφ′ 1 +q′2 δ [1 − f(ϵqφ + uqφ,φ′) +q′ +]δ {g[Eq′φ′,f(ϵqφ + uqφ,φ′)]} +× {[−δ(ω − Ek,−σ − uq) + δ(ω − Ek,−σ − 2uq)]θ(ϵq + uq)θ(−ϵq)(1 − nk,−σ) ++[δ(ω − Ek,−σ + uq) − δ(ω − Ek,−σ + 2uq)]θ(−ϵq)θ(ϵq + uq)nk,−σ}, +(A12) +where +∫ dφdφ′ 1 +q′2 δ [1 − f(ϵqφ + uqφ,φ′) +q′ +]δ {φ′ − g[Eq′φ′,f(ϵqφ + uqφ,φ′)]} +(A13) +gives a quantity of the order 1/(Fermi momentum)2 and thus of the order O(1). We then transform ∫ qdqq′dq′ to +∫ D(E,φ)D(E′,φ′)dEdE′ where E = ϵq and E′ = ϵq + uq such that uq = E − E′, and we let D(E,φ)D(E′,φ′) ≈ D2. +Eq. (A12) becomes +Im Σ6f +σ (ω,k) ≈ i6πV 2D2 ∫ dEdE′ {[−δ(ω − Ek,−σ − E′ + E) + δ(ω − Ek,−σ − 2E′ − 2E)]θ(E′)θ(−E)(1 − nk,−σ) +− [δ(ω − Ek,−σ + E′ − E) − δ(ω − Ek,−σ + 2E′ − 2E)]θ(−E)θ(E′)nk,−σ}. +(A14) +This gives +Σ6f +σ (ω,k) ≈i6πV 2D2 [Ek,−σ − ω +2 +(1 − nk,−σ) + −ω + Ek,−σ +2 +nk,−σ], +(A15) + +12 +Figure 8 +which can be further simplified: +Im Σ6f +σ (ω,k) ≈i6 π +2 V 2D2(Ek,−σ − ω). +(A16) +As ω → Ekσ, we obtain +Im Σ6f +σ (k) ≈ − π +2 V 2D2uk(nkσ − nk,−σ). +(A17) +We consider Fig. 6c, 6g, 6d, and 6h together. We find the equality in Fig. 8. So here we only consider Fig. 6c +and 6g. We start with Fig. 6c first +Σ6c +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π G0 +σ(s0,k)G0 +σ(s0 − q0,k − q)G0 +−σ(k′0,k) +× G0 +−σ(k′0 − ω + s0,k)G0 +−σ(k′0 + q0,k + q)Γpp(s0 + k′0,k). +(A18) +�⇒ Σ6c +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π [ +(1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ +(s0 − Ekσ + iη)(s0 − q0 − Ek−q,σ − iη)(k′0 − Ek,−σ + iη) +× +1 +(k′0 − ω + s0 − Ek,−σ + iη)(k′0 + q0 − Ek+q,−σ − iη)( 1 +uk − +1 +s0+k′0−Ek,σ−Ek,−σ+iη) ++ +nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) +(s0 − Ekσ − iη)(s0 − q0 − Ek−q,σ + iη)(k′0 − Ek,−σ − iη) +× +1 +(k′0 − ω + s0 − Ek,−σ − iη)(k′0 + q0 − Ek+q,−σ + iη)( 1 +uk + +1 +s0+k′0−Ek,σ−Ek,−σ−iη) +⎤⎥⎥⎥⎥⎥⎦ +. +(A19) + +Fig. +3c += Fig. +3d +Fig. 3g +Fig. +3h13 +The frequency integral gives +Fig. 6c +Fig. 6g +Σ6c +σ (ω,k) =i6V 2uk ∫ +d2q +(2π)2 +× [− +(1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ +(−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ + iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ − uk + iη) ++ +nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) +(−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ − iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ + uk − iη)]. +(A20) +The imaginary part is +Im Σ6c +σ (ω,k) = − πV 2uk ∫ +d2q +(2π)2 δ(ω − Ek−q,σ − Ek+q,−σ + Ek,−σ) +× [sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Ek+q,−σ − uk) +(1 − nk,−σ)(1 − nkσ)nk+q,−σnq,σ +∣ − Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ + uk∣ ++sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Ek+q,−σ + uk) +nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) +∣ − Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ − uk∣]. +(A21) +We k + q → q′ and treat q′ as an additional integration variable. We rewrite the integrals in terms of the polar +coordinates such that (qx,qy) = (q cosφ,q sinφ) and (q′ +x,q′ +y) = (q′ cosφ′,q′ sinφ′). Eq. (A21) becomes +− πV 2uk +1 +(2π)2 ∫ qdqq′dq′dφdφ′ 1 +q′ δ(q′ − ∣k + q∣)δ(φ′ − φk+q)δ(ω − Ek−q,σ − Eq′,−σ + Ek,−σ) +× [sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Eq′,−σ − uk) (1 − nk,−σ)(1 − nkσ)θ(−Eq′,−σ)θ(−Ek−q,σ) +∣ − Ek,−σ + Eq′,−σ + Ek−q,σ − Ekσ + uk∣ ++sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Eq′,−σ + uk) +nk,−σnkσθ(Ek−q,σ)θ(−Eq′,−σ) +∣ − Ek,−σ + Eq′,−σ + Ek−q,σ − Ekσ − uk∣]. +(A22) +We transform the integral ∫ qdqq′dq′ → ∫ dEdE′J(E,E′), where E = Ek−q,σ, E′ = Eq′,−σ, and the Jacobian +J(E,E′,φ,φ′) = q ∂q +∂E q′ ∂q′ +∂E′ = D(E,φ)D(E′,φ′), +(A23) +with the angle-dependent density of states +D(E,φ) = q ∂q +∂E , +(A24) +D(E′,φ′) = q′ ∂q′ +∂E′ . +(A25) + +0 +0 +s+m- +.k +w-sk +0 +k-q +-q,k-q +s-m14 +As done in Section IV, we treat them as a constant in terms of density of states D: +D(E,φ)D(E′,φ′) ≈ (2πD)2 . +(A26) +Eq. (A22) then becomes +− πV 2ukD2 ∫ dEdE′dφdφ′ 1 +q′ δ(q′ − ∣k + q∣)δ(φ′ − φk+q)δ(ω − E − E′ + Ek,−σ) +× [sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ − uk) (1 − nk,−σ)(1 − nkσ)θ(−E′)θ(E) +∣ − Ek,−σ + E′ + E − Ekσ + uk∣ ++sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ + uk) +nk,−σnkσθ(E)θ(−E′) +∣ − Ek,−σ + E′ + E − Ekσ − uk∣]. +(A27) +Letting δ(q′ − ∣k + q∣) → 1 +q′ δ (1 − ∣k+q∣ +q′ ), we consider the angle integrals first: +∫ dφdφ′ 1 +q′2 δ (1 − ∣k + q∣ +q′ +)δ(φ′ − φk+q). +(A28) +In terms of the integral variables E, E′, φ′ and the angle dependence φk+q, this can be expressed as +∫ dφdφ′ +1 +f 2(E′,φ′)δ [1 − f(E,φk+q) +f(E′,φ′) ]δ(φ′ − φk+q), +(A29) +where function f is the solution of E(k,φk) = E in favor of k. Performing the angle integral over φ′ yields +∫ dφ +1 +f 2(E′,φk+q)δ [1 − f(E,φk+q) +f(E′,φk+q)]. +(A30) +The quantity φk+q can by further replaced by another function g in terms of k, E and φ. The integral becomes +∫ dφ +1 +f 2(E′,g(k,E,φ))δ {1 − f[E,g(k,E,φ)] +f[E′,g(k,E,φ)]}. +(A31) +This gives us a quantity of order 1/(Fermi momentum)2, which is of order O(1) for generic lattice filling. +The +expression (A27) then becomes +ImΣ6c +σ (ω,k) ≈ +− πV 2ukD2 ∫ dEdE′δ(ω − E − E′ + Ek,−σ) +× [sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ − uk) (1 − nk,−σ)(1 − nkσ)θ(−E′)θ(−E) +∣ − Ek,−σ + E′ + E − Ekσ − uk∣ ++sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ + uk) +nk,−σnkσθ(E)θ(E′) +∣ − Ek,−σ + E′ + E − Ekσ + uk∣]. +(A32) +We then perform the integrals over E and E′: +Im Σ6c +σ (ω,k) ≈ −πV 2ukD2 ∫ dE [sgn(ω − Ekσ − uk) (1 − nk,−σ)(1 − nkσ)θ(−ω + E − Ek,−σ)θ(−E) +∣ω − Ekσ − uk∣ ++sgn(ω − Ekσ + uk) nk,−σnkσθ(E)θ(ω − E + Ek,−σ) +∣ω − Ekσ + uk∣ +]. +(A33) +The last integral on E gives +Im Σ6c +σ (ω,k) ≈Im Σ6d +σ (k) ≈ −πV 2ukD2 [sgn(ω − Ekσ − uk) (1 − nk,−σ)(1 − nkσ)(−ω − Ek,−σ) +∣ω − Ekσ − uk∣ ++sgn(ω − Ekσ + uk) nk,−σnkσ(ω + Ek,−σ) +∣ω − Ekσ + uk∣ +]. +(A34) + +15 +After simplification, this results in +ImΣ6c +σ (ω,k) ≈ Im Σ6d +σ (k) ≈ +− πV 2D2 [−sgn(ω − Ekσ − uk) (1 − nk,−σ)(1 − nkσ) +∣ω − Ekσ − uk∣ ++ sgn(ω − Ekσ + uk) +nk,−σnkσ +∣ω − Ekσ + uk∣]uk(Ek,−σ + ω). +(A35) +As uk approaches to 0, the quantity (A35) vanishes. We let ω → Ekσ. The sign functions become sgn(−uk) and +sgn(uk). Hence, (A35) becomes +− πV 2D2 [−sgn(−uk)(1 − nk,−σ)(1 − nkσ) +∣ − uk∣ ++ sgn(uk)nk,−σnkσ +∣uk∣ +]uk(Ek,σ + Ek,−σ) +(A36) += − 2πV 2D2 uk +∣uk∣sgn(uk)[ϵk(1 − nk,−σ)(1 − nkσ) + (ϵk + uk)nk,−σnkσ] +(A37) += − 2πV 2D2 [ϵk(1 − nk,−σ)(1 − nkσ) + (ϵk + uk)nk,−σnkσ], +(A38) +where the value of Ekσ and Ek,−σ depend on corresponding occupation number: for nkσ = nk,−σ = 0, Ekσ = Ek,−σ = ϵk; +with nkσ = nk,−σ = 1, Ekσ = Ek,−σ = ϵk + uk. +We examine the conditions with two different occupations: as +nkσ = nk,−σ = 0, +Im Σ6c +σ (k) ≈ Im Σ6d +σ (k) ≈ −2πV 2D2ϵk. +(A39) +According to Eq. (8), when we approach both the Fermi-arcs and the pseudo-Fermi surfaces from nk = 0 region, +ϵk → 0 and thus (A40) vanishes. As nkσ = nk,−σ = 2, +Im Σ6c +σ (k) ≈ Im Σ6d +σ (k) ≈ −2πV 2D2(ϵk + uk). +(A40) +According to Eq. (8) and Eq. (9), when we approach the Fermi-arcs from nk = 2 region, ϵk + uk → 0, and when +approaching to the pseudo-Fermi surfaces, we have ϵk +uk → uk. Hence, we conclude that overall the result of (A38) +is zero or linear in uk. +We then consider the self-energy diagram in Fig. 6g. +Σ6g +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π G0 +σ(s0,k)G0 +σ(s0 − q0,k − q)G0 +−σ(k′0,k) +× G0 +−σ(k′0 + ω − s0,k)G0 +−σ(k′0 − q0,k′ − q)Γph(s0 − k′0,k). +(A41) +�⇒ Σ6g +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π [ +nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ +(s0 − Ekσ + iη)(s0 − q0 − Ek−q,σ − iη)(k′0 − Ek,−σ − iη) +× +1 +(k′0 + ω − s0 − Ek,−σ − iη)(k′0 − q0 − Ek−q,−σ + iη)( 1 +uk − +1 +s0−k′0−Ekσ+Ek,−σ+iη) ++ +(1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) +(s0 − Ekσ − iη)(s0 − q0 − Ek−q,σ + iη)(k′0 − Ek,−σ + iη) +× +1 +(k′0 + ω − s0 − Ek,−σ + iη)(k′0 − q0 − Ek−q,−σ − iη)( 1 +uk + +1 +s0−k′0−Ekσ+Ek,−σ−iη) +⎤⎥⎥⎥⎥⎥⎦ +. +(A42) +The frequency integral gives +Σ6g +σ (ω,k) =i6V 2uk ∫ +d2q +(2π)2 +× [ +nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ +(ω + Ek−q,−σ − Ek−q,σ − Ek,−σ − iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk + iη) ++ +(1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) +(ω + Ek−q,σ − Ek−q,σ − Ek,−σ + iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk − iη)]. +(A43) + +16 +The imaginary part is +Im Σ6g +σ (ω,k) =i6πV 2uk ∫ +d2q +(2π)2 δ(ω + Ek−q,−σ − Ek−q,σ − Ek,−σ) +× [−sgn(ω + Ekσ − 2Ek,−σ + 2Ek−q,−σ − 2Ek−q,σ + uk) +nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ +∣Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk∣ ++sgn(ω + Ekσ − 2Ek,−σ + 2Ek−q,−σ − 2Ek−q,σ − uk) +(1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) +∣Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk∣]. +(A44) +After simplification we have +ImΣ6g +σ (ω,k) =i6πV 2uk ∫ +d2q +(2π)2 +× [−sgn(ω + Ekσ − 2Ek,−σ − 2uk−q + uk)δ(ω + uk−q − Ek,−σ)nk,−σ(1 − nkσ)θ(ϵk−q + uk−q)θ(−ϵk−q) +∣Ek,−σ − uk−q − Ekσ − uk∣ ++sgn(ω + Ekσ − 2Ek,−σ + 2uk−q − uk)δ(ω − uk−q − Ek,−σ)(1 − nk,−σ)nkσθ(ϵk−q + uk−q)θ(−ϵk−q) +∣Ek,−σ + uk−q − Ekσ + uk∣ +]. +(A45) +We let Eq′ = ϵk−q + uk−q. In polar coordinates, we have the solutions +q′ = f(ϵkqφ + ukqφ,φ′), +(A46) +φ′ = g(ϵkqφ + ukqφ,q′) = g[ϵkqφ + ukqφ,f(ϵkqφ + ukqφ,φ′)]. +(A47) +Eq. (A45) becomes +ImΣ6g +σ (ω,k) =i6 +π +(2π)2 V 2uk ∫ qdqq′dq′dφdφ′ 1 +q′2 δ [1 − f(ϵkqφ + ukqφ,φ′) +q′ +]δ{φ′ − g[ϵkqφ + ukqφ,f(ϵkqφ + ukqφ,φ′)]} +× [−sgn(ω + Ekσ − 2Ek,−σ − 2Eq′φ′ + 2ϵkqφ + uk)δ(ω + Eq′φ′ − ϵkqφ − Ek,−σ)nk,−σ(1 − nkσ)θ(Eq′φ′)θ(−ϵkqφ) +∣Ek,−σ − Eq′φ′ + ϵkqφ − Ekσ − uk∣ ++sgn(ω + Ekσ − 2Ek,−σ + 2Eq′φ′ − 2ϵkqφ − uk)δ(ω − Eq′φ′ + ϵkqφ − Ek,−σ)(1 − nk,−σ)nkσθ(Eq′φ′)θ(−ϵkqφ) +∣Ek,−σ + Eq′φ′ − ϵkqφ − Ekσ + uk∣]. +(A48) +After we transform ∫ qdqq′dq′ → (2πD)2 ∫ dEdE′ where ϵkqφ → E and Eq′φ′ → E′, and we ignore the quantities of +the order O(1), we have +ImΣ6g +σ (ω,k) ≈i6 +π +(2π)2 (2π)2D3V 2uk ∫ dEdE′ +× [−D1sgn(ω + Ekσ − 2Ek,−σ − 2E′ + 2E + uk)δ(ω + E′ − E − Ek,−σ)nk,−σ(1 − nkσ)θ(E′)θ(−E) +∣Ek,−σ − E′ + E − Ekσ − uk∣ ++D2sgn(ω + Ekσ − 2Ek,−σ − 2E′ + 2E − uk)δ(ω − E′ + E − Ek,−σ)(1 − nk,−σ)nkσθ(E′)θ(−E) +∣Ek,−σ + E′ − E − Ekσ + uk∣], +(A49) +where D1, D2, and D3 are densities of states. This gives +ImΣ6g +σ (ω,k) ≈i6πV 2D3uk [−sgn(3ω + Ekσ − 4Ek,−σ + uk)D1 +nk,−σ(1 − nkσ)(−ω + Ek,−σ) +∣ω − Ekσ − uk∣ +(A50) ++sgn(3ω + Ekσ − 4Ek,−σ − uk)D2 +(1 − nk,−σ)nkσ(ω − Ek,−σ) +∣ω − Ekσ + uk∣ +]. +(A51) + +17 +We let ω → Ekσ and D1 ≈ D2 ≈ D3 ≈ D, +ImΣ6g +σ (k) ≈ − πV 2D2uk(nk,−σ − nkσ)2. +(A52) +Fig. 6e +Fig. 6i +Finally we consider the self-energy diagrams in Fig. 6e and Fig. 6i. The expression for Fig. 6e is +Σ6e +σ (ω,k) = i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π G0 +σ(s0,k)G0 +σ(s0 − q0,k − q)G0 +σ(s0 − v0 + k′0,k)G0 +−σ(k′0,k)G0 +−σ(v0,k) +× +G0 +−σ(s0 − ω + k′0,k)G0 +−σ(k′0 + q0,k + q)(Γpp(s0 + k′0,k))2. +(A53) +�⇒ Σ6e +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π [ +(1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ +(s0 − Ekσ + iη)(s0 − q0 − Ek−q,σ − iη)(k′0 − Ek,−σ + iη) +× +1 +(s0 − v0 + k′0 − Ekσ + iη)(v0 − Ek,−σ + iη) +× +1 +(k′0 − ω + s0 − Ek,−σ + iη)(k′0 + q0 − Ek+q,−σ − iη)( 1 +uk − +1 +s0+k′0−Ek,σ−Ek,−σ+iη) +2 ++ +nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) +(s0 − Ekσ − iη)(s0 − q0 − Ek−q,σ + iη)(k′0 − Ek,−σ − iη) +× +1 +(s0 − v0 + k′0 − Ekσ − iη)(v0 − Ek,−σ − iη) +× +1 +(k′0 − ω + s0 − Ek,−σ − iη)(k′0 + q0 − Ek+q,−σ + iη)( 1 +uk + +1 +s0+k′0−Ek,σ−Ek,−σ−iη) +2 +⎤⎥⎥⎥⎥⎥⎥⎦ +. +(A54) + ++3- +k+q +v°kq +0 +S +k-g +k18 +The frequency gives +i6V 2u2 +k ∫ +d2q +(2π)2 +× [ +(1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ +(−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ + iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ − uk + iη)2 ++ +nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) +(−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ − iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ + uk − iη)2 ]. +(A55) +The expression for Fig. 6i is +Σ6i +σ (ω,k) = i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π G0 +σ(s0,k)G0 +σ(s0 − q0,k − q)G0 +σ(s0 − v0 + k′0,k)G0 +−σ(k′0,k)G0 +−σ(v0,k) +× +G0 +−σ(−s0 + ω + k′0,k)G0 +−σ(k′0 − q0,k − q)(Γph(s0 − k′0,k))2. +(A56) +�⇒ Σ6i +σ (ω,k) =i3V 2 ∫ +d2q +(2π)2 ∫ +ds0 +2π +dq0 +2π +dk′0 +2π [ +nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ +(s0 − Ekσ + iη)(s0 − q0 − Ek−q,σ − iη)(k′0 − Ek,−σ − iη) +× +1 +(s0 + v0 − k′0 − Ekσ + iη)(v0 − Ek,−σ − iη) +× +1 +(k′0 + ω − s0 − Ek,−σ − iη)(k′0 − q0 − Ek−q,−σ + iη)( 1 +uk − +1 +s0−k′0−Ek,σ+Ek,−σ+iη) +2 ++ +(1 − nk,−σ)nkσnk−q,−σ)(1 − nk−q,σ) +(s0 − Ekσ − iη)(s0 − q0 − Ek−q,σ + iη)(k′0 − Ek,−σ + iη) +× +1 +(s0 + v0 − k′0 − Ekσ − iη)(v0 − Ek,−σ + iη) +× +1 +(k′0 + ω − s0 − Ek,−σ + iη)(k′0 − q0 − Ek−q,−σ − iη)( 1 +uk + +1 +s0−k′0−Ek,σ+Ek,−σ−iη) +2 +⎤⎥⎥⎥⎥⎥⎥⎦ +. +(A57) +The frequency integral gives +i6V 2u2 +k ∫ +d2q +(2π)2 +× [− +nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ +(ω + Ek−q,−σ − Ek−q,σ − Ek,−σ − iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk + iη)2 ++ +(1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) +(ω + Ek−q,−σ − Ek−q,σ − Ek,−σ + iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk − iη)2 ]. +(A58) +The imaginary parts are +Im Σ6i +σ (ω,k) =i6V 2u2 +k ∫ +d2q +(2π)2 δ(ω − Ek−q,σ − Ek+q,−σ + Ek,−σ) +× {−sgn[(−Ek,−σ + Ek−q,σ + Ek+q,−σ − Ekσ − uk)(−2ω − Ekσ − uk − 3Ek,−σ + 3Ek−q,σ + 3Ek+q,−σ)] +× +(1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ +(−Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ − uk)2 ++sgn[(−Ek,−σ + Ek−q,σ + Ek+q,−σ − Ekσ + uk)(−2ω − Ekσ + uk − 3Ek,−σ + 3Ek−q,σ + 3Ek+q,−σ)] +× +nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) +(−Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ + uk)2 +⎫⎪⎪⎬⎪⎪⎭ +. +(A59) + +19 +and +Im Σ6e +σ (ω,k) =i6V 2u2 +k ∫ +d2q +(2π)2 δ(ω + Ek−q,−σ − Ek−q,σ − Ek,−σ) +× {sgn[(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk)(2ω + Ekσ − uk − 3Ek,−σ + 3Ek−q,−σ − 3Ek−q,σ)] +× +nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ +(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk)2 ++sgn[(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk)(2ω + Ekσ + uk − 3Ek,−σ + 3Ek−q,−σ − 3Ek−q,σ)] +× +(1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) +(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk)2 +⎫⎪⎪⎬⎪⎪⎭ +, +(A60) +The substitution and the transformation on the integral variable are the same as we did for Fig. 6c and Fig. 6g. +The final results are +Im Σ6e +σ (ω,k) ≈i6V 2D2 {sgn[(ω − Ek,−σ − uk)2]−(1 − nk,−σ)(1 − nkσ) +(ω − Ekσ − uk)2 ++ sgn[(ω − Ek,−σ + uk)2] +nk,−σnkσ +(ω − Ekσ + uk)2 } +× u2 +k(Ek,−σ + ω), +(A61) +Im Σ6i +σ (ω,k) ≈i6V 2D2 {−sgn[(ω − Ek,−σ − uk)2] (1 − nkσ)nk,−σ +(ω − Ekσ − uk)2 + sgn[(ω − Ek,−σ + uk)2] nkσ(1 − nk,−σ) +(ω − Ekσ + uk)2 } +× u2 +k(ω − Ek,−σ). +(A62) +As ω → Ekσ, +Im Σ6e +σ (ω,k) ≈ −2πV 2D2 [−ϵk(1 − nk,−σ)(1 − nkσ) + (ϵk + uk)nk,−σnkσ], +(A63) +Im Σ6i +σ (ω,k) ≈ −πV 2D2uk(nk,−σ − nkσ)2. +(A64) +Here we notice that (A64) is linear in uk and thus it vanishes as uk → 0. As for the result (A63), we perform the +same analysis as we did for Fig. 6c and 6d, and hence we conclude: as it approaches to Fermi-arcs from either nk = 0 +or nk = 2 regions, and approaches to pseudo-Fermi surfaces from nk = 0 region, it always vanishes; as it approaches +to pseudo-Fermi surfaces from nk = 2 region, it results in a quantity proportional to uk. + diff --git a/gtE3T4oBgHgl3EQffwrX/content/tmp_files/load_file.txt b/gtE3T4oBgHgl3EQffwrX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b13011e87aa55955f584a1ed8a8a2a589ee07af5 --- /dev/null +++ b/gtE3T4oBgHgl3EQffwrX/content/tmp_files/load_file.txt @@ -0,0 +1,423 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf,len=422 +page_content='Non-Fermi liquid behavior in a simple model of Fermi arcs and pseudogap Ruojun Wang and Kun Yang Department of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306, USA (Dated: January 12, 2023) We consider a perturbed version of a very simple and exactly solvable model that supports Fermi arcs and pseudogap in its ground state and excitation spectrum, which includes Hubbard-like in- teractions in both momentum and real spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We find the combined effects give rise to non-Fermi liquid behavior in the electron self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Comparison will be made with phenomenology of high temperature cuprate superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' INTRODUCTION The normal state of high transition temperature (Tc) cuprate superconductors are known to exhibit non-Fermi liquid behavior in the underdoped regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Among their mysterious properties [1], they support pseudogaps and Fermi arcs [2] instead of closed Fermi surfaces of ordinary Fermi liquid [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Non-Fermi liquid behavior is also mani- fested in the lack of coherence in quasiparticle excitations and unusual transport properties [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Understanding such non-Fermi liquid physics is an exciting challenge we face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In an earlier paper [4] one of us introduced an ex- tremely simple and exactly solvable model, and showed that Fermi arcs and pseudogap appear very naturally (and hand-in-hand) in its ground state and excitation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' That model is a variant of a model intro- duced by Hatsugai and Kohmoto (HK) [5] (a model sim- ilar to that of HK was considered earlier by Baskaran [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' An unusual property of this model [4] (which we refer to as HKY model from now on) is that all quasi- particle and quasihole excitations are sharp, albeit be- ing gapped in the perdogap region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This is, of course, opposite to non-Fermi liquids where quasiparticle and quasihole excitations are incoherent, rendering the elec- tron spectral functions very broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The sharpness of the electron spectral function in the HKY model is the consequence of the fact that its interaction is local in momentum space and only gives rise to forward scatter- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' To remove this artifact, in the present paper we perturb the HKY model with a (real space) Hubbard in- teraction, and calculate its contribution to electron self- energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We demonstrate that the combined effects of the Hubbard and HKY interactions render the quasiparticle and quasihole excitations incoherent, consistent with the cuprate phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The rest of the paper is organized as what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' II we introduce the HKY model perturbed by the Hubbard interaction, and its meanfield solution which gives rise to Fermi arcs and pseudogap regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' III we set up the Feynman rules for pertur- bative treatments of interactions, and demonstrate that all non-vanishing diagrams involving HKY interactions form particle-particle and particle-hole ladders that can be summed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' IV we calculate the electron self-energy to the 2nd order in Hubbard interaction, and demonstrate its imaginary part remains finite in the low- energy limit, resulting in non-Fermi liquid behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' A brief summary is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' MODEL AND MEAN-FIELD SOLUTION We start by considering the HKY model: HHKY = ∑ k [ϵk(ˆnk↑ + ˆnk↓) + ukˆnk↑ˆnk↓], (1) where ϵk is the single particle energy, nkσ = c† kσckσ is the fermion occupation for momentum k and spin σ =↑,↓, and uk is the interaction energy between the spin-up and spin-down particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' When uk is a constant, the model is reduced to the HK model [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' While (1) is exactly solvable, in preparation for the breakdown of solvability once the (real space) Hubbard (or any other generic) interaction is introduced we first introduce a meanfield solution to (1), which we will use as the starting point for perturbation theory later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Note this meanfield solution is exact for the ground state and single particle/hole excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We separate the operator ˆnkσ into its expectation value and fluctuation: ˆnkσ = nkσ + δˆnkσ, (2) where nkσ = ⟨ˆnkσ⟩, and write the Hamiltonian as: HHKY = H0 + H1 + H2, (3) such that H0 = ∑kσ Ekσˆnkσ, (4) H1 = −∑kσ nk,−σukˆnkσ, (5) H2 = ∑k ukˆnk↑ˆnk↓, (6) where Ekσ = ϵk + nk,−σuk, is the single-particle energy within the Hartree approximation and we denote −σ as the opposite spin of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The ground state of H0, which is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='04556v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='str-el] 11 Jan 2023 2 also the exact ground state of HHKY, has the following occupation pattern: nk = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 0, ϵk > 0 and ϵk + uk > 0 1, ϵk < 0 and ϵk + uk > 0 2, ϵk < 0 and ϵk + uk < 0 (7) and those regions are distinguished by the surfaces de- fined by ϵk = 0, (8) ϵk + uk = 0, (9) uk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (10) As pointed out in [4], we have pseudo-Fermi surfaces across which occupation numbers change by 2 (∆nk = 2) where there is a single-particle energy gap of ∣uk∣/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=', pseudogap), and Fermi-arcs across which occupa- tion numbers change by 1 (∆nk = 1) with no such gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In the region with nk = 1, each state can be occupied by either spin-up or spin-down fermions, resulting in a massive degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In order to remove this degeneracy, we can introduce an infinitesimal Zeeman splitting ∆Z between the spin-up and spin-down fermions so that the nk = 1 regions are occupied by the spin-down fermions only in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The Fermi arcs are then the Fermi surfaces for spin-up and spin-down fermions re- spectively, albeit they are not closed (hence arcs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The single-particle Green’s function of H0, again the same as the exact Green’s function of HHKY, is G0 σ(ωk) =(1 − nkσ)(1 − nk,−σ) ω + iη − ̵h−1ϵk + (1 − nkσ)nk,−σ ω + iη − ̵h−1 (ϵk + uk) + nkσ(1 − nk,−σ) ω − iη − ̵h−1ϵk + nkσnk,−σ ω − iη − ̵h−1 (ϵk + uk) = (1 − nkσ) ω + iη − ̵h−1Ekσ + nkσ ω − iη − ̵h−1Ek,−σ , (11) where η is an infinitesimal positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In addition to the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (3), we consider a perturbing Hubbard interaction: HHubbard =V ∑ i ˆni↑ˆni↓ (12) = V N ∑ kk′q c† k+q↑c† k′−q↓ck′↓ck↑, (13) where V is the interaction strength, i is site index and N is system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Hence, the full Hamiltonian we would like to consider is the sum of the HKY Hamiltonian and the Hubbard interaction, H = HHKY + HHubbard = H0 + H′, (14) where we treat H1, H2, and HHubbard perturbatively such that H′ = H1 + H2 + HHubbard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (15) (a) (b) (c) Figure 1: Vertices given by the interaction resulting from (a) H1, (b) H2, and (c) HHubbard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Figure 2: An example of diagrams that are not part of the particle-particle or particle-hole ladder of HKY interaction (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' All such diagrams vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' FEYNMAN RULES AND LADDER SUMS The Feynman rules can be established following stan- dard text books [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We introduce a solid line to denote the unperturbed Green’s function G0 σ(ω,k) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (11), a cross symbol to denote the interaction given by H1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (5), a dashed line to denote the one given by H2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (6), and a wavy line to denote the Hubbard in- teraction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Hence, it is necessary to consider those three kinds of vertices in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In terms of these diagrammatic components, we find the diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 2 cancel each other, where the second one is the Hartree diagram in terms of the H2 interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Note this cancel- lation occurs not only when they stand alone in the first order diagram illustrated here, but also when they are embedded in higher order diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This cancellation, guaranteed by the self-consistent Hartree condition, is a significant simplification, as we can now drop all di- agrams that involve the cross symbol given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 1a and/or a Hartree bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Other than this simplification, the Feynman rules are the same as the usual ones [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=" wko wko^w+sk Aw'-sk↓ uk wk ↑ ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content="0 w'k↓b." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content="b wk ↑ w'k' ↓wko." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' wko wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='-o + = 0 wkoL wko3 Figure 3: Particle-particle ladder diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In addition to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 2, another major simplification is all other contributions from H2 interaction can be organized as particle-particle and particle-hole ladder diagrams illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The first line of the equation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 4 includes all particle-particle crossing diagrams, which are equivalent to the particle-hole ladder diagrams on the following line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 3 and 4 are the only nonzero contributions given by the H2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This is because H2 only gives rise to forward scattering, and cannot create particle-hole pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Also for this reason these ladder diagrams can be summed up easily because they form geometric series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' To see this we inspect the corresponding Bethe-Salpeter equations [7] Γpp(p0 1 + p0 2,p) = u(p) + u(p)Γ(p0 1 + p0 2,p)∫ dq0 2π G0 σ (p0 1 + p0 2 2 + q0,p)G0 −σ (p0 1 + p0 2 2 − q0,p), (16) and Γph(p0 1 − p0 4,p) = u(p) + u(p)Γ(p0 1 − p0 4,p)∫ dq0 2π G0 σ (p0 1 − p0 4 2 + q0,p)G0 −σ (q0 − p0 1 − p0 4 2 ,p), (17) where p0 1, p0 2 and p0 4 are the frequencies carried by the external propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Due to the forward-scattering nature there is no momentum integral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' as a result Γ does not enter the integrals on the RHS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' allowing the integrals to be carried out explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' yielding Γpp(p0 1 − p0 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='p) = u(p)(1 − npσ)(1 − np,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) 1 − u(p) ̵h(p0 1+p0 2)−(Epσ+Ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)+iη + u(p)npσnp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ 1 + u(p) ̵h(p0 1+p0 2)−(Epσ+Ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)−iη ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (18) Γph(p0 1 − p0 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='p) = u(p)(1 − npσ)np,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ 1 − u(p) ̵h(p0 1−p0 4)−(Epσ−Ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)+iη + u(p)npσ(1 − np,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) 1 + u(p) ̵h(p0 1−p0 4)−(Epσ−Ep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)−iη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (19) All other diagrams (which inevitably mix particle-particle and particle-hole ladders) vanish, an example of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This is due to the restrictions on the occupations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (18) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (19), where we only have the combinations of npσnp,−σ, (1 − npσ)(1 − np,−σ) for the former, and (1 − npσ)np,−σ, npσ(1 − np,−σ) for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' THE SELF-ENERGY DIAGRAMS In this section we study the electron self-energy Σσ(ω,k), especially its imaginary part, which tells us the decay rate and the broadening of the electron spec- TpP p3 p2p dm ↑ dyd4 Figure 4: Particle-hole ladder diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Figure 5: An example of diagrams that are not part of the particle-particle or particle-hole ladder of HKY interaction (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' All such diagrams vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' tral function measured in the angle-resolved photoemis- sion spectroscopy (ARPES): 1 τ = ImΣσ(ω,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (20) We evaluate the self-energy diagrams to the 2nd order of Hubbard interaction (V 2), which is the lowest order that gives rise to an imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We will, however, include all contributions from HKY interaction, using the ladder sums performed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The simplest diagram is the one with Hubbard inter- action only (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Its imaginary part is Im Σ6a σ (ω,k) =V 2 ∫ d2k′ (2π)2 d2q (2π)2 δ(ω + Ek′,−σ + Ek′−q,−σ − Ek+q,σ) × [(1 − nkσ)nk+q,σ(1 − nk′,−σ)nk′−q,−σ − nkσ(1 − nk+q,σ)nk′,−σ(1 − nk′−q,−σ)], (21) p2p Iph = pgp 个 p4p pp dn dn p2p↓5 which yields the familiar Fermi liquid result near the (pseudo) Fermi surface: ImΣ6a σ (ω,k) ≈ −D3V 2(̵hω)2, (22) where D is the density of states at the non-interacting Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We note however this results in much more broadening in the pseudogap region as the quasiparti- cle energy ̵hω ∼ ∣u∣ is bounded below by the size of the pseudogap, compared to that near the Fermi arcs where ̵hω can be arbitrary small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This is consistent with the cuprate phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We now turn to the diagrams that involve HKY interaction (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Those diagrams are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Here we present the calculation on the self-energy diagram given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6b as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' A full calculation on each diagram is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 6: Self-energy Feynman diagrams to the second order in Hubbard interaction (V 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (a) is the self-energy diagrams in the second order with Hubbard interaction only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (b)-(i) are the self-energy diagrams with both Hubbard and the HKY (H2) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6 After performing the frequency integrals, the corresponding imaginary part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6b takes the form Im Σ6b σ (ω,k) =i6πV 2 ∫ d2q (2π)2 × {[δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ) − δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ − uq](1 − nqσ)(1 − nq,−σ)n−k+2q,−σ −[δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ) − δ(ω + E−k+2q,−σ − Eqσ − Eq,−σ + uq)]nqσnq,−σ(1 − n−k+2q,−σ)}, (23) Note compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (21), we have one fewer momentum integral to perform, despite the extra loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This is due to the fact HKY interaction forces the propagators coupled by it to have the same momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This simplification changes the phase space constraints significantly and enhances Im Σ, as we demonstrate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' To bring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (23) to a form closer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (21), letting −k + 2q → q′, we treat q′ as an additional integration variable, compensated by an additional delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Rewrite the integrals in terms of the polar coordinates such that (qx,qy) = (q cosφ,q sinφ) and (q′ x,q′ y) = (q′ cosφ′,q′ sinφ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (23) becomes Im Σ6b σ (ω,k) =i6πV 2 1 (2π)2 ∫ qdqq′dq′dφdφ′ 1 q′ δ(q′ − ∣ − k + 2q∣)δ(φ′ − φ−k+2q) × {[δ(ω + Eq′,−σ − 2ϵq) − δ(ω + Eq′,−σ − 2ϵq − uq)]θ(ϵq)θ(−Eq′,−σ) −[δ(ω + Eq′,−σ − 2ϵq − 2uq) − δ(ω + Eq′,−σ − 2ϵq − 2uq + uq)]θ(−ϵq − uq)θ(Eq′,−σ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (24) We transform the integral from ∫ qdqq′dq′ to ∫ dϵdE′J(ϵ,E′,φ,φ′) and ∫ dEdE′dφdφ′, where ϵ = ϵq, E = ϵq + uq, and E′ = Eq′,−σ, and the Jacobian J(ϵ,E′,φ,φ′) = q ∂q ∂ϵ q′ ∂q′ ∂E′ = D(ϵ,φ)D(E′,φ′), (25) J(E,E′,φ,φ′) = q ∂q ∂E q′ ∂q′ ∂E′ = D(E,φ)D(E′,φ′), (26) with the angle-dependent density of states D(ϵ,φ) = q ∂q ∂ϵ , (27) D(E,φ) = q ∂q ∂E , (28) D(E′,φ′) = q′ ∂q′ ∂E′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (29) In a two-dimensional system it is a good approximation to treat them as a constant in terms of density of states D: D(ϵ,φ)D(E′,φ′) ≈ D(E,φ)D(E′,φ′) ≈ (2πD)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (30) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (24) then becomes Im Σ6b σ (ω,k) ≈i6πD2 {∫ dϵdE′dφdφ′ 1 q′2 δ (1 − ∣ − k + 2q∣ q′ )δ(φ′ − φ−k+2q)[δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uq)]θ(ϵ)θ(−E′) −∫ dEdE′dφdφ′ 1 q′2 δ (1 − ∣ − k + 2q∣ q′ )δ(φ′ − φ−k+2q)[δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uq)]θ(−E)θ(E′)}, (31) where we have four integrals and three delta functions, and we let δ(q′ − ∣ − k + 2q∣) → 1 q′ δ (1 − ∣−k+2q∣ q′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We then consider the angle integral on φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In order to express ∣ − k + 2q∣, q, q′ in terms of the integral variables and their corresponding angle dependence, we solve the equation E(k,φk) = E so that k = f(E,φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Rewriting uq = uqφ, the 7 approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (31) becomes Im Σ6b σ (ω,k) ≈i6πD2 ∫ dφ′δ(φ′ − φ−k+2q) × {∫ dϵdE′dφ 1 f 2(E′,φ′)δ [1 − f(ϵ,φ−k+2q) f(E′,φ′) ][δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf(ϵ,φ),φ)]θ(ϵ)θ(−E′) −∫ dEdE′dφ 1 f 2(E′,φ′)δ [1 − f(E,φ−k+2q) f(E′,φ′) ][δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf(E,φ),φ)]θ(−E)θ(E′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (32) Performing the angle integral over φ′ yields Im Σ6b σ (ω,k) ≈ i6πD2 {∫ dϵdE′dφ 1 f 2(E′,φ−k+2q)δ [1 − f(ϵ,φ−k+2q) f(E′,φ−k+2q)][δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf(ϵ,φ),φ)]θ(ϵ)θ(−E′) −∫ dEdE′dφ 1 f 2(E′,φ−k+2q)δ [1 − f(E,φ−k+2q) f(E′,φ−k+2q)][δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf(E,φ),φ)]θ(−E)θ(E′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (33) We notice that φ−k+2q can by further replaced by another function g in terms of k, ϵ, E and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (33) becomes Im Σ6b σ (ω,k) ≈ i6πD2 {∫ dϵdE′dφ 1 f 2[E′,g(k,ϵ,φ)]δ {1 − f[ϵ,g(k,ϵ,φ)] f[E′,g(k,ϵ,φ)]}[δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf(ϵ,φ),φ)]θ(ϵ)θ(−E′) −∫ dEdE′dφ 1 f 2[E′,g(k,E,φ)]δ {1 − f[E,g(k,E,φ)] f[E′,g(k,E,φ)]}[δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf(E,φ),φ)]θ(−E)θ(E′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (34) We first perform the integral over φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The remaining delta functions give us φ’s dependence on k, ϵ, E and E′, and we denote it as a function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Hence, (34) becomes Im Σ6b σ (ω,k) ≈ i6πD2 {∫ dϵdE′ 1 f 2{E′,g[k,ϵ,h(k,ϵ,E′)]} [δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ + uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′))]θ(ϵ)θ(−E′) −∫ dEdE′ 1 f 2{E′,g[k,E,h(k,E,E′)]} [δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf[E,h(k,E,E′)],h(k,E,E′))]θ(−E)θ(E′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (35) While we do not know the exact form of the terms 1/f 2, they give us quantities of order 1/(Fermi momentum)2, which is of order O(1) for generic lattice filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Its dependence on ϵ, E, E′ and k is unimportant due to the phase space constraints, as will become clear soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We are then left with integrals with energy variables ϵ, E and E′: Im Σ6b σ (ω,k) ≈i6πV 2D2 {∫ dϵdE′ [δ(ω + E′ − 2ϵ) − δ(ω + E′ − 2ϵ − uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′))]θ(ϵ)θ(−E′) −∫ dEdE′ [δ(ω + E′ − 2E) − δ(ω + E′ − 2E + uf[E,h(k,E,E′)],h(k,E,E′))]θ(−E)θ(E′)}, (36) We can then carry out the integral by assuming uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′) ≈ uf[E,h(k,E,E′)],h(k,E,E′) ≈ ∣u∣, where ∣u∣ is some constant average over uf[ϵ,h(k,ϵ,E′)],h(k,ϵ,E′) or uf[E,h(k,E,E′)],h(k,E,E′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The integrals over E′ give Im Σ6b σ (ω,k) ≈ i6πV 2D2 {∫ dϵθ(ϵ)[θ(−2ϵ + ω) − θ(−2ϵ + ω − ∣u∣)] − ∫ dEθ(E)[θ(−2E + ω) − θ(−2E + ω − ∣u∣)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (37) 8 The integrals on ϵ, E give Im Σ6b σ (ω,k) ≈i6πV 2 [(ω 2 − ω − ∣u∣ 2 ) − (ω 2 − ω + ∣u∣ 2 )] (38) =i6πV 2D2 [∣u∣ 2 − (−∣u∣ 2 )] (39) =i6πV 2D2∣u∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (40) Here we note that the linearity in ∣u∣ comes from the energy integral per the phase space restrictions given by the step functions θ(−2ϵ + ω), θ(−2ϵ + ω − uq), θ(E), and θ(−2E + ω + uq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The final result would not be exactly linear in ∣u∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Hence, we append a function f(k) varying in k: Im Σ6b σ (ω,k) ≈ − πV 2D2∣u∣f(k), (41) where f(k) is a dimensionless quantity of order O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Letting ω → Ekσ, we obtain the self-energy results from other diagrams: Im Σ6f σ (k) ≈ −π 2 V 2D2uk(nkσ − nk,−σ), (42) Im Σ6g σ (k) ≈ ImΣ6h σ (k) ≈ Im Σ6i σ (k) ≈ −πV 2D2uk(nkσ − nk,−σ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (43) The rest imaginary parts resulting from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c, 6d, and 6e share the same form but appear more complicated as we presented later in the expressions (A38) and (A63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' However, since we are only interested in the cases that occur near Fermi-arcs and pseudo-Fermi surfaces, further simplifications can be done so that overall these imaginary parts results in zero or a quantity linear in uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' SUMMARY AND DISCUSSIONS In this paper we studied the model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [4] (referred to as HKY model) which gives rise to Fermi arcs and pseudogap, perturbed by Hubbard interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We found the combination of Hubbard and HKY inter- actions gives rise to a non-zero imaginary part to the electron self-energy in the low-energy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The origin of such non-Fermi liquid behavior lies in the singular na- ture of HKY interaction, which has infinite range in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' While our work was motivated by the cuprates, and gives rise to results that are qualitatively consistent with its phenomenology, the specific (and certainly over- simplified) model we studied should not be taken as a realistic description of the physics of cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Its value lies, instead, in its simplicity, which demonstrates not only the possibility of Fermi arcs and pseudogap, but also that they go hand-in-hand with each other and with the observed non-Fermi liquid behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In fact recent years have witnessed increasing activities in research us- ing models that are extensions of the HK model [8–11] aimed at understanding cuprate phenomenology, includ- ing superconductivity itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' It is our hope that our work provides a starting point to build more realistic models for cuprates and other strongly correlated electron sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the National Science Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' DMR-1932796, and performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' DMR-1644779, and the State of Florida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [1] For a recent review, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=', Cyril Proust and Louis Taillefer, The Remarkable Underlying Ground States of Cuprate Superconductors, Annual Review of Condensed Matter Physics Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 10:409-429 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [2] See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [7] Alexander L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Fetter, and John Dirk Walecka, Quantum theory of many-particle systems (Courier Corporation, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [8] Philip W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Phillips, Luke Yeo, and Edwin W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Exact theory for superconductivity in a doped Mott in- sulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Nature Physics 16, 12: 1175-1180 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [9] Ryan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Nesselrodt and James K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Freericks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Exact so- lution of two simple non-equilibrium electron-phonon and electron-electron coupled systems: The atomic limit of the Holstein-Hubbard model and the generalized Hatsugai-Komoto model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Physical Review B 104, 15: 155104 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [10] Yu Li, Vivek Mishra, Yi Zhou, and Fu-Chun Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Two-stage superconductivity in the Hatsugai- Kohomoto-BCS model, New Journal of Physics (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' [11] Yin Zhong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Solvable periodic Anderson model with infinite-range Hatsugai-Kohmoto interaction: Ground- states and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Physical Review B 106, 15: 155119 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Appendix A: Calculations on the self-energy diagrams Here in the appendix we present more calculation details for each self energy diagram in the section .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6f The self-energy terms given by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6b are Σ6b σ (ω,k) =i3V 2 ∫ d2q (2π)2 ds0 (2π) dq0 (2π) dk′0 (2π)G0 σ(q0q)G0 σ(s0q)G0 −σ(k′0q) × G0 −σ(k′0 + q0 − s0,q)G0 −σ(q0 + k′0 − ω,2q − k)Γpp(q0 + k′0,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A1) k-g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='b w,2-k 0 0 w-s°,k-qK b w- k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='010 Once the Green’s functions are plugged in, it becomes Σ6b σ (ω,k) =i3V 2 ∫ d2q (2π)2 ds0 (2π) dq0 (2π) dk′0 (2π) [ (1 − nqσ)(1 − nq,−σ)n−k+2q,−σ (q0 − Eqσ + iη)(s0 − Eqσ + iη)(k′0 − Eq,−σ + iη) × 1 (k′0 + q0 − s0 − Eq,−σ + iη)(q0 + k′0 − ω − E2q−k,−σ − iη)( 1 uq − 1 q0+k′0−Eqσ−Eq,−σ+iη) + nqσnq,−σ(1 − n−k+2q,−σ) (q0 − Eqσ − iη)(s0 − Eqσ − iη)(k′0 − Eq,−σ − iη) × 1 (k′0 + q0 − s0 − Eq,−σ − iη)(q0 + k′0 − ω − E2q−k,−σ + iη)( 1 uq + 1 q0+k′0−Eqσ−Eq,−σ−iη) ⎤⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A2) The frequency integral gives Σ6b σ (ω,k) =i6V 2 ∫ d2q (2π)2 × [nqσnq,−σ(1 − n−k+2q,−σ)( 1 ω + E−k+2q,−σ − Eqσ − Eq,−σ − iη − 1 ω + E−k+2q,−σ − Eqσ − Eq,−σ + uq − iη ) +(1 − nqσ)(1 − nq,−σ)n−k+2q,−σ ( 1 ω + E−k+2q,−σ − Eqσ − Eq,−σ + iη − 1 ω + E−k+2q,−σ − Eqσ − Eq,−σ − uq + iη )] (A3) The corresponding imaginary part and a sample calculation on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6b is shown in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The final result is Im Σ6b σ (ω,k) ≈ − πV 2D2∣u∣f(k), (A4) where f(k) is a dimensionless quantity of order O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Next we would like to present the calculation on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Σ6f σ (ω,k) =i3V 2 ∫ d2q (2π)2 ds0 (2π) dq0 (2π) dk′0 (2π)G0 σ(q0q)G0 σ(s0q)G0 −σ(k′0q) × G0 −σ(k′0 − q0 + s0,q)G0 −σ(−q0 + k′0 + ω,k)Γph(q0 − k′0,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A5) Σ6f σ (ω,k) =i3V 2 ∫ d2q (2π)2 ds0 (2π) dq0 (2π) dk′0 (2π) [ (1 − nqσ)nq,−σ(1 − n−k+2q,−σ) (q0 − Eqσ + iη)(s0 − Eqσ + iη)(k′0 − Eq,−σ − iη) × 1 (k′0 − q0 + s0 − Eq,−σ − iη)(−q0 + k′0 + ω − Ek,−σ + iη)( 1 uq − 1 q0−k′0−Eqσ+Eq,−σ+iη) + nqσ(1 − nq,−σ)n−k+2q,−σ (q0 − Eqσ − iη)(s0 − Eqσ − iη)(k′0 − Eq,−σ + iη) × 1 (k′0 − q0 + s0 − Eq,−σ + iη)(−q0 + k′0 + ω − Ek,−σ − iη)( 1 uq + 1 q0−k′0−Eqσ+Eq,−σ−iη) ⎤⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A6) 11 The frequency integral gives Σ6f σ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='k) =i6V 2 ∫ d2q (2π)2 × [(1 − nqσ)nq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ(1 − nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)( 1 ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη − 1 ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − uq + iη ) +nqσ(1 − nq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ ( 1 ω − E−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη − 1 ω + Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + uq − iη )] (A7) The corresponding imaginary part is Im Σ6f σ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='k) =i6πV 2 ∫ d2q (2π)2 {[−δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) + δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − uq)](1 − nqσ)nq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ(1 − nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) +[δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) − δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Eqσ + Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + uq)]nqσ(1 − nq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ} =πV 2 ∫ d2q (2π)2 {[−δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − ϵq − uq + ϵq) + δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − ϵq − uq + ϵq − uq)]θ(ϵq + uq)θ(−ϵq)(1 − nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) +[δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − ϵq + ϵq + uq) − δ(ω − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − ϵq + uq + ϵq + uq)]θ(−ϵq)θ(ϵq + uq)nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A8) After simplification we have Im Σ6f σ (ω,k) =i3πV 2 ∫ d2q (2π)2 {[−δ(ω − Ek,−σ − uq) + δ(ω − Ek,−σ − 2uq)]θ(ϵq + uq)θ(−ϵq)(1 − nk,−σ) −[δ(ω − Ek,−σ + uq) − δ(ω − Ek,−σ + 2uq)]θ(−ϵq)θ(ϵq + uq)nk,−σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A9) We let Eq′ = ϵq + uq such that Eq′φ′ = ϵqφ + uqφ in polar coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We solve this equation in favor of q′ and φ′: q′ = f(ϵqφ + uqφ,φ′), (A10) φ′ = g(ϵqφ + uqφ,q′) = g[ϵqφ + uqφ,f(ϵqφ + uqφ,φ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A11) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A9) becomes Im Σ6f σ (ω,k) =i6 π (2π)2 V 2 ∫ qdqq′dq′dφdφ′ 1 q′2 δ [1 − f(ϵqφ + uqφ,φ′) q′ ]δ {g[Eq′φ′,f(ϵqφ + uqφ,φ′)]} × {[−δ(ω − Ek,−σ − uq) + δ(ω − Ek,−σ − 2uq)]θ(ϵq + uq)θ(−ϵq)(1 − nk,−σ) +[δ(ω − Ek,−σ + uq) − δ(ω − Ek,−σ + 2uq)]θ(−ϵq)θ(ϵq + uq)nk,−σ}, (A12) where ∫ dφdφ′ 1 q′2 δ [1 − f(ϵqφ + uqφ,φ′) q′ ]δ {φ′ − g[Eq′φ′,f(ϵqφ + uqφ,φ′)]} (A13) gives a quantity of the order 1/(Fermi momentum)2 and thus of the order O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We then transform ∫ qdqq′dq′ to ∫ D(E,φ)D(E′,φ′)dEdE′ where E = ϵq and E′ = ϵq + uq such that uq = E − E′, and we let D(E,φ)D(E′,φ′) ≈ D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A12) becomes Im Σ6f σ (ω,k) ≈ i6πV 2D2 ∫ dEdE′ {[−δ(ω − Ek,−σ − E′ + E) + δ(ω − Ek,−σ − 2E′ − 2E)]θ(E′)θ(−E)(1 − nk,−σ) − [δ(ω − Ek,−σ + E′ − E) − δ(ω − Ek,−σ + 2E′ − 2E)]θ(−E)θ(E′)nk,−σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A14) This gives Σ6f σ (ω,k) ≈i6πV 2D2 [Ek,−σ − ω 2 (1 − nk,−σ) + −ω + Ek,−σ 2 nk,−σ], (A15) 12 Figure 8 which can be further simplified: Im Σ6f σ (ω,k) ≈i6 π 2 V 2D2(Ek,−σ − ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A16) As ω → Ekσ, we obtain Im Σ6f σ (k) ≈ − π 2 V 2D2uk(nkσ − nk,−σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A17) We consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c, 6g, 6d, and 6h together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We find the equality in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' So here we only consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c and 6g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We start with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c first Σ6c σ (ω,k) =i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π G0 σ(s0,k)G0 σ(s0 − q0,k − q)G0 −σ(k′0,k) × G0 −σ(k′0 − ω + s0,k)G0 −σ(k′0 + q0,k + q)Γpp(s0 + k′0,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A18) �⇒ Σ6c σ (ω,k) =i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π [ (1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ (s0 − Ekσ + iη)(s0 − q0 − Ek−q,σ − iη)(k′0 − Ek,−σ + iη) × 1 (k′0 − ω + s0 − Ek,−σ + iη)(k′0 + q0 − Ek+q,−σ − iη)( 1 uk − 1 s0+k′0−Ek,σ−Ek,−σ+iη) + nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) (s0 − Ekσ − iη)(s0 − q0 − Ek−q,σ + iη)(k′0 − Ek,−σ − iη) × 1 (k′0 − ω + s0 − Ek,−σ − iη)(k′0 + q0 − Ek+q,−σ + iη)( 1 uk + 1 s0+k′0−Ek,σ−Ek,−σ−iη) ⎤⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A19) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 3c = Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 3d Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 3g Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 3h13 The frequency integral gives Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6g Σ6c σ (ω,k) =i6V 2uk ∫ d2q (2π)2 × [− (1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ (−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ + iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ − uk + iη) + nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) (−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ − iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ + uk − iη)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A20) The imaginary part is Im Σ6c σ (ω,k) = − πV 2uk ∫ d2q (2π)2 δ(ω − Ek−q,σ − Ek+q,−σ + Ek,−σ) × [sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Ek+q,−σ − uk) (1 − nk,−σ)(1 − nkσ)nk+q,−σnq,σ ∣ − Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ + uk∣ +sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Ek+q,−σ + uk) nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) ∣ − Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ − uk∣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A21) We k + q → q′ and treat q′ as an additional integration variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We rewrite the integrals in terms of the polar coordinates such that (qx,qy) = (q cosφ,q sinφ) and (q′ x,q′ y) = (q′ cosφ′,q′ sinφ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A21) becomes − πV 2uk 1 (2π)2 ∫ qdqq′dq′dφdφ′ 1 q′ δ(q′ − ∣k + q∣)δ(φ′ − φk+q)δ(ω − Ek−q,σ − Eq′,−σ + Ek,−σ) × [sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Eq′,−σ − uk) (1 − nk,−σ)(1 − nkσ)θ(−Eq′,−σ)θ(−Ek−q,σ) ∣ − Ek,−σ + Eq′,−σ + Ek−q,σ − Ekσ + uk∣ +sgn(−ω − Ekσ − 2Ek,−σ + 2Ek−q,σ + 2Eq′,−σ + uk) nk,−σnkσθ(Ek−q,σ)θ(−Eq′,−σ) ∣ − Ek,−σ + Eq′,−σ + Ek−q,σ − Ekσ − uk∣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A22) We transform the integral ∫ qdqq′dq′ → ∫ dEdE′J(E,E′), where E = Ek−q,σ, E′ = Eq′,−σ, and the Jacobian J(E,E′,φ,φ′) = q ∂q ∂E q′ ∂q′ ∂E′ = D(E,φ)D(E′,φ′), (A23) with the angle-dependent density of states D(E,φ) = q ∂q ∂E , (A24) D(E′,φ′) = q′ ∂q′ ∂E′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A25) 0 0 s+m- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='k w-sk 0 k-q q,k-q s-m14 As done in Section IV, we treat them as a constant in terms of density of states D: D(E,φ)D(E′,φ′) ≈ (2πD)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A26) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A22) then becomes − πV 2ukD2 ∫ dEdE′dφdφ′ 1 q′ δ(q′ − ∣k + q∣)δ(φ′ − φk+q)δ(ω − E − E′ + Ek,−σ) × [sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ − uk) (1 − nk,−σ)(1 − nkσ)θ(−E′)θ(E) ∣ − Ek,−σ + E′ + E − Ekσ + uk∣ +sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ + uk) nk,−σnkσθ(E)θ(−E′) ∣ − Ek,−σ + E′ + E − Ekσ − uk∣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A27) Letting δ(q′ − ∣k + q∣) → 1 q′ δ (1 − ∣k+q∣ q′ ), we consider the angle integrals first: ∫ dφdφ′ 1 q′2 δ (1 − ∣k + q∣ q′ )δ(φ′ − φk+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A28) In terms of the integral variables E, E′, φ′ and the angle dependence φk+q, this can be expressed as ∫ dφdφ′ 1 f 2(E′,φ′)δ [1 − f(E,φk+q) f(E′,φ′) ]δ(φ′ − φk+q), (A29) where function f is the solution of E(k,φk) = E in favor of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Performing the angle integral over φ′ yields ∫ dφ 1 f 2(E′,φk+q)δ [1 − f(E,φk+q) f(E′,φk+q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A30) The quantity φk+q can by further replaced by another function g in terms of k, E and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The integral becomes ∫ dφ 1 f 2(E′,g(k,E,φ))δ {1 − f[E,g(k,E,φ)] f[E′,g(k,E,φ)]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A31) This gives us a quantity of order 1/(Fermi momentum)2, which is of order O(1) for generic lattice filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The expression (A27) then becomes ImΣ6c σ (ω,k) ≈ − πV 2ukD2 ∫ dEdE′δ(ω − E − E′ + Ek,−σ) × [sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ − uk) (1 − nk,−σ)(1 − nkσ)θ(−E′)θ(−E) ∣ − Ek,−σ + E′ + E − Ekσ − uk∣ +sgn(−ω − Ekσ − 2Ek,−σ + 2E + 2E′ + uk) nk,−σnkσθ(E)θ(E′) ∣ − Ek,−σ + E′ + E − Ekσ + uk∣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A32) We then perform the integrals over E and E′: Im Σ6c σ (ω,k) ≈ −πV 2ukD2 ∫ dE [sgn(ω − Ekσ − uk) (1 − nk,−σ)(1 − nkσ)θ(−ω + E − Ek,−σ)θ(−E) ∣ω − Ekσ − uk∣ +sgn(ω − Ekσ + uk) nk,−σnkσθ(E)θ(ω − E + Ek,−σ) ∣ω − Ekσ + uk∣ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A33) The last integral on E gives Im Σ6c σ (ω,k) ≈Im Σ6d σ (k) ≈ −πV 2ukD2 [sgn(ω − Ekσ − uk) (1 − nk,−σ)(1 − nkσ)(−ω − Ek,−σ) ∣ω − Ekσ − uk∣ +sgn(ω − Ekσ + uk) nk,−σnkσ(ω + Ek,−σ) ∣ω − Ekσ + uk∣ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A34) 15 After simplification, this results in ImΣ6c σ (ω,k) ≈ Im Σ6d σ (k) ≈ − πV 2D2 [−sgn(ω − Ekσ − uk) (1 − nk,−σ)(1 − nkσ) ∣ω − Ekσ − uk∣ + sgn(ω − Ekσ + uk) nk,−σnkσ ∣ω − Ekσ + uk∣]uk(Ek,−σ + ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A35) As uk approaches to 0, the quantity (A35) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We let ω → Ekσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The sign functions become sgn(−uk) and sgn(uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Hence, (A35) becomes − πV 2D2 [−sgn(−uk)(1 − nk,−σ)(1 − nkσ) ∣ − uk∣ + sgn(uk)nk,−σnkσ ∣uk∣ ]uk(Ek,σ + Ek,−σ) (A36) = − 2πV 2D2 uk ∣uk∣sgn(uk)[ϵk(1 − nk,−σ)(1 − nkσ) + (ϵk + uk)nk,−σnkσ] (A37) = − 2πV 2D2 [ϵk(1 − nk,−σ)(1 − nkσ) + (ϵk + uk)nk,−σnkσ], (A38) where the value of Ekσ and Ek,−σ depend on corresponding occupation number: for nkσ = nk,−σ = 0, Ekσ = Ek,−σ = ϵk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' with nkσ = nk,−σ = 1, Ekσ = Ek,−σ = ϵk + uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We examine the conditions with two different occupations: as nkσ = nk,−σ = 0, Im Σ6c σ (k) ≈ Im Σ6d σ (k) ≈ −2πV 2D2ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A39) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (8), when we approach both the Fermi-arcs and the pseudo-Fermi surfaces from nk = 0 region, ϵk → 0 and thus (A40) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' As nkσ = nk,−σ = 2, Im Σ6c σ (k) ≈ Im Σ6d σ (k) ≈ −2πV 2D2(ϵk + uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A40) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (9), when we approach the Fermi-arcs from nk = 2 region, ϵk + uk → 0, and when approaching to the pseudo-Fermi surfaces, we have ϵk +uk → uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Hence, we conclude that overall the result of (A38) is zero or linear in uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' We then consider the self-energy diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' Σ6g σ (ω,k) =i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π G0 σ(s0,k)G0 σ(s0 − q0,k − q)G0 −σ(k′0,k) × G0 −σ(k′0 + ω − s0,k)G0 −σ(k′0 − q0,k′ − q)Γph(s0 − k′0,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A41) �⇒ Σ6g σ (ω,k) =i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π [ nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ (s0 − Ekσ + iη)(s0 − q0 − Ek−q,σ − iη)(k′0 − Ek,−σ − iη) × 1 (k′0 + ω − s0 − Ek,−σ − iη)(k′0 − q0 − Ek−q,−σ + iη)( 1 uk − 1 s0−k′0−Ekσ+Ek,−σ+iη) + (1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) (s0 − Ekσ − iη)(s0 − q0 − Ek−q,σ + iη)(k′0 − Ek,−σ + iη) × 1 (k′0 + ω − s0 − Ek,−σ + iη)(k′0 − q0 − Ek−q,−σ − iη)( 1 uk + 1 s0−k′0−Ekσ+Ek,−σ−iη) ⎤⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A42) The frequency integral gives Σ6g σ (ω,k) =i6V 2uk ∫ d2q (2π)2 × [ nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ (ω + Ek−q,−σ − Ek−q,σ − Ek,−σ − iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk + iη) + (1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) (ω + Ek−q,σ − Ek−q,σ − Ek,−σ + iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk − iη)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A43) 16 The imaginary part is Im Σ6g σ (ω,k) =i6πV 2uk ∫ d2q (2π)2 δ(ω + Ek−q,−σ − Ek−q,σ − Ek,−σ) × [−sgn(ω + Ekσ − 2Ek,−σ + 2Ek−q,−σ − 2Ek−q,σ + uk) nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ ∣Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk∣ +sgn(ω + Ekσ − 2Ek,−σ + 2Ek−q,−σ − 2Ek−q,σ − uk) (1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) ∣Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk∣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A44) After simplification we have ImΣ6g σ (ω,k) =i6πV 2uk ∫ d2q (2π)2 × [−sgn(ω + Ekσ − 2Ek,−σ − 2uk−q + uk)δ(ω + uk−q − Ek,−σ)nk,−σ(1 − nkσ)θ(ϵk−q + uk−q)θ(−ϵk−q) ∣Ek,−σ − uk−q − Ekσ − uk∣ +sgn(ω + Ekσ − 2Ek,−σ + 2uk−q − uk)δ(ω − uk−q − Ek,−σ)(1 − nk,−σ)nkσθ(ϵk−q + uk−q)θ(−ϵk−q) ∣Ek,−σ + uk−q − Ekσ + uk∣ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A45) We let Eq′ = ϵk−q + uk−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' In polar coordinates, we have the solutions q′ = f(ϵkqφ + ukqφ,φ′), (A46) φ′ = g(ϵkqφ + ukqφ,q′) = g[ϵkqφ + ukqφ,f(ϵkqφ + ukqφ,φ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A47) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A45) becomes ImΣ6g σ (ω,k) =i6 π (2π)2 V 2uk ∫ qdqq′dq′dφdφ′ 1 q′2 δ [1 − f(ϵkqφ + ukqφ,φ′) q′ ]δ{φ′ − g[ϵkqφ + ukqφ,f(ϵkqφ + ukqφ,φ′)]} × [−sgn(ω + Ekσ − 2Ek,−σ − 2Eq′φ′ + 2ϵkqφ + uk)δ(ω + Eq′φ′ − ϵkqφ − Ek,−σ)nk,−σ(1 − nkσ)θ(Eq′φ′)θ(−ϵkqφ) ∣Ek,−σ − Eq′φ′ + ϵkqφ − Ekσ − uk∣ +sgn(ω + Ekσ − 2Ek,−σ + 2Eq′φ′ − 2ϵkqφ − uk)δ(ω − Eq′φ′ + ϵkqφ − Ek,−σ)(1 − nk,−σ)nkσθ(Eq′φ′)θ(−ϵkqφ) ∣Ek,−σ + Eq′φ′ − ϵkqφ − Ekσ + uk∣].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A48) After we transform ∫ qdqq′dq′ → (2πD)2 ∫ dEdE′ where ϵkqφ → E and Eq′φ′ → E′, and we ignore the quantities of the order O(1), we have ImΣ6g σ (ω,k) ≈i6 π (2π)2 (2π)2D3V 2uk ∫ dEdE′ × [−D1sgn(ω + Ekσ − 2Ek,−σ − 2E′ + 2E + uk)δ(ω + E′ − E − Ek,−σ)nk,−σ(1 − nkσ)θ(E′)θ(−E) ∣Ek,−σ − E′ + E − Ekσ − uk∣ +D2sgn(ω + Ekσ − 2Ek,−σ − 2E′ + 2E − uk)δ(ω − E′ + E − Ek,−σ)(1 − nk,−σ)nkσθ(E′)θ(−E) ∣Ek,−σ + E′ − E − Ekσ + uk∣], (A49) where D1, D2, and D3 are densities of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' This gives ImΣ6g σ (ω,k) ≈i6πV 2D3uk [−sgn(3ω + Ekσ − 4Ek,−σ + uk)D1 nk,−σ(1 − nkσ)(−ω + Ek,−σ) ∣ω − Ekσ − uk∣ (A50) +sgn(3ω + Ekσ − 4Ek,−σ − uk)D2 (1 − nk,−σ)nkσ(ω − Ek,−σ) ∣ω − Ekσ + uk∣ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A51) 17 We let ω → Ekσ and D1 ≈ D2 ≈ D3 ≈ D, ImΣ6g σ (k) ≈ − πV 2D2uk(nk,−σ − nkσ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A52) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6e Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6i Finally we consider the self-energy diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6e and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The expression for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6e is Σ6e σ (ω,k) = i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π G0 σ(s0,k)G0 σ(s0 − q0,k − q)G0 σ(s0 − v0 + k′0,k)G0 −σ(k′0,k)G0 −σ(v0,k) × G0 −σ(s0 − ω + k′0,k)G0 −σ(k′0 + q0,k + q)(Γpp(s0 + k′0,k))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A53) �⇒ Σ6e σ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='k) =i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π [ (1 − nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)(1 − nkσ)nk+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σnk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ (s0 − Ekσ + iη)(s0 − q0 − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − iη)(k′0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη) × 1 (s0 − v0 + k′0 − Ekσ + iη)(v0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη) × 1 (k′0 − ω + s0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη)(k′0 + q0 − Ek+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη)( 1 uk − 1 s0+k′0−Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ−Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ+iη) 2 + nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σnkσ(1 − nk+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)(1 − nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ) (s0 − Ekσ − iη)(s0 − q0 − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ + iη)(k′0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη) × 1 (s0 − v0 + k′0 − Ekσ − iη)(v0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη) × 1 (k′0 − ω + s0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη)(k′0 + q0 − Ek+q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη)( 1 uk + 1 s0+k′0−Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ−Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ−iη) 2 ⎤⎥⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A54) +3- k+q v°kq 0 S k-g k18 The frequency gives i6V 2u2 k ∫ d2q (2π)2 × [ (1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ (−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ + iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ − uk + iη)2 + nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) (−ω + Ek−q,−σ + Ek+q,−σ − Ek,−σ − iη)(−Ekσ + Ek−q,−σ + Ek+q,−σ − Ek,−σ + uk − iη)2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A55) The expression for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6i is Σ6i σ (ω,k) = i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π G0 σ(s0,k)G0 σ(s0 − q0,k − q)G0 σ(s0 − v0 + k′0,k)G0 −σ(k′0,k)G0 −σ(v0,k) × G0 −σ(−s0 + ω + k′0,k)G0 −σ(k′0 − q0,k − q)(Γph(s0 − k′0,k))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A56) �⇒ Σ6i σ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='k) =i3V 2 ∫ d2q (2π)2 ∫ ds0 2π dq0 2π dk′0 2π [ nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ(1 − nkσ)(1 − nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ (s0 − Ekσ + iη)(s0 − q0 − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − iη)(k′0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη) × 1 (s0 + v0 − k′0 − Ekσ + iη)(v0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη) × 1 (k′0 + ω − s0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη)(k′0 − q0 − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη)( 1 uk − 1 s0−k′0−Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ+Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ+iη) 2 + (1 − nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)nkσnk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)(1 − nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ) (s0 − Ekσ − iη)(s0 − q0 − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ + iη)(k′0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη) × 1 (s0 + v0 − k′0 − Ekσ − iη)(v0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη) × 1 (k′0 + ω − s0 − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + iη)(k′0 − q0 − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − iη)( 1 uk + 1 s0−k′0−Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ+Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ−iη) 2 ⎤⎥⎥⎥⎥⎥⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A57) The frequency integral gives i6V 2u2 k ∫ d2q (2π)2 × [− nk,−σ(1 − nkσ)(1 − nk−q,−σ)nk−q,σ (ω + Ek−q,−σ − Ek−q,σ − Ek,−σ − iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ − uk + iη)2 + (1 − nk,−σ)nkσnk−q,−σ(1 − nk−q,σ) (ω + Ek−q,−σ − Ek−q,σ − Ek,−σ + iη)(Ek,−σ − Ek−q,−σ + Ek−q,σ − Ekσ + uk − iη)2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A58) The imaginary parts are Im Σ6i σ (ω,k) =i6V 2u2 k ∫ d2q (2π)2 δ(ω − Ek−q,σ − Ek+q,−σ + Ek,−σ) × {−sgn[(−Ek,−σ + Ek−q,σ + Ek+q,−σ − Ekσ − uk)(−2ω − Ekσ − uk − 3Ek,−σ + 3Ek−q,σ + 3Ek+q,−σ)] × (1 − nk,−σ)(1 − nkσ)nk+q,−σnk−q,σ (−Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ − uk)2 +sgn[(−Ek,−σ + Ek−q,σ + Ek+q,−σ − Ekσ + uk)(−2ω − Ekσ + uk − 3Ek,−σ + 3Ek−q,σ + 3Ek+q,−σ)] × nk,−σnkσ(1 − nk+q,−σ)(1 − nk−q,σ) (−Ek,−σ + Ek+q,−σ + Ek−q,σ − Ekσ + uk)2 ⎫⎪⎪⎬⎪⎪⎭ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A59) 19 and Im Σ6e σ (ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='k) =i6V 2u2 k ∫ d2q (2π)2 δ(ω + Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ) × {sgn[(Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − Ekσ − uk)(2ω + Ekσ − uk − 3Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + 3Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − 3Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ)] × nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ(1 − nkσ)(1 − nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ (Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − Ekσ − uk)2 +sgn[(Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − Ekσ + uk)(2ω + Ekσ + uk − 3Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + 3Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − 3Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ)] × (1 − nk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ)nkσnk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ(1 − nk−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ) (Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ − Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='−σ + Ek−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content='σ − Ekσ + uk)2 ⎫⎪⎪⎬⎪⎪⎭ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A60) The substitution and the transformation on the integral variable are the same as we did for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' The final results are Im Σ6e σ (ω,k) ≈i6V 2D2 {sgn[(ω − Ek,−σ − uk)2]−(1 − nk,−σ)(1 − nkσ) (ω − Ekσ − uk)2 + sgn[(ω − Ek,−σ + uk)2] nk,−σnkσ (ω − Ekσ + uk)2 } × u2 k(Ek,−σ + ω), (A61) Im Σ6i σ (ω,k) ≈i6V 2D2 {−sgn[(ω − Ek,−σ − uk)2] (1 − nkσ)nk,−σ (ω − Ekσ − uk)2 + sgn[(ω − Ek,−σ + uk)2] nkσ(1 − nk,−σ) (ω − Ekσ + uk)2 } × u2 k(ω − Ek,−σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A62) As ω → Ekσ, Im Σ6e σ (ω,k) ≈ −2πV 2D2 [−ϵk(1 − nk,−σ)(1 − nkσ) + (ϵk + uk)nk,−σnkσ], (A63) Im Σ6i σ (ω,k) ≈ −πV 2D2uk(nk,−σ − nkσ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' (A64) Here we notice that (A64) is linear in uk and thus it vanishes as uk → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' As for the result (A63), we perform the same analysis as we did for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' 6c and 6d, and hence we conclude: as it approaches to Fermi-arcs from either nk = 0 or nk = 2 regions, and approaches to pseudo-Fermi surfaces from nk = 0 region, it always vanishes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} +page_content=' as it approaches to pseudo-Fermi surfaces from nk = 2 region, it results in a quantity proportional to uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE3T4oBgHgl3EQffwrX/content/2301.04556v1.pdf'} diff --git a/iNAyT4oBgHgl3EQfXvei/content/tmp_files/2301.00190v1.pdf.txt b/iNAyT4oBgHgl3EQfXvei/content/tmp_files/2301.00190v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..153e7b38c69617c9de2e23ffd3e8dc41baecfb75 --- /dev/null +++ b/iNAyT4oBgHgl3EQfXvei/content/tmp_files/2301.00190v1.pdf.txt @@ -0,0 +1,2836 @@ +Tracking Passengers and Baggage Items using +Multiple Overhead Cameras at Security Checkpoints +Abubakar Siddique, Student Member, IEEE, Henry Medeiros, Senior Member, IEEE*†‡§¶ +Abstract—We introduce a novel framework to track multiple +objects in overhead camera videos for airport checkpoint security +scenarios where targets correspond to passengers and their +baggage items. We propose a Self-Supervised Learning (SSL) +technique to provide the model information about instance +segmentation uncertainty from overhead images. Our SSL ap- +proach improves object detection by employing a test-time data +augmentation and a regression-based, rotation-invariant pseudo- +label refinement technique. Our pseudo-label generation method +provides multiple geometrically-transformed images as inputs to +a Convolutional Neural Network (CNN), regresses the augmented +detections generated by the network to reduce localization errors, +and then clusters them using the mean-shift algorithm. The self- +supervised detector model is used in a single-camera tracking +algorithm to generate temporal identifiers for the targets. Our +method also incorporates a multi-view trajectory association +mechanism to maintain consistent temporal identifiers as pas- +sengers travel across camera views. An evaluation of detection, +tracking, and association performances on videos obtained from +multiple overhead cameras in a realistic airport checkpoint +environment demonstrates the effectiveness of the proposed +approach. Our results show that self-supervision improves object +detection accuracy by up to 42% without increasing the inference +time of the model. Our multi-camera association method achieves +up to 89% multi-object tracking accuracy with an average +computation time of less than 15 ms. +Index Terms—Self-supervised Learning, Detection, Tracking, +Tracklet Association, Multi-camera Tracking, Surveillance. +I. INTRODUCTION +A +UTOMATED video surveillance requires the detection, +tracking, and recognition of objects of interest in a scene. +Accurate and precise surveillance in crowded scenes is one of +the most challenging computer vision applications. To address +the problem of visual surveillance in the domain of airport +checkpoint security, the Department of Homeland Security +*Manuscript received August 22, 2022; accepted November 11, 2022. +Date of publication December 14, 2022. +†This material is based upon work supported by the U.S. Department of +Homeland Security, Science and Technology Directorate, Office of University +Programs, under Award Number 2013-ST-061-E0001-04. The views and +conclusions contained in this document are those of the authors and should not +be interpreted as necessarily representing the official policies, either expressed +or implied, of the U.S. Department of Homeland Security. +‡Abubakar +Siddique +is +with +the +Department +of +Electrical +and +Computer Engineering, Marquette University, Milwaukee, USA, e-mail: +abubakar.siddique@marquette.edu +§Henry Medeiros is with the Department of Agricultural and Bi- +ological Engineering, University of Florida, Gainesville, USA, e-mail: +hmedeiros@ufl.edu +¶Digital Object Identifier (DOI): 10.1109/TSMC.2022.3225252 +(DHS) ALERT (Awareness and Localization of Explosives- +Related Threats) center of excellence at Northeastern Univer- +sity initiated the CLASP (Correlating Luggage and Specific +Passengers) project. This initiative aims to help the Transporta- +tion Security Administration (TSA) detect security incidents, +such as theft of items and abandoned bags. +Current approaches for detecting and tracking passengers +and luggage in airport checkpoints divide the image area +within each camera’s field of view into regions of inter- +est where certain passenger behaviors are expected (e.g., +passengers divest their items near the roller conveyor) [1], +[2]. While these approaches are effective within individual +regions of interest, they cannot detect and track passengers and +their belongings throughout an entire checkpoint. Moreover, +most recent detection algorithms [3]–[6] are unable to detect +multiple objects in realistic overhead camera scenarios due +to the unavailability of large-scale datasets obtained using +unconventional camera perspectives. +Fine-tuning pre-trained models using human annotated la- +bels is a common approach in computer vision methods. +However, this strategy hinders the applicability of state-of- +the-art algorithms in scenarios where images are obtained +from perspectives that are not commonly observed in existing +publicly available datasets. The dramatic variability of video +surveillance systems used in airport checkpoints would require +deployment-specific fine-tuning of models, and in some sce- +narios, even camera-specific adjustments. To overcome this +challenge, we leverage the fact that models pre-trained on +large-scale datasets can build upon their initial predictions to +adapt to new scenarios using SSL strategies. Our proposed +SSL framework obviates the tedious and expensive human +annotation procedure by automatically generating pseudo- +labels to update the model. +To generate pseudo-labels, we cluster multiple detections +obtained from geometrically transformed images using the +mean-shift algorithm [7]. Each cluster corresponds to the +detection of one object observed at different orientations on +several augmented input images. The cluster modes with +the corresponding bounding boxes, segmentation masks, and +confidence scores are used to update the model. Thus, our +model learns from rotation-invariant pseudo-labels and can +be integrated with a tracking-by-detection algorithm [8] to +generate accurate target tracklets from overhead perspectives. +Our SSL algorithm is inspired by the methods described in +[9]–[13]. However, unlike [9], instead of resorting to multi-task +1 +arXiv:2301.00190v1 [cs.CV] 31 Dec 2022 + +2 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +Fig. 1. Proposed SSL framework. The augmented proposal generation stage uses multiple rotated versions of the unlabeled input images to generate augmented +detections from an instance segmentation model and then remaps these predictions into their original coordinates. The clustering algorithm leverages the model’s +regression ability to reduce localization errors using the augmented predictions as region proposals. The regressed cluster modes are then used to generate +augmented pseudo-labels to update the model. +strategies to guide the learning process, we employ a multi- +inference approach similar in spirit to the self-consistency +method based on equivariant transformations proposed in +[10]. Our method differs from [10] in that, rather than using +the uncertainties from multiple model predictions to select +image patches for additional training, it aggregates multiple +inferences into accurate pseudo-labels that are used to refine +the model. Our method departs significantly from unsupervised +model adaptation [11] and knowledge distillation approaches +[14] in that we only use automatically generated labels and +avoid human annotations altogether during model update. +We also propose a Multi-Camera Tracklet Association +(MCTA) algorithm to maintain the temporal identifiers of pas- +sengers across cameras. We leverage the fact that our system +is comprised of overhead cameras with partially overlapping +fields of view to employ a simple but effective geometry- +based trajectory association method. Our algorithm compares +the projected centroids of target detections on neighboring +cameras using the homographies between their image planes. +We track passengers and bags across multiple views and +generate global tracks by combining pairwise associations +from the partially overlapping camera views. +We evaluate the detection and tracking performance of our +algorithms on videos from a simulated airport checkpoint and +demonstrate that our approach performs on par with a model +trained in an entirely supervised manner and substantially +outperforms the pre-trained detection model. Our multi-camera +evaluation shows that our MCTA method effectively handles +the problem of passenger identity hand-off across cameras. +In summary, the key contributions of this work are: +• A novel self-supervised object detection algorithm that +generates pseudo-labels based on instance segmentation +uncertainties. +• A new data augmentation and regression-based clustering +mechanism that substantially improves the quality of +pseudo-labels for self-supervised training. +• A new recursive tracklet association algorithm to address +the identity hand-off issue during transitions between +crowded overhead camera views. +• We provide an extensive evaluation of our methods on +a dataset collected using multiple overhead cameras in a +realistic airport checkpoint scenario. +• Our SSL models and the corresponding source code are +available at https://github.com/siddiquemu/SCT MCTA. +To our knowledge, this is the first approach to solve the +overhead multi-view association problem in a network of +cameras with partially overlapping fields of view using a self- +supervised detection strategy. +II. RELATED WORK +Multiple target tracking using camera networks is an active +research topic with several potential applications [15]–[18]. +Most works on camera networks focus on the multi-camera +aspect of the problem and do not consider the challenges +associated with camera perspectives. Although generic object +tracking algorithms could be used in surveillance systems (e.g. +[19]–[21]), when object categories are known, trackers based +on specialized detectors are more accurate and less prone +to model drift [22], [23]. This observation has led to the +development of a variety of multiple target tracking algorithms +that specialize in tracking humans [24]–[34] or vehicles [35]– +[38]. However, in many scenarios, it is desirable to track +additional objects of known categories. In these cases, more +flexible detection algorithms are needed, but the effectiveness +of modern object detection models is highly dependent upon +the characteristics of the training datasets [3]–[5]. +Previous works have used SSL techniques to improve visual +feature learning [39]–[41], reducing dependency on human an- +notations for training backbone models. However, transferring +knowledge from pre-trained backbones to downstream tasks is +a far less explored topic. Unlike our proposed approach, SSL +techniques for detection [9], [11] and semantic segmentation +[10] rely on annotations to initialize the model before iterative +learning can take place. +Data augmentation is an effective mechanism to improve +the robustness of CNNs in scenarios not available during + +亚(t) +DS(t) +R +Instance +Prediction +Segmentation +Remapping +Model +1 +ROI +Backbone +HeadSIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS +3 +Fig. 2. Visualization of our data augmentation approach. The first and second columns show the segmentation masks and detections at θ = 0◦ and θ = 186◦, +respectively. The third column shows the remapped detections in the set SC on the original image (using Alg. 1) with the best detections (blue) from Alg. 2. +training [14], [42], but little attention has been given so far +to approaches for combining the response of the network +to augmented samples. In multi-target tracking applications, +multiple detections mapped to a common coordinate system +can be interpreted as the probability of occupancy of the area +observed by the cameras [43]. Although it is possible to use +clustering techniques to map the modes of this distribution to +unique target detections, bounding box alignment errors pose +a challenge to the generation of high-quality pseudo-labels for +SSL. Hence, we propose a test-time regression technique that +leverages instance segmentation information for pseudo-label +generation. +A systematic solution to the data association problem +is another important component of multi-target tracking-by- +detection methods [44]–[57]. Single-camera trackers [8], [58] +use detectors trained on multiple datasets [59] to generate +bounding boxes and form track hypotheses for all the targets +in each frame. In this work, we employ a state-of-the-art +single-camera tracker [8] using a detector based on our self- +supervised models, which achieves unprecedented tracking +performance in previously unseen airport surveillance videos. +Finally, multi-camera tracking systems require sophisticated +trajectory association mechanisms to maintain target identities +across cameras [60]–[62]. Even within a single camera, occlu- +sions must be handled using similar strategies [63], [64]. Most +association approaches compute trajectory similarity scores +based on a combination of appearance and motion features +[60]–[64]. These features are learned using a large number of +continuous trajectories, which are difficult to obtain with typi- +cal ceiling-height overhead cameras due to their limited fields +of view. Some methods use camera calibration information to +project tracks onto a common plane and perform association +using occlusion modeling [32] or re-identification techniques +[33]–[37]. Dependency on camera calibration further limits +the applicability of these methods to security systems since +calibrating multiple cameras with partially overlapping fields +of view is a complex task [65]–[68]. +III. PROPOSED MODEL +Our system consists of two main components: i) a detection +algorithm trained using SSL and ii) a multi-camera tracking- +by-detection mechanism. A single-camera tracking algorithm +uses SSL detections to generate tracklets for passengers and +baggage items. We then employ a novel multi-camera target +trajectory association algorithm to uniquely identify passen- +gers throughout the checkpoint. +A. Self-Supervised Learning +We use the PANet model [5] with a ResNet-50 backbone +[69], [70] as the baseline detector. Since the categories of +interest are persons and their belongings, we use a model +pre-trained on the COCO dataset [71], which includes object +classes related to these categories (i.e., person, handbag, back- +pack, and suitcase). Because the COCO dataset consists mostly +of images captured at roughly eye-level, detectors trained using +that dataset do not perform well on overhead perspectives. + +4 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +To address this limitation, our SSL framework updates the +baseline model using rotation-invariant pseudo-labels. As Fig. +1 shows, our SSL framework consists of three main steps: +i) augmented region proposals generation, ii) pseudo-label +generation and refinement through cluster regression, and iii) +iterative model update. +Algorithm 1 Augmented Proposals Generation +1: function AUGMENTEDPROPOSALS(I(t), r) +2: +SC(t) = ∅, Θ = {i · ∆θ}r +i=1 +3: +for θi ∈ Θ do +4: +Ψθi(t) = Rθi(I(t)) +5: +DC +θi(t) = DPANet(Ψθi(t)) +6: +SC +θi(t) = R−θi(DC +θi(t)) +7: +SC(t) = SC(t) ∪ SC +θi(t) +8: +end for +9: +return SC(t) +10: end function +1) Augmented Proposals Generation: Our data augmenta- +tion method, summarized in Alg. 1, uses the PANet model +to detect and segment multiple instances of objects of in- +terest. During the first iteration of SSL training, we retain +only the outputs of the pre-trained model for the person, +handbag, backpack, and suitcase classes. The person class +corresponds to passengers and detections of handbag, back- +pack, and suitcase items are treated as baggage items. In +subsequent iterations of SSL training, we modify the model +to generate only the object categories C ∈ {pax, bag}, where +pax corresponds to passengers and bag to baggage items. Let +DC(t) be the set of detections on image I(t) at time t. That +is, DC(t) = {d1, . . . , dnC +t }, where dj ∈ R5 is the detection of +the j-th object and nC +t is the number of objects of class C in +frame I(t). Each detection dj consists of the coordinates and +dimensions of the target’s bounding box, bC +j ∈ R4, as well as +its detection confidence score sj ∈ [0, 1]. +We noticed that the detector performs better when objects +are observed at more commonly occurring angles (e.g., up- +right). Therefore, to reduce the negative effect of the overhead +perspective, we generate multiple rotated copies of the input +image Ψθi(t) = Rθi(I(t)) (line 4 in Alg. 1), where Rθi(·) is +the rotation operator, which rotates the image by an angle θi. +The angle of rotation θi varies between 0 and 2π at intervals +of ∆θ = +� 2π +r +� +, i.e., θi = ∆θ, . . . , 2π, where r determines +without regression +with regression +regressed pseudo-labels +Fig. 3. +Regression on test-time augmented bounding boxes (middle) and +cluster modes (right) to generate pseudo-labels for SSL training. +Fig. 4. Probability of occupancy of passengers at one frame of our evaluation +datasets (Fig. 2). +the rotation resolution. At each rotation step, we compute +the detection set DC +θi(t) for both classes C ∈ {pax, bag} +using a single call to the function DPANet(·) (line 5). We +then remap the resulting detections to the coordinate frame +of the original image by applying the inverse rotation to each +of the detections in DC +θi(t) (line 6). To avoid localization +errors introduced by rotating axis-aligned bounding boxes, we +apply the rotation operation to the binary segmentation masks +produced by PANet and compute the corresponding bounding +boxes using the rotated masks. At the end of Alg. 1, the set +SC(t) = ∪r +i=1SC +θi(t) contains the detections at all the rotation +angles θi. Fig. 2 illustrates the detections at two rotation +angles and the result of mapping detections at 20 different +orientations back to the original coordinate system. +Algorithm 2 Cluster Regression +1: function CLUSTERREGRESSION(SC(t)) +2: +DC(t) = ∅ +3: +Refine the augmented detections using SC(t) as region +proposals for the DPANet model +4: +OC(t) = mean − shift(SC(t)) +5: +for Q ∈ OC(t) do +6: +Compute the cluster score ¯ηQ using Eq. 2 +7: +if ¯ηQ ≥ λ then +8: +d = argmax +di∈Q +(si) +9: +DC(t) = DC(t) ∪ {d} +10: +end if +11: +end for +12: +return DC(t) +13: end function +2) Cluster Regression: Alg. 2 summarizes our approach to +combine the set of augmented detections SC(t) into a set +of refined target detections DC(t). To reduce discrepancies +among bounding boxes caused by segmentation errors, we +leverage the pre-trained model to regress the set of augmented +detections SC(t). As shown in Fig. 1, our cluster regression +method uses the backbone features [72] with the augmented +detections SC(t) as region proposals (instead of proposals +generated using the region proposal network [73]) to the + +person 1.00 +bag 0.99 +bag 1.00 +person +04/16/2019 09:06:49:4571.01 +0.98 +C0.90 +0.99 +1.00 +04/16/2019 09:06:49:4570.99 +0.99 +1.00 +0.9B +1.00 +001 +04/16/2019 09:06:49:457ProbabilityDensity +400 +200 +0 +1 +1 +0.8 +0.6 +0.5 +0.4 +0.2 +y +0 +0SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS +5 +Algorithm 3 Pseudo-Label Generation +1: function PSEUDOLABELS(DC(t), r) +2: +PC(t) = ∅, Θ = {i · ∆θ}r +i=1 +3: +for dj ∈ DC(t) do +4: +for θi ∈ Θ do +5: +Generate the augmented region proposals +di,j = Rθi(dj) +6: +end for +7: +Generate the pseudo-label (ˆbi, ˆmi, αi) using the +region proposals di,j +8: +PC(t) = PC(t) ∪ +� +(ˆbi, ˆmi, αi) +� +9: +end for +10: +return PC(t) +11: end function +downstream box and mask heads. To avoid disregarding low- +confidence detections that might correspond to relevant region +proposals, we do not apply non-maximum suppression to +the model predictions. Fig. 3 shows that cluster regression +significantly increases the accuracy of the bounding boxes +generated using the augmented input, and the corresponding +segmentation masks are consequently also more accurate. +a) Cluster Mode Detection: As Fig. 4 illustrates, detec- +tions and their corresponding confidence scores form a non- +parametric distribution of the image’s occupancy probability. +We use the mean-shift algorithm [43] to identify the modes +of that distribution and cluster detections corresponding to +common targets. We cluster detections according to their +bounding boxes bj using a multivariate Gaussian kernel [43] +with bandwidth hC. We use the sample variances of the +object bounding boxes at each frame to determine the kernel +bandwidth, i.e., +hC = diag +� +� +nC +t +� +j=1 +(bC +j − ¯bC +j )(bC +j − ¯bC +j )T +� +� , +(1) +where ¯bC +j is the sample mean of bC +j and diag (·) is the diagonal +of the covariance matrix. The correlations among the elements +of bj are negligible and can be safely ignored. Each call to +the mean-shift algorithm (line 4 in Alg. 2) produces a set of +clusters OC(t) whose elements are sets of detections assigned +to the same target. We consider the detections of passengers +and baggage items separately. Hence, two separate invocations +of the mean-shift procedure are required to produce the sets +Opax(t) and Obag(t). The confidence score ¯ηQ of cluster +Q ∈ OC(t) is defined as the ratio between the total score +of detections within that cluster and the number of rotation +angles considered in the augmentation process, i.e., +¯ηQ = 1 +r +� +dj∈Q +sj. +(2) +Lines 6-10 of Alg. 2 show that we discard clusters with scores +lower than a threshold λ to remove false positive detections. +3) Self-Supervised Model Update: Alg. 3 shows the proce- +dure to generate the pseudo-labels used to update the model. +Since our goal is to train the model using labels generated from +multiple perspectives, we rotate both the original image and +the corresponding predicted modes to generate pseudo-label +proposals at each orientation. That is, for each mode dj ∈ +DC(t), we generate the pseudo-label mask ˆmj by using the +rotated cluster modes di,j = Rθi(dj), i = 1, . . . , r as region +proposals for the segmentation head, using the same approach +described in Section III-A2. We then find the bounding box +ˆbj corresponding to ˆmj. The confidence ˆαj of the resulting +pseudo-label is given by its corresponding cluster score. The +set of pseudo-labels PC(t) = +� +(ˆbj, ˆmj, ˆαj) | dj ∈ DC(t) +� +thus contains accurate annotations even for targets that the +model is unable to detect at certain orientations. +a) Rotation-Invariant Loss: To update the model using +rotation-invariant pseudo-labels in a robust and efficient man- +ner, we propose a novel uncertainty-aware, multi-task loss +function given by +L = +� +ˆc∈C +� +(ˆbj, ˆmj,ˆαj)∈PC(t) +ˆαj(Lc(ˆc, ˜c) + Lb(ˆbj,˜bj) ++ Lm( ˆmj, ˜mj)) + Lrpn, +(3) +where +˜c, ˜bj, and ˜mj are the object class, bounding box, +and segmentation mask predicted by the network; Lc, Lb, and +Lm are the classification and bounding box regression losses +defined in [73] and the pixel-wise binary cross entropy mask +loss described in [3]; and Lrpn is the region proposal network +loss from [73]. In Eq. 3, the instance head losses are weighted +by their corresponding cluster scores. This strategy ensures +that instances with low cluster scores that might correspond to +incorrect pseudo-labels have little impact on the update of the +network parameters. As Alg. 4 indicates, a new set of pseudo- +labels is generated at each SSL iteration using the updated +model from the previous iteration. +Algorithm 4 Self-Supervised Detection Model Update +Input: Image sequence I(t), t = 1, . . . , T +Output: Updated detection model DPANet +1: repeat +2: +for t = 1, . . . , T do +3: +SC(t) = AUGMENTEDPROPOSALS(I(t)) +4: +DC(t) = CLUSTERREGRESSION(SC(t)) +5: +PC(t) = PSEUDOLABELS(DC(t)) +6: +end for +7: +Fine-tune the DPANet model using the pseudo-labels +� +PC(t) +�T +t=1 according to the loss function in Eq. 3 +8: until Convergence criterion is met +B. Multi-View Passenger and Baggage Tracking +Our multi-camera tracking framework comprises two main +steps: i) a single-camera, multiple-target tracking-by-detection +algorithm, and ii) a multi-camera trajectory association mech- +anism. Our single-camera tracker uses the detections generated +by our SSL framework and a Single-Camera Trajectory Asso- +ciation (SCA) method to keep track of the identities of individ- +ual passengers and baggage items within the field of view of +each camera. Our MCTA strategy then projects the trajectories +of passengers observed in cameras with overlapping fields of + +6 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +view onto a common image plane. These trajectories are then +compared using the Fr´echet distance and associated using a +recursive graph-based mechanism. +1) Single-camera Tracking: We use the Tracktor algorithm +[8] as our baseline single-camera tracker. The output of the +algorithm at each image frame is a set T C(t) = {ω1, . . . , ωnC +t }, +where ωj = [bj, lj], with lj corresponding to a unique identi- +fier label for each passenger and baggage item in the frame. +These labels remain the same throughout the video sequence +and hence perform temporal association among detections. +The tracklet for the k-th object is thus given by the set of +detections over the entire video sequence whose temporal +identifier is lj = k, i.e., τk = ∪T +t=1 +� +ωj | ωj ∈ T C(t), lj = k +� +. +Temporary occlusions between passengers may lead to the +fragmentation of trajectories within the field of view of a +camera. Tracktor’s simple re-identification strategy is unable +to accommodate the longer occlusions, appearance variations, +and somewhat erratic motion patterns commonly observed in +airport checkpoints. Thus, we incorporate an SCA mechanism +to resolve this issue. Our method associates new tracklets with +recently terminated tracklets such that the Euclidean distance +between the centroids of the last detection of the previous +tracklet and the first detection of the new tracklet is minimized. +That is, let τm and τn be two distinct tracklets, and bti +m, +btf +n be the first detection of τm and the last detection of τn, +respectively. Defining δe = ||bti +m − btf +n ||2, the association cost +between τm and τn is given by +Csc(τm, τn) = +� +δe +if 0 < ti − tf ⩽ tth, δe < δmax +∞ +otherwise, +(4) +where tth, δmax are the maximum temporal offset and max- +imum distance to consider two tracklets for association. We +then compute the optimal tracklet assignment using the Hun- +garian algorithm based on the costs Csc(τm, τn). +2) Multi-Camera Tracklet Association: Since passengers +may temporarily leave and later re-enter the fields of view +of individual cameras, their corresponding trajectories may +be fragmented into multiple segments. To associate tracklets +across camera views, we consider the fact that two tracklets +corresponding to the same target include temporally over- +lapping detections. Let the camera whose partial tracklets +we wish to complete be our primary camera, and let the +auxiliary camera be the one whose tracklets will be used to +complement the tracklets observed by the primary camera. +Further, let Tp and Ta be the sets of tracklets in the primary +and auxiliary cameras, respectively. As Alg. 5 shows, we use +the homography Hp,a to project detections from the auxiliary +camera onto the primary camera. However, due to projective +distortions, the corresponding bounding boxes in the two +cameras may not necessarily overlap. Hence, we compute +the optimal association cost using the Fr´echet distance [74] +between the centroids of the detections in each tracklet as +follows +Cmc(τa, τp) = +� +f(˜τp, ˜τa) +if ˜τa ̸= ∅, ˜τp ̸= ∅, f < fmax +∞ +otherwise, +(5) +Algorithm 5 Multi-Camera Tracklet Association Algorithm +Input: Set of tracklets from the primary camera Tp and the +auxiliary camera Ta, homography Hp,a mapping the aux- +iliary camera image plane to that of the primary camera +Output: Updated set of primary tracklet labels +1: Project the detections of tracklets in Ta onto the image +plane of the primary camera using Hp,a +2: Compute the association costs Cmc(τa, τp) ∀τp ∈ Tp, +∀τa ∈ Ta according to Eq. 5 +3: Initialize the graph Gmc = (V, E), E = ∅, V = {τ|τ ∈ +Tp ∪ Ta} +4: while minτp∈Tp,τa∈Ta (Cmc(τa, τp)) < ∞ do +5: +Associate tracklet segments using the Hungarian algo- +rithm based on the costs Cmc +6: +Update the costs of the tracklets τ ∈ Ta and τ ′ ∈ Tp +for which τ ∩ τa /∈ ∅ and τ ′ ∩ τp /∈ ∅ to Cmc(τ, τp) = +Cmc(τa, τ ′) = ∞ +7: +E = E ∪ (τa, τp) +8: end while +9: for each τp ∈ Tp do +10: +Np = DFS(τp, Gmc) +11: +Update the labels of tracklets in Np using Eq. 6 +12: +E = E − {(τi, τj)|(τi, τj) ∈ Np} +13: end for +where ˜τp and ˜τa are the temporally overlapping segments of +tracklets τp ∈ Tp and τa ∈ Ta, f(˜τp, ˜τa) is the Fr´echet distance +of the centroids of the corresponding detections, and fmax is +the maximum distance threshold that allows tracklet pairs to +be considered for association. +We use the Hungarian algorithm again to determine optimal +tracklet associations according to the costs Cmc(τa, τp). How- +ever, since the trajectory of a passenger that re-enters the field +of view of a camera multiple times consists of a sequence of +tracklets, we iteratively update the association costs until no +further associations are possible. We keep track of indirectly +associated tracklets by constructing the reachability graph +Gmc = (V, E), which contains one edge for each pair of +associated tracklets. We then set the temporal identifiers of +all the tracklets in Tp associated with a common tracklet τa +to the first identifier among them. That is, the temporal label +of a tracklet τ is given by +lτ = +min +(τi,τj)∈Np (lτi), +(6) +where lτi is the temporal label of tracklet τi, and Np is the +set of tracklets that can be reached from tracklet τp on Gmc, +which we obtain through Depth-First Search (DFS). +IV. RESULTS AND DISCUSSION +In this section, we first discuss the datasets that we used to +evaluate our algorithms. We then present an assessment of the +proposed SSL approach in terms of passenger and baggage +detection, followed by an evaluation of the single-camera +tracking and multi-view tracklet association algorithms. Our +evaluation is based on the Multi-Object Detection (MOD) +and Tracking (MOT) metrics [59], [75]. Additional results are +presented in the Supplementary Materials. + +SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS +7 +Fig. 5. Document checking station and divestiture area at the Kostas Research +Institute simulated airport checkpoint. +A. Datasets +The video datasets used in this work were recorded at the +Kostas Research Institute (KRI) video analytics laboratory at +Northeastern University. As shown in Fig. 5, the laboratory +is configured to emulate a realistic airport checkpoint. It is +equipped with 14 standard IP surveillance cameras (Bosch +NDN-832-V03P) with 1920 × 1080 resolution and focal +lengths between 3 mm and 9 mm. The cameras are installed +approximately three meters from the floor with partially over- +lapping fields of view. Fig. 6 shows a panoramic perspective +of the fields of view of the cameras. +Several actors traverse the checkpoint with baggage items +while performing a variety of activities commonly observed +in real airports.1 These activities range from simple scenarios +in which just a few passengers pass through the checkpoint in +sequential order to crowded scenes in which multiple passen- +gers divest and retrieve their items in a more erratic manner. +We collected two separate video datasets: CLASP1, which +includes relatively simple scenarios, and CLASP2, which is +more complex. Fig. 7 shows sample frames of videos from +the two datasets. Of the 14 cameras in the laboratory, most +passenger interactions take place on cameras 9 and 11. Camera +9 monitors the divestiture area and camera 11 observes the +baggage retrieval area. Passengers place their belongings into +bins or directly on the conveyor belt in the divestiture area. +Then, after passing through the metal detector, they collect +their belongings in the baggage retrieval area. +As Table I shows, a total of 146 passengers carrying 126 +baggage items leave and re-enter the fields of view of the +cameras several times. We manually annotate the videos with +uniquely identified axis-aligned bounding boxes. Given the +1The datasets are available upon request at alert-coe@northeastern.edu. +Northeastern University’s Institutional Review Board (IRB) and the Com- +pliance Assurance Program Office (CAPO) within the DHS Science and +Technology Directorate have reviewed the referenced human subjects research +protocol and related research documentation. No compliance issues or con- +cerns related to the use of human subjects in this protocol have been identified +through the review, and the DHS policy requirements for human subjects +research protocol review has been met. +TABLE I +DATASETS USED TO EVALUATE OUR ALGORITHMS. FOR EACH VIDEO +SEQUENCE, THE TABLE SHOWS THE NUMBER OF PASSENGERS, BAGGAGE +ITEMS, VIDEO FRAMES, ANNOTATED FRAMES, AND THE TOTAL NUMBER +OF ANNOTATED BOUNDING BOXES. +Dataset +Video +Pass- +Bag- +Video +Annotated +Bounding +seq. +engers +gage +frames +frames/rate [fps] +boxes +CLASP1 +A +12 +10 +6,030 +288 (1) +995 +B +12 +10 +6,180 +564 (2) +1,720 +C +8 +9 +6,030 +491 (2) +853 +D +12 +8 +6,030 +523 (2) +1,197 +E +9 +9 +4,719 +1,648 (10) +4,254 +CLASP2 +F +20 +20 +12,910 +179 (0.01) +737 +G +38 +31 +10,390 +1,346 (3) +4,826 +H +35 +29 +11,200 +198 (0.01) +900 +Total +– +146 +126 +63,489 +5,237 +15,482 +large number of video frames available in the datasets, the an- +notation rate for the video sequences varies between 0.01 and +10 frames per second (fps). We randomly partition each dataset +into a training set containing 80% of the video frames and a +test set with the remaining 20%. For a fair comparison, the +Supervised Learning (SL) and SSL models are trained using +only the frames from the training set, but the SSL models are +fully self-supervised and do not use any manual annotations. +However, disregarding every video sequence that includes +annotated frames would substantially limit the amount of +data available for the computation of tracking performance +measures. Hence, to assess tracking performance, we consider +all the annotations listed in Table I. The only method that uses +the training set annotations is the SL approach. Although this +evaluation strategy favors that method, it also more accurately +reflects the generalization performance of the SSL approaches +to unseen data. Due to space limitations, the results presented +in this section were obtained using the aggregated CLASP1 +and CLASP2 test sets. Dataset-specific results are given in the +Supplementary Materials. +B. Self-Supervised Learning Detection Performance +During training, we freeze the network weights up to the +region proposal network layer so that the pre-trained backbone +features are effectively used in the downstream task. We use +an initial learning rate of 5e−3, mini-batch size per image +N += 256, r = 20 different orientations, and a cluster +confidence threshold λ = 0.1. Similar to the baseline model, +we use stochastic gradient descent with a momentum of 0.9, +weight decay of 1e−4. At each SSL iteration, we fine-tune the +model for 20k iterations, reducing the learning rate by a factor +of 10 at every 5k iterations. In our evaluation, we use an IoU +threshold of 0.5, and a non-maximum suppression threshold +ηnms = 0.3 for all the models. The detection threshold for +region proposal generation is ηdet = 0.5. +Fig. 8 shows the Multi-Object Detection Accuracy (MODA) +of our model as a function of the number of SSL iterations. To +illustrate the impact of the cluster confidence score, we also +evaluate a model in which the samples are not weighed by their +scores (SSL-wo-α). Instead, this model uses a hard threshold +λ ≤ 0.4 to discard noisy detections during training. The figure +also shows the performance of the Multiple-Inference (MI) + +8 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +Fig. 6. Panoramic overview of the camera views at the Kostas Research Institute simulated airport checkpoint. +Fig. 7. +Sample images from the datasets collected at the simulated airport +checkpoint (left: CLASP2 and right: CLASP1 in Table I). The images show +the divestiture area (right: camera 9) and item retrieval area (left: camera 11). +0 +2 +4 +6 +8 +Iteration +40 +60 +80 +100 +MODA +MI-wo- +MI +SSL-wo- +SSL +0 +2 +4 +6 +8 +Iteration +40 +60 +80 +100 +MODA +MI-wo- +MI +SSL-wo- +SSL +Fig. 8. MODA measures for person (left) and baggage (right) classes during +SSL training. +strategy used to generate the pseudo-labels, which reflects +the quality of the pseudo-labels before SSL training. That is, +in the MI model, the pseudo-labels themselves are used as +model predictions. As the figure indicates, the SSL models +gradually approach the performance of the MI strategy. The +incorporation of cluster confidences not only increases the +speed of convergence of the models but also leads to noticeable +performance gains, particularly for baggage items. +Fig. 9 shows the precision-recall curves for passenger and +baggage detection using four detector models: pre-trained +PANet (baseline), PANet trained using SL, SSL-wo-α, and +SSL. Even though the SSL models are trained without manual +annotations, they perform on par with the SL model for pas- +sengers. For baggage items, the maximum average precision +for the baseline model is less than half of the performance of +the SSL models. As illustrated in Fig. 14, the performance +0 +0.2 +0.4 +0.6 +0.8 +1 +Recall +0 +0.2 +0.4 +0.6 +0.8 +1 +Precision +Base (0.77) +SSL-wo- + (0.94) +SSL (0.95) +SL (0.95) +0 +0.2 +0.4 +0.6 +0.8 +1 +Recall +0 +0.2 +0.4 +0.6 +0.8 +1 +Precision +Base (0.35) +SSL-wo- + (0.74) +SSL (0.82) +SL (0.93) +Fig. 9. Precision-recall curves for person (left) and baggage (right) detection. +The legend shows the average precision of the models. +difference between the SL and SSL models is due to two +main issues: i) appearance similarities among bags and certain +garments/items placed inside security bins, and ii) baggage +items that can only be partially observed before being placed +on the conveyor belt. +Table VII demonstrates the benefits of incorporating clus- +ter uncertainties in the SSL loss function (column α) and +of the proposed cluster regression technique (column reg.). +The method that incorporates both cluster uncertainty and +regression is equivalent to the approach identified as SSL +in Figs. 8 and 9 whereas the method that does not include +cluster confidences corresponds to SSL-wo-α. The results +in the table correspond to the point that maximizes the F1 +score of the curves in Fig. 9 at the best performing SSL +iteration. The top-performing method in Table VII and in +the remainder of this section is highlighted in boldface, the +second-best is underlined, and ties are broken according to +the MODA/MOTA results. +In comparison with the baseline model, our SSL algorithm +substantially increases the recall (Rcll) and precision (Prcn) +for passenger detection, which is a result of improvements in +true positive (TP), false positive (FP), and false negative (FN) +detections. The cluster confidence scores substantially reduce +the contribution of low-confidence pseudo-labels, especially +for baggage items, leading to a noticeable increase in the +number of true positives. Cluster regression corrects pseudo- + +2019 19278440SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS +9 +Fig. 10. +Sample results showing failure cases for baggage detection using the SSL model in the CLASP2 dataset. The magenta arrows indicate bag-like +object detections that are not annotated (false positives), the red arrows indicate annotated baggage items the model fails to detect (false negatives), the green +bounding boxes show passenger detections, and the red bounding boxes represent the manual annotations for both classes. +TABLE II +PASSENGER AND BAGGAGE DETECTION EVALUATION. +Model +Method +↑Rcll +↑Prcn +↑TP +↓FP +↓FN +↑MODA +α +reg. +person +bag +person +bag +person +bag +person +bag +person +bag +person +bag +Baseline + + +73.8 +37.1 +87.0 +82.9 +1560 +426 +228 +85 +552 +724 +62.8 +29.3 +SSL + + +93.8 +73.8 +92.3 +82.5 +1989 +858 +155 +194 +123 +291 +86.0 +57.6 +SSL +✓ + +93.5 +75.9 +93.5 +86.1 +1985 +863 +134 +144 +127 +286 +87.1 +62.9 +SSL + +✓ +93.6 +73.1 +96.2 +92.5 +1985 +844 +73 +71 +127 +305 +90.1 +67.1 +SSL +✓ +✓ +95.7 +78.6 +96.0 +91.8 +2025 +903 +79 +83 +87 +246 +91.8 +71.5 +SL + + +95.6 +91.4 +96.4 +92.8 +2022 +1048 +70 +83 +90 +101 +92.1 +84.4 +(a) +(c) +(b) +Fig. 11. Qualitative detection results on the CLASP2 dataset using (a) Baseline, (b) SSL, and (c) SL models (the SL model only predicts bounding boxes). +label errors caused by inaccurate bounding boxes generated +from poor segmentation results. As a result, the reduction +in false positives for both classes is even more pronounced +when cluster regression is incorporated. Overall, our SSL +framework shows a relative MODA score improvement of 46% +for passengers and 144% for baggage items with respect to the +baseline model. +Fig. 11 shows qualitative results for the models under con- +sideration. In comparison with the SL model, the SSL models +not only improve the accuracy of the predicted bounding boxes +but also generate improved segmentation masks since they are +trained using instance segmentation pseudo-labels. +C. Single-Camera Tracking +We compare the performance of our single-camera tracking +algorithm using the proposed SSL detectors with the pre- +trained baseline detector and the SL detector. We also evaluate +the impact of our SCA algorithm, described in Section III-B1, +where we use tth = 3 seconds and δmax = 200 in Eq. 4. To +preserve the entirely self-supervised nature of our pipeline, +we refrain from fine-tuning the re-identification module of +the baseline tracker, which is pre-trained on the MOT17 [59] +dataset. To dissociate the evaluation of the tracking method +from our MCTA approach, we use a modified version of the +annotations in Table I where a passenger that re-enters the +field of view of a camera receives a new identifier. Thus, the +number of unique ground truth passenger identifiers (column +GT in Table III) is much higher than those listed in Table I. We +evaluate our system’s ability to maintain consistent passenger +identifiers across multiple perspectives in Section IV-D. +As Table III shows, the SSL-wo-α and SSL approaches out- +perform the tracker using the baseline detector by a large mar- +gin. The notable improvements in identity-based F1 (IDF1), +recall (IDR), and precision (IDP) [76] as well as in standard +recall and precision are primarily a result of the reduction in +false positives and false negatives generated by the SSL model. +Self-supervision also improves the tracking-specific metrics +of mostly tracked (MT), mostly lost (ML), identity switches +(IDs), and fragmented (FM) trajectories [75]. As a result, our +method produces substantial gains in MOTA. Again, both SSL +models perform on par with the SL model for person tracking. +For baggage items, we see similar performance improvements, +but the challenges illustrated in Fig. 14 again preclude the +SSL models from reaching the performance of the SL strategy. +Finally, our SCA algorithm leads to further performance gains, +particularly in terms of IDs. +D. Multi-Camera Tracklet Association +We evaluate the performance of our MCTA algorithm using +the same experimental procedure described in the previous +section, with the exception that passengers are now assigned +unique identifiers as they leave and re-enter the fields of + +backpack 0.89 +packpack 0.94 +suitcase 0.78 +suitcase 0.70 +person 0.98rson 0.65 +04/16/2019:09:03:46:476bag 1.00bag 1.00 +bag 1.00 +bag 1.00 +person 0.98 +person 0.91 +person 0.97 +person 1.00 +04/16/2019 09:03:46:476bag 1.00 +bag 1.00 +person 1.00 +person 1.00 +person 0.98 +person 1.00 +04/16/2019 09:03:46:47610 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +TABLE III +SINGLE-CAMERA TRACKING EVALUATION FOR PERSON AND BAGGAGE CLASSES. +Class +Model +α +SCA +GT +↑IDF1 +↑IDR +↑IDP +↑Rcll +↑Prcn +↓FP +↓FN +↑MT +↓ML +↓IDs +↓FM +↑MOTA +↑MOTP +Person +Baseline + + +391 +84.5 +83.3 +86.1 +91.2 +94.6 +750 +753 +283 +42 +93 +152 +84.0 +85.5 +SSL + + +391 +87.8 +87.2 +88.3 +95.1 +96.4 +350 +554 +319 +31 +80 +123 +90.1 +85.2 +SSL +✓ + +391 +88.4 +87.9 +88.9 +95.6 +96.6 +354 +438 +326 +27 +86 +122 +90.7 +85.2 +SSL +✓ +✓ +391 +88.5 +88.1 +88.9 +95.6 +96.5 +358 +435 +326 +26 +76 +123 +90.8 +85.2 +SL + + +391 +87.0 +86.4 +87.7 +95.2 +96.9 +357 +457 +332 +24 +86 +121 +90.5 +85.2 +Bag +Baseline + + +255 +67.5 +57.0 +86.4 +61.1 +92.3 +431 +1800 +108 +73 +31 +89 +54.3 +81.0 +SSL + + +255 +78.9 +74.2 +85.4 +81.0 +93.7 +308 +1014 +159 +38 +71 +105 +72.9 +80.4 +SSL +✓ + +255 +81.3 +78.2 +85.5 +84.4 +92.3 +401 +822 +169 +28 +68 +97 +75.1 +80.4 +SSL +✓ +✓ +255 +84.3 +81.1 +88.5 +84.4 +92.3 +401 +822 +169 +28 +48 +97 +75.6 +80.4 +SL + + +255 +85.3 +86.6 +84.1 +94.7 +91.7 +387 +339 +226 +10 +103 +69 +83.2 +80.0 +#1357 +C11 +(a) +(b) +C05 +C11 +#2232 +C11 +#0627 +C09 +#1652 +C09 +Fig. 12. Sample results showing (a) cross-camera passenger association between cameras 5 and 11 using MCTA, and (b) tracking and association between +passengers and baggage items where the top and bottom rows show image sequences from cameras 9 and 11 respectively. We associate passenger tracklets +in cameras 9 and 11 by leveraging the associations between cameras 2 and 5 (passengers flow in Fig. 6: C9→C2→C5→C11). Baggage items are associated +using temporally constrained distance-based matching when each item receives a unique identifier PiBj, representing the j-th item from the i-th passenger. +TABLE IV +MCTA EVALUATION. THE COLUMN LABELED DIST. INDICATES WHETHER +WE EMPLOY THE HAUSDORFF (dh) OR FR´ECHET (df ) DISTANCE TO +EVALUATE TRACKLET SIMILARITY. +Dist. SL SSL-wo-α SSL MCTA ↑IDF1 ↑IDR ↑IDP ↓IDs ↑MOTA +- + +✓ + + +82.1 +83.2 +81.0 +157 +88.2 + + +✓ + +82.0 +83.1 +80.9 +170 +88.5 +✓ + + + +81.5 +82.8 +80.4 +170 +88.0 +dh + +✓ + +✓ +87.4 +88.9 +86.4 +115 +88.8 + + +✓ +✓ +84.8 +88.6 +86.2 +140 +89.0 +✓ + + +✓ +86.2 +87.5 +85.0 +134 +88.6 +df + +✓ + +✓ +88.0 +89.3 +86.8 +115 +88.9 + + +✓ +✓ +88.2 +89.5 +87.0 +132 +89.1 +✓ + + +✓ +86.7 +88.1 +85.5 +122 +89.0 +view of the cameras. Based on the overall flow of passengers +through our simulated checkpoint, cameras 9 and 11 are the +primary cameras for our tracklet association method (Alg. 5). +Cameras 2 and 5, the cameras immediately below them in Fig. +6, are the respective auxiliary cameras. For a fair comparison +among the detectors, we generate tracklets in the auxiliary +cameras using the corresponding SL or SSL model used in +the primary cameras (i.e., trained using only frames from the +primary camera). To provide a set of reference performance +measures, we first evaluate our tracking algorithms in the ab- +sence of a MCTA mechanism. We then assess the performance +of our association method when tracklet similarity is computed +using the Fr´echet distance and the more traditional Hausdorff +distance [64] with fmax = 0.25 in Eq. 5. +Table VIII shows that tracklet association improves the +IDF1 measure by up to 6.2%. This is mainly a consequence +of the dramatic reduction in the number of identity switches. +Using the Fr´echet distance to determine tracklet similarity pro- +vides consistent performance improvements in all the metrics +under consideration, especially for the SSL strategy. The more +modest gains in MOTA (up to 1.0%) demonstrate the need +for measures that focus specifically on the impact of identity +switches on tracking performance. +Fig. 12(a) illustrates the tracklet association procedure be- +tween cameras 5 and 11. As the passengers with identities +P2 and P3, whose trajectories are represented in green and +yellow, move from the field of view of camera 5 to camera +11, their tracklets are projected from the former camera to +the latter. The projected trajectories (red for P2 and pink for +P3) are successfully associated with the tracklets from camera + +P1B1B2P1 transfers P1B1 +Pl carrying P1B1 +P2 carrying P2B2 +P2 enters into C9 +P1B1 +P1 enters into C9 +P2B2P3B3 +P2 +P2B2P4B4 left C9 +P2 left C9 +P4 transfers P4B4 +P4 carrying P4B4 +P3B3 +P3 transfers P3B3 +P4B4 +P3 carrying P3B3 +p4 +P4 enters into C9 +P3 enters into C9SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS +11 +11 based on the Fr´echet distances among their temporally +overlapping segments. In the instant shown in the figure, +passenger P2 is re-entering the field of view of camera 5, +and the corresponding tracklet is also correctly associated with +that passenger’s tracklet in camera 11. Hence, the passenger’s +identity is successfully handed off between the cameras. Fig. +12(b) demonstrates a potential application of the proposed +system. Baggage items are associated with passengers when +they are divested in camera 9 and their identifiers can be +verified at retrieval time, which is observed in camera 11. +V. CONCLUSION +We propose a multistage tracking-by-detection framework +to overcome performance limitations of object detection and +tracking algorithms in overhead camera videos for which +limited training data is available. Our framework is composed +of an SSL mechanism to fine-tune object detection models to +specific camera views without the need for manual annotations +and an MCTA method that only requires the homographies +among neighboring cameras. 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Stiefelhagen, “Evaluating multiple object tracking +performance: The CLEAR MOT metrics.” EURASIP J. on Image and +Video Processing, vol. 2008, 2008. +[76] E. Ristani, F. Solera, R. Zou, R. Cucchiara, and C. Tomasi, “Performance +measures and a data set for multi-target, multi-camera tracking,” in +European Conf. on Comp. Vis. Workshops, 2016. +[77] M. Xu, Z. Zhang et al., “End-to-end semi-supervised object detection +with soft teacher,” In IEEE/CVF Int. Conf. on Comp. Vis., 2021. + +Supplementary Materials: Tracking Passengers and Baggage Items using +Multiple Overhead Cameras at Security Checkpoints +Abubakar Siddique, Student Member, IEEE, Henry Medeiros, Senior Member, IEEE +Abstract—This document supplements our main paper with +additional experimental results on the CLASP1 and CLASP2 +datasets. We extend our Self-Supervised Learning (SSL) ap- +proach into a Semi-Supervised Learning (Semi-SL) mechanism +to further improve target detection performance, especially for +baggage items. We also investigate the impact of additional data +augmentation strategies, rotation resolution, and the computa- +tional requirements of our proposed technique. These additional +evaluation results show that our algorithm outperforms the +baseline as well as state-of-the-art supervised and semi-supervised +approaches. +Index Terms—Detection, Tracking, Association, Homography, +Tracklet, Multi-camera, Surveillance. +VI. SINGLE CAMERA DETECTIONS +This section presents a breakdown on the performance of +our SSL detector for individual cameras in the CLASP1 and +CLASP2 datasets. It also evaluates the performance impact of +additional data augmentation strategies, number of rotation an- +gles used for data augmentation, and incorporation of labeled +data in a semi-supervised approach. +A. Self-Supervised Learning +Fig. 13 shows a detailed breakdown of the performance of +our SSL detection model for individual camera views in the +CLASP1 and CLASP2 datasets. The high recall, precision, and +MODA values indicate that our SSL approach detects most +passengers correctly in these video sequences. The average +precision (AP) for passenger detection is slightly higher for +camera 11 in both datasets. The main factor contributing to +this performance difference is that in camera 9, passengers +are only partially visible most of the time, whereas camera +11 has a better view of the region where the passengers +stand next to the conveyor belt. On the other hand, this also +contributes to the lower baggage detection performance in +*Manuscript received August 22, 2022; accepted November 11, 2022. +Date of publication December 14, 2022. +†This material is based upon work supported by the U.S. Department of +Homeland Security, Science and Technology Directorate, Office of University +Programs, under Award Number 2013-ST-061-E0001-04. The views and +conclusions contained in this document are those of the authors and should not +be interpreted as necessarily representing the official policies, either expressed +or implied, of the U.S. Department of Homeland Security. +‡Abubakar +Siddique +is +with +the +Department +of +Electrical +and +Computer Engineering, Marquette University, Milwaukee, USA, e-mail: +abubakar.siddique@marquette.edu +§Henry Medeiros is with the Department of Agricultural and Bi- +ological Engineering, University of Florida, Gainesville, USA, e-mail: +hmedeiros@ufl.edu +¶Digital Object Identifier (DOI): 10.1109/TSMC.2022.3225252 +camera 11. That is, in camera 11, partially observed baggage +items being carried by passengers (see Fig. 14) are much +more common than in camera 9. As with passenger detection, +we observed similar baggage detection improvements in the +camera-specific performance comparisons. This performance +could be further improved by using additional unlabelled video +frames available in the CLASP1 and CLASP2 datasets to train +the SSL models. +B. Additional Data Augmentation Strategies +We investigate the impact of other data augmentation strate- +gies during SSL training, including color jittering and motion +blur along with multiple rotations. For color jittering, we +increase/decrease image brightness, contrast, saturation, and +hue by a factor sampled uniformly from the range [0, maxjit], +where maxjit is 0.4 for brightness, 0.5 for contrast, 0.2 for +saturation, and 0.05 for hue. To emulate motion blur, we use +Gaussian blur with kernel size uniformly sampled from the set +{5, . . . , 9} and standard deviation sampled from the interval +[0.1, 5]. We observe that applying color jittering and mo- +tion blur on the pseudo-label augmentation further improves +MODA scores by up to 2.9% and 4.8% for passengers and +baggage items, respectively. For a fair comparison, we reduced +the number of rotation angles used for augmentation such that +the total number of augmented images remains the same in +both scenarios. Maintaining the original number of rotations +would further increase performance gains. +TABLE V +PERFORMANCE IMPACT OF ADDITIONAL DATA AUGMENTATION +STRATEGIES IN THE SSL ITERATIONS. +Dataset +Method +↑AP +↑ F1 +↑MODA +Rot. C-Jit. Mot.-Blur person bag person bag person bag +CLASP1 +✓ + + +89.2 +43.4 +92.0 +59.7 +83.9 +41.6 +✓ +✓ +✓ +91.5 +48.3 +92.3 +64.5 +84.3 +46.4 +CLASP2 +✓ + + +79.4 +47.4 +86.2 +62.5 +73.6 +42.2 +✓ +✓ +✓ +84.2 +47.8 +88.0 +63.0 +76.5 +43.6 +C. Impact of Rotation Resolution +Table VI shows the impact of rotation resolution r on the +generation of pseudo-labels. One SSL iteration with r = 20 +improves the MODA scores by up to 3.1% for passengers and +5.6% for baggage items. The inference time for a single frame +increases linearly with the number of rotations, contributing +to longer SSL training iterations. If training time is a concern, +r = 10 offers a reasonable speed vs. performance trade-off. We +13 + +14 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +Fig. 13. Passenger and baggage detection performance in cameras 9 and 11 for the CLASP1 (CL1) and CLASP2 (CL2) datasets. Here, P +stands for passenger and B for baggage. A description of the methods under consideration is given in Section IV.B of the main paper. +Fig. 14. Additional illustrative failure cases for baggage detection using the SSL model in the CLASP1 dataset (see Fig. 10 in the main paper for failures +in the more challenging CLASP2 dataset). The magenta arrows indicate bag-like object detections that are not annotated (false positives), the red arrows +indicate annotated baggage items the model fails to detect (false negatives), the green bounding boxes show passenger detections, and the red bounding boxes +represent manual annotations for both classes. +use r = 20 for all the SSL models to demonstrate the potential +performance of our framework. As Table VI indicates, further +increasing the value of r would likely lead to minor additional +performance gains. +TABLE VI +PERFORMANCE IMPACT OF THE NUMBER OF ROTATION ANGLES USED IN +THE SSL ITERATIONS. +Dataset +r ↓Infer. Time +↑ F1 +↑MODA +(secs) +person bag person bag +CLASP1 +1 +0.3 +94.5 +69.5 +89.0 +51.7 +5 +2.2 +95.0 +70.4 +90.1 +52.8 +10 +4.5 +95.3 +70.8 +90.6 +53.6 +20 +9.1 +95.8 +70.8 +91.5 +53.7 +CLASP2 +1 +0.3 +91.0 +74.5 +82.3 +56.8 +5 +2.6 +92.1 +76.4 +84.6 +59.6 +10 +4.0 +92.1 +76.5 +84.5 +59.8 +20 +11.7 +92.2 +76.5 +84.9 +60.0 +D. Semi-Supervised Learning +As Table VII indicates, the performance of our SSL algo- +rithm is limited by the initial accuracy of the baseline model. +Thus, we extend our method to a semi-supervised approach +where we use a certain amount of manual annotations to +initialize our model before initiating SSL training. For the +labeled frames, we employ the same data augmentation pro- +cedure used to generate augmented labels. Fig. 15 shows that +training the SSL model using 10% of the manual labels leads +to a performance comparable to the SL model, outperforming +SoftTeacher [77], a state-of-the-art Semi-SL technique. Our +method is particularly effective when small amounts of anno- +tations are used. For example, using only 1% of the manual +labels, our Semi-SL approach outperforms SoftTeacher by +104% and is only 1.6% behind the SL method (Table VII) +for baggage items. Furthermore, we observe a 5.7% MODA +improvement over the SL method when we use all the manual +annotations during training. +VII. SINGLE-CAMERA TRACKING +Fig. 16 shows the Single-Camera Tracking (SCT) perfor- +mance of our algorithm for passengers and baggage items in +the individual cameras of the CLASP1 and CLASP2 datasets. + +100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wO-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +↑AP100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个 Precision100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个F1100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +↑ Recall100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +TMODA100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个MODPSIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS +15 +TABLE VII +PASSENGER AND BAGGAGE DETECTION EVALUATION MEASURES ON THE CLASP1 AND CLASP2 TEST SETS. +Dataset +Model +Method +↑Rcll +↑Prcn +↑TP +↓FP +↓FN +↑MODA +α +reg. +person +bag +person +bag +person +bag +person +bag +person +bag +person +bag +CLASP1 +Baseline + + +73.8 +36.2 +89.0 +87.9 +886 +233 +110 +32 +314 +411 +64.7 +31.2 +SSL + + +96.4 +80.4 +95.8 +78.6 +1157 +518 +51 +141 +43 +126 +92.2 +58.5 +SSL +✓ + +96.9 +70.4 +94.2 +90.9 +1163 +451 +71 +45 +37 +193 +91.0 +63.0 +SSL + +✓ +96.0 +76.1 +97.7 +90.7 +1152 +490 +27 +50 +48 +154 +93.8 +68.3 +SSL +✓ +✓ +96.8 +78.6 +97.3 +90.2 +1162 +506 +32 +55 +38 +138 +94.2 +70.0 +SL + + +96.7 +89.4 +98.1 +91.4 +1160 +576 +23 +54 +40 +68 +94.8 +81.1 +CLASP2 +Baseline + + +73.9 +38.0 +85.1 +78.0 +674 +192 +118 +53 +238 +313 +61.0 +27.5 +SSL + + +91.2 +67.3 +88.9 +86.5 +832 +340 +104 +53 +80 +165 +79.8 +56.8 +SSL +✓ + +90.1 +81.6 +92.9 +81.3 +822 +412 +63 +95 +90 +93 +83.2 +62.8 +SSL + +✓ +91.3 +70.1 +94.8 +94.4 +833 +354 +46 +21 +79 +151 +86.5 +65.9 +SSL +✓ +✓ +94.6 +78.6 +94.8 +93.4 +863 +397 +47 +28 +49 +108 +89.5 +73.1 +SL + + +94.5 +93.5 +94.8 +94.2 +862 +472 +47 +29 +50 +33 +89.4 +87.7 +Fig. 15. Semi-SL model performance on CLASP2 using a semi-supervised +extension of our proposed SSL method versus SoftTeacher (ST) [77]. Here, P +and B stand for the passenger and baggage categories. The SSL model uses +no labeled data and the SL model is trained with 100% of the samples. +For passenger tracking, the SSL methods outperform the SL +approach in terms of IDF1, IDP, and IDR in all the scenarios +under consideration. In both datasets, the SL approach shows +slightly higher MT results for camera 9, largely due to +the partial passenger detection problem. Since CLASP1 has +lower object density, we observe more consistent performance +among different methods for both cameras in that dataset. +While all the methods perform better on the CLASP1 dataset, +the benefits of SSL training compared to the baseline detector +are particularly evident in the MT results on the CLASP2 +dataset. +Regarding baggage items, although the SSL models lead to +a moderate increase in the number of IDs, these switches are +offset by substantial gains in MT. As a matter of fact, the SL +model shows a much more significant degradation in IDs for +the more complex CLASP2 dataset. This is particularly evident +for camera 9, and it explains the lower IDP obtained by the SL +method in that dataset. The most evident performance gains for +baggage tracking are observed in camera 11 on the CLASP2 +dataset because the of the difficulty of partially visible baggage +items using the baseline model. +VIII. MULTI-CAMERA TRACKLET ASSOCIATION +Regarding our Multi-camera Tracklet Association (MCTA) +method, Table VIII shows that the Fr´echet distance metric is +particularly useful in crowded scenarios. Although we obtain +comparable results using the Hausdorff distance on the easier +CLASP1 dataset, we achieve noticeable improvements in all +the evaluation criteria on CLASP2 using the Fr´echet distance. +The single-camera trackers in the auxiliary cameras are trained +using frames from the primary cameras. Hence, in crowded +scenarios they sometimes fails to keep alive trajectories of +targets that are temporarily outside the field of view of the +primary camera. This is the main reason behind the overall +lower tracking performance on the CLASP2 dataset. Training +camera-specific detectors using our SSL framework would +mitigate this issue. +TABLE VIII +MCTA EVALUATION. THE COLUMN LABELED DIST. INDICATES WHETHER +WE EMPLOY THE HAUSDORFF (dh) OR FR´ECHET (df ) DISTANCE TO +EVALUATE TRACKLET SIMILARITY. +Data. +Dist. SL SSL-wo-α SSL MCTA ↑IDF1 ↑IDR ↑IDP ↓IDs ↑MOTA +CLASP1 +- + +✓ + + +87.4 +87.8 86.9 +45 +94.4 +- + + +✓ + +87.0 +87.4 86.6 +48 +95.1 +- +✓ + + + +87.3 +87.5 87.2 +42 +94.4 +dh + +✓ + +✓ +94.0 +94.5 93.5 +21 +94.9 + + +✓ +✓ +92.7 +93.1 92.2 +30 +95.5 +✓ + + +✓ +93.0 +93.1 92.8 +25 +94.8 +df + +✓ + +✓ +93.8 +94.3 93.3 +22 +94.9 + + +✓ +✓ +92.8 +93.3 92.4 +27 +95.6 +✓ + + +✓ +93.4 +93.6 93.3 +23 +94.8 +CLASP2 +- + +✓ + + +76.9 +78.7 75.1 112 +82.0 +- + + +✓ + +77.0 +78.8 75.3 122 +81.9 +- +✓ + + + +75.8 +78.2 73.6 128 +81.6 +dh + +✓ + +✓ +81.2 +83.3 79.3 +94 +82.8 + + +✓ +✓ +82.1 +84.2 80.3 110 +82.5 +✓ + + +✓ +79.5 +82.0 77.2 109 +82.5 +df + +✓ + +✓ +82.3 +84.4 80.4 +93 +83.0 + + +✓ +✓ +83.6 +85.7 81.7 105 +82.7 +✓ + + +✓ +80.1 +82.7 77.8 +99 +83.2 +IX. COMPUTATIONAL COMPLEXITY +In this section, we analyze the theoretical computational +complexity of our SSL strategy and measure the computation +time and memory utilization of each step of our algorithm. All +our experiments were performed on a workstation equipped +with two RTX-2090Ti GPUs and an Intel® Xeon® Silver 4112 +CPU @2.6GHz. +A. Self-Supervised Learning +The computational complexity of our approach increases +linearly with the number of rotation angles used for augmenta- +tion in the pseudo-label generation step. That is, for a baseline +detection algorithm with computational complexity Θ(f(I(t)), + +100 +80 +60 +40 +P-SL +P-ST +P-SSL +P-Semi-SL +20 +B-SL +B-ST +B-SSL +B-Semi-SL +0 +1% +5% +10% +50% +100% +↑AP100 +80 +60 +40 +P-SL +P-ST +P-SSL +P-Semi-SL +20 +B-SL +B-ST +B-SSL +B-Semi-SL +0 +1% +5% +10% +50% +100% +TMODA16 +IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. X, NO. Y, DECEMBER 2022 +Fig. 16. Comparison of SCT performance of person and baggage classes in individual cameras of the CLASP1 and CLASP2 datasets using the Baseline, +SSL-wo-α, SSL, and SL detectors. +the complexity of our approach is Θ(r·f(I(t)), where r is the +number of rotation angles. For example, for r = 20, the run- +time is 20 times that of a single iteration without augmentation. +However, these operations are parallelizable as long as the +hardware resources support the simultaneous processing of +multiple frames. With our unoptimized implementation, the +total time to complete one SSL iteration is approximately +six hours for both model training and pseudo-label genera- +tion. However, we have observed that hardware resources are +severely underutilized, which indicates substantial room for +reduction in overall computation time. +B. Inference Performance +Table IX shows the computation time of the proposed +tracking-by-detection algorithm, employing a PANet detector +with a ResNet-50 backbone. The SCT uses the detector re- +sults and a ResNet-50-based Re-Identification (Re-ID) model +trained on MOT17 to re-label tracklets lost due to short-term +occlusions. Hence, the computation time and memory utiliza- +tion for the SCT are similar to those for the detector model. +Since we are processing single images individually instead +of image batches, the inference time for the detector and the +SCT are far from optimal. Preliminary experiments indicate +that processing batches of 10 images simultaneously leads to +an approximate six-fold reduction in detector inference time +without exceeding the memory capacity of the GPUs. Reusing +the backbone features from the detector in the Re-ID model +should also lead to a dramatic reduction in SCT time, since +The execution time of the proposed MCTA algorithm +depends on the average length of the overlapping tracklet +segments in each camera pair. In the CLASP1 dataset, which +contains fewer and shorter tracklets, the algorithm can be +executed in real time. In CLASP2, it can run at approximately +TABLE IX +COMPUTATION TIME OF THE PROPOSED TRACKING-BY-DETECTION +FRAMEWORK. +Data +Model +Infer. Time (ms) Memory (MB) +CLASP1 +Detector +333.3 +1,850 +SCT +142.8 +1,748 +MCTA +25.6 +9.1 +CLASP2 +Detector +333.3 +1,850 +SCT +166.6 +1,750 +MCTA +83.3 +22.7 +feature generation is the most computationally demanding +element of the tracking algorithm. +12 fps. However, the current implementation of the proposed +system uses the full life-span of a tracklet to compute the +Fr´echet association distance in the MCTA algorithm. It is pos- +sible to substantially reduce computation time by limiting the +length of single-camera tracklets compared by the algorithm. +Table X shows that if we limit the length of the tracklets to 240 +frames (or eight seconds), it is possible to achieve real-time +performance for both datasets without degrading the accuracy +of the algorithm. +TABLE X +COMPUTATION TIME OF THE PROPOSED MCTA. +Data. +Dist. Metric Max. Size Infer. Time (ms) Memory (MB) MOTA +CLASP1 +Hausdorff +– +0.52 +4.5 +94.5 +240 +0.50 +4.5 +94.5 +Fr´echet +– +25.6 +9.1 +94.6 +240 +8.20 +4.3 +94.6 +CLASP2 +Hausdorff +– +0.64 +16.3 +78.4 +240 +0.61 +16.3 +78.4 +Fr´echet +– +83.3 +22.7 +78.5 +240 +19.6 +16.3 +78.7 + +100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个 IDE1100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个 IDP100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wO-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个 IDR100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wO-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个 MT100 +P-baseline +B-baseline +P-SSL-wo-α +B-SSL-wo-α +80 +P-SSL +B-SSL +P-SL +B-SL +60 +40 +20 +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +↓ IDs100 +80 +60 +40 +P-baseline +B-baseline +20 +P-SSL-wo-α +B-SSL-wo-α +P-SSL +B-SSL +P-SL +B-SL +0 +9:CL1 +11:CL1 +9:CL2 +11:CL2 +个 MOTA \ No newline at end of file diff --git a/iNAyT4oBgHgl3EQfXvei/content/tmp_files/load_file.txt b/iNAyT4oBgHgl3EQfXvei/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40876ad4aa7f246610191d58fb6cb8931e83e5e3 --- /dev/null +++ b/iNAyT4oBgHgl3EQfXvei/content/tmp_files/load_file.txt @@ -0,0 +1,1960 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf,len=1959 +page_content='Tracking Passengers and Baggage Items using Multiple Overhead Cameras at Security Checkpoints Abubakar Siddique, Student Member, IEEE, Henry Medeiros, Senior Member, IEEE*†‡§¶ Abstract—We introduce a novel framework to track multiple objects in overhead camera videos for airport checkpoint security scenarios where targets correspond to passengers and their baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We propose a Self-Supervised Learning (SSL) technique to provide the model information about instance segmentation uncertainty from overhead images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our SSL ap- proach improves object detection by employing a test-time data augmentation and a regression-based, rotation-invariant pseudo- label refinement technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our pseudo-label generation method provides multiple geometrically-transformed images as inputs to a Convolutional Neural Network (CNN), regresses the augmented detections generated by the network to reduce localization errors, and then clusters them using the mean-shift algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The self- supervised detector model is used in a single-camera tracking algorithm to generate temporal identifiers for the targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our method also incorporates a multi-view trajectory association mechanism to maintain consistent temporal identifiers as pas- sengers travel across camera views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' An evaluation of detection, tracking, and association performances on videos obtained from multiple overhead cameras in a realistic airport checkpoint environment demonstrates the effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our results show that self-supervision improves object detection accuracy by up to 42% without increasing the inference time of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our multi-camera association method achieves up to 89% multi-object tracking accuracy with an average computation time of less than 15 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Index Terms—Self-supervised Learning, Detection, Tracking, Tracklet Association, Multi-camera Tracking, Surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' INTRODUCTION A UTOMATED video surveillance requires the detection, tracking, and recognition of objects of interest in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Accurate and precise surveillance in crowded scenes is one of the most challenging computer vision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To address the problem of visual surveillance in the domain of airport checkpoint security, the Department of Homeland Security Manuscript received August 22, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' accepted November 11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Date of publication December 14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' †This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Award Number 2013-ST-061-E0001-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Department of Homeland Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ‡Abubakar Siddique is with the Department of Electrical and Computer Engineering, Marquette University, Milwaukee, USA, e-mail: abubakar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='siddique@marquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='edu §Henry Medeiros is with the Department of Agricultural and Bi- ological Engineering, University of Florida, Gainesville, USA, e-mail: hmedeiros@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='edu ¶Digital Object Identifier (DOI): 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1109/TSMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3225252 (DHS) ALERT (Awareness and Localization of Explosives- Related Threats) center of excellence at Northeastern Univer- sity initiated the CLASP (Correlating Luggage and Specific Passengers) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This initiative aims to help the Transporta- tion Security Administration (TSA) detect security incidents, such as theft of items and abandoned bags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Current approaches for detecting and tracking passengers and luggage in airport checkpoints divide the image area within each camera’s field of view into regions of inter- est where certain passenger behaviors are expected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', passengers divest their items near the roller conveyor) [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' While these approaches are effective within individual regions of interest, they cannot detect and track passengers and their belongings throughout an entire checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Moreover, most recent detection algorithms [3]–[6] are unable to detect multiple objects in realistic overhead camera scenarios due to the unavailability of large-scale datasets obtained using unconventional camera perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fine-tuning pre-trained models using human annotated la- bels is a common approach in computer vision methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, this strategy hinders the applicability of state-of- the-art algorithms in scenarios where images are obtained from perspectives that are not commonly observed in existing publicly available datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The dramatic variability of video surveillance systems used in airport checkpoints would require deployment-specific fine-tuning of models, and in some sce- narios, even camera-specific adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To overcome this challenge, we leverage the fact that models pre-trained on large-scale datasets can build upon their initial predictions to adapt to new scenarios using SSL strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our proposed SSL framework obviates the tedious and expensive human annotation procedure by automatically generating pseudo- labels to update the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To generate pseudo-labels, we cluster multiple detections obtained from geometrically transformed images using the mean-shift algorithm [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Each cluster corresponds to the detection of one object observed at different orientations on several augmented input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The cluster modes with the corresponding bounding boxes, segmentation masks, and confidence scores are used to update the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Thus, our model learns from rotation-invariant pseudo-labels and can be integrated with a tracking-by-detection algorithm [8] to generate accurate target tracklets from overhead perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our SSL algorithm is inspired by the methods described in [9]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, unlike [9], instead of resorting to multi-task 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00190v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='CV] 31 Dec 2022 2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Proposed SSL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The augmented proposal generation stage uses multiple rotated versions of the unlabeled input images to generate augmented detections from an instance segmentation model and then remaps these predictions into their original coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The clustering algorithm leverages the model’s regression ability to reduce localization errors using the augmented predictions as region proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The regressed cluster modes are then used to generate augmented pseudo-labels to update the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' strategies to guide the learning process, we employ a multi- inference approach similar in spirit to the self-consistency method based on equivariant transformations proposed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our method differs from [10] in that, rather than using the uncertainties from multiple model predictions to select image patches for additional training, it aggregates multiple inferences into accurate pseudo-labels that are used to refine the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our method departs significantly from unsupervised model adaptation [11] and knowledge distillation approaches [14] in that we only use automatically generated labels and avoid human annotations altogether during model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We also propose a Multi-Camera Tracklet Association (MCTA) algorithm to maintain the temporal identifiers of pas- sengers across cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We leverage the fact that our system is comprised of overhead cameras with partially overlapping fields of view to employ a simple but effective geometry- based trajectory association method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our algorithm compares the projected centroids of target detections on neighboring cameras using the homographies between their image planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We track passengers and bags across multiple views and generate global tracks by combining pairwise associations from the partially overlapping camera views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We evaluate the detection and tracking performance of our algorithms on videos from a simulated airport checkpoint and demonstrate that our approach performs on par with a model trained in an entirely supervised manner and substantially outperforms the pre-trained detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our multi-camera evaluation shows that our MCTA method effectively handles the problem of passenger identity hand-off across cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In summary, the key contributions of this work are: A novel self-supervised object detection algorithm that generates pseudo-labels based on instance segmentation uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A new data augmentation and regression-based clustering mechanism that substantially improves the quality of pseudo-labels for self-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A new recursive tracklet association algorithm to address the identity hand-off issue during transitions between crowded overhead camera views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We provide an extensive evaluation of our methods on a dataset collected using multiple overhead cameras in a realistic airport checkpoint scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our SSL models and the corresponding source code are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='com/siddiquemu/SCT MCTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To our knowledge, this is the first approach to solve the overhead multi-view association problem in a network of cameras with partially overlapping fields of view using a self- supervised detection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' RELATED WORK Multiple target tracking using camera networks is an active research topic with several potential applications [15]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Most works on camera networks focus on the multi-camera aspect of the problem and do not consider the challenges associated with camera perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Although generic object tracking algorithms could be used in surveillance systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' [19]–[21]), when object categories are known, trackers based on specialized detectors are more accurate and less prone to model drift [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This observation has led to the development of a variety of multiple target tracking algorithms that specialize in tracking humans [24]–[34] or vehicles [35]– [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, in many scenarios, it is desirable to track additional objects of known categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In these cases, more flexible detection algorithms are needed, but the effectiveness of modern object detection models is highly dependent upon the characteristics of the training datasets [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Previous works have used SSL techniques to improve visual feature learning [39]–[41], reducing dependency on human an- notations for training backbone models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, transferring knowledge from pre-trained backbones to downstream tasks is a far less explored topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Unlike our proposed approach, SSL techniques for detection [9], [11] and semantic segmentation [10] rely on annotations to initialize the model before iterative learning can take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Data augmentation is an effective mechanism to improve the robustness of CNNs in scenarios not available during 亚(t) DS(t) R Instance Prediction Segmentation Remapping Model 1 ROI Backbone HeadSIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Visualization of our data augmentation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The first and second columns show the segmentation masks and detections at θ = 0◦ and θ = 186◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The third column shows the remapped detections in the set SC on the original image (using Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1) with the best detections (blue) from Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' training [14], [42], but little attention has been given so far to approaches for combining the response of the network to augmented samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In multi-target tracking applications, multiple detections mapped to a common coordinate system can be interpreted as the probability of occupancy of the area observed by the cameras [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Although it is possible to use clustering techniques to map the modes of this distribution to unique target detections, bounding box alignment errors pose a challenge to the generation of high-quality pseudo-labels for SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence, we propose a test-time regression technique that leverages instance segmentation information for pseudo-label generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A systematic solution to the data association problem is another important component of multi-target tracking-by- detection methods [44]–[57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Single-camera trackers [8], [58] use detectors trained on multiple datasets [59] to generate bounding boxes and form track hypotheses for all the targets in each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In this work, we employ a state-of-the-art single-camera tracker [8] using a detector based on our self- supervised models, which achieves unprecedented tracking performance in previously unseen airport surveillance videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Finally, multi-camera tracking systems require sophisticated trajectory association mechanisms to maintain target identities across cameras [60]–[62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Even within a single camera, occlu- sions must be handled using similar strategies [63], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Most association approaches compute trajectory similarity scores based on a combination of appearance and motion features [60]–[64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' These features are learned using a large number of continuous trajectories, which are difficult to obtain with typi- cal ceiling-height overhead cameras due to their limited fields of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Some methods use camera calibration information to project tracks onto a common plane and perform association using occlusion modeling [32] or re-identification techniques [33]–[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dependency on camera calibration further limits the applicability of these methods to security systems since calibrating multiple cameras with partially overlapping fields of view is a complex task [65]–[68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' PROPOSED MODEL Our system consists of two main components: i) a detection algorithm trained using SSL and ii) a multi-camera tracking- by-detection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A single-camera tracking algorithm uses SSL detections to generate tracklets for passengers and baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then employ a novel multi-camera target trajectory association algorithm to uniquely identify passen- gers throughout the checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Self-Supervised Learning We use the PANet model [5] with a ResNet-50 backbone [69], [70] as the baseline detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Since the categories of interest are persons and their belongings, we use a model pre-trained on the COCO dataset [71], which includes object classes related to these categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', person, handbag, back- pack, and suitcase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Because the COCO dataset consists mostly of images captured at roughly eye-level, detectors trained using that dataset do not perform well on overhead perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 4 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 To address this limitation, our SSL framework updates the baseline model using rotation-invariant pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1 shows, our SSL framework consists of three main steps: i) augmented region proposals generation, ii) pseudo-label generation and refinement through cluster regression, and iii) iterative model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Algorithm 1 Augmented Proposals Generation 1: function AUGMENTEDPROPOSALS(I(t), r) 2: SC(t) = ∅, Θ = {i · ∆θ}r i=1 3: for θi ∈ Θ do 4: Ψθi(t) = Rθi(I(t)) 5: DC θi(t) = DPANet(Ψθi(t)) 6: SC θi(t) = R−θi(DC θi(t)) 7: SC(t) = SC(t) ∪ SC θi(t) 8: end for 9: return SC(t) 10: end function 1) Augmented Proposals Generation: Our data augmenta- tion method, summarized in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1, uses the PANet model to detect and segment multiple instances of objects of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' During the first iteration of SSL training, we retain only the outputs of the pre-trained model for the person, handbag, backpack, and suitcase classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The person class corresponds to passengers and detections of handbag, back- pack, and suitcase items are treated as baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In subsequent iterations of SSL training, we modify the model to generate only the object categories C ∈ {pax, bag}, where pax corresponds to passengers and bag to baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Let DC(t) be the set of detections on image I(t) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, DC(t) = {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , dnC t }, where dj ∈ R5 is the detection of the j-th object and nC t is the number of objects of class C in frame I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Each detection dj consists of the coordinates and dimensions of the target’s bounding box, bC j ∈ R4, as well as its detection confidence score sj ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We noticed that the detector performs better when objects are observed at more commonly occurring angles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', up- right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Therefore, to reduce the negative effect of the overhead perspective, we generate multiple rotated copies of the input image Ψθi(t) = Rθi(I(t)) (line 4 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1), where Rθi(·) is the rotation operator, which rotates the image by an angle θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The angle of rotation θi varies between 0 and 2π at intervals of ∆θ = � 2π r � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', θi = ∆θ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , 2π, where r determines without regression with regression regressed pseudo-labels Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Regression on test-time augmented bounding boxes (middle) and cluster modes (right) to generate pseudo-labels for SSL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Probability of occupancy of passengers at one frame of our evaluation datasets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' the rotation resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' At each rotation step, we compute the detection set DC θi(t) for both classes C ∈ {pax, bag} using a single call to the function DPANet(·) (line 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then remap the resulting detections to the coordinate frame of the original image by applying the inverse rotation to each of the detections in DC θi(t) (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To avoid localization errors introduced by rotating axis-aligned bounding boxes, we apply the rotation operation to the binary segmentation masks produced by PANet and compute the corresponding bounding boxes using the rotated masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' At the end of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1, the set SC(t) = ∪r i=1SC θi(t) contains the detections at all the rotation angles θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2 illustrates the detections at two rotation angles and the result of mapping detections at 20 different orientations back to the original coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Algorithm 2 Cluster Regression 1: function CLUSTERREGRESSION(SC(t)) 2: DC(t) = ∅ 3: Refine the augmented detections using SC(t) as region proposals for the DPANet model 4: OC(t) = mean − shift(SC(t)) 5: for Q ∈ OC(t) do 6: Compute the cluster score ¯ηQ using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2 7: if ¯ηQ ≥ λ then 8: d = argmax di∈Q (si) 9: DC(t) = DC(t) ∪ {d} 10: end if 11: end for 12: return DC(t) 13: end function 2) Cluster Regression: Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2 summarizes our approach to combine the set of augmented detections SC(t) into a set of refined target detections DC(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To reduce discrepancies among bounding boxes caused by segmentation errors, we leverage the pre-trained model to regress the set of augmented detections SC(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1, our cluster regression method uses the backbone features [72] with the augmented detections SC(t) as region proposals (instead of proposals generated using the region proposal network [73]) to the person 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 bag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='99 bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 person 04/16/2019 09:06:49:4571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='01 0.' metadata={'source': 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+page_content='2 y 0 0SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS 5 Algorithm 3 Pseudo-Label Generation 1: function PSEUDOLABELS(DC(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' r) 2: PC(t) = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Θ = {i · ∆θ}r i=1 3: for dj ∈ DC(t) do 4: for θi ∈ Θ do 5: Generate the augmented region proposals di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='j = Rθi(dj) 6: end for 7: Generate the pseudo-label (ˆbi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ˆmi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' αi) using the region proposals di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='j 8: PC(t) = PC(t) ∪ � (ˆbi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ˆmi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' αi) � 9: end for 10: return PC(t) 11: end function downstream box and mask heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To avoid disregarding low- confidence detections that might correspond to relevant region proposals, we do not apply non-maximum suppression to the model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 3 shows that cluster regression significantly increases the accuracy of the bounding boxes generated using the augmented input, and the corresponding segmentation masks are consequently also more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' a) Cluster Mode Detection: As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 4 illustrates, detec- tions and their corresponding confidence scores form a non- parametric distribution of the image’s occupancy probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We use the mean-shift algorithm [43] to identify the modes of that distribution and cluster detections corresponding to common targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We cluster detections according to their bounding boxes bj using a multivariate Gaussian kernel [43] with bandwidth hC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We use the sample variances of the object bounding boxes at each frame to determine the kernel bandwidth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', hC = diag � � nC t � j=1 (bC j − ¯bC j )(bC j − ¯bC j )T � � , (1) where ¯bC j is the sample mean of bC j and diag (·) is the diagonal of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The correlations among the elements of bj are negligible and can be safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Each call to the mean-shift algorithm (line 4 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2) produces a set of clusters OC(t) whose elements are sets of detections assigned to the same target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We consider the detections of passengers and baggage items separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence, two separate invocations of the mean-shift procedure are required to produce the sets Opax(t) and Obag(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The confidence score ¯ηQ of cluster Q ∈ OC(t) is defined as the ratio between the total score of detections within that cluster and the number of rotation angles considered in the augmentation process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', ¯ηQ = 1 r � dj∈Q sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' (2) Lines 6-10 of Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2 show that we discard clusters with scores lower than a threshold λ to remove false positive detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 3) Self-Supervised Model Update: Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 3 shows the proce- dure to generate the pseudo-labels used to update the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Since our goal is to train the model using labels generated from multiple perspectives, we rotate both the original image and the corresponding predicted modes to generate pseudo-label proposals at each orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, for each mode dj ∈ DC(t), we generate the pseudo-label mask ˆmj by using the rotated cluster modes di,j = Rθi(dj), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , r as region proposals for the segmentation head, using the same approach described in Section III-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then find the bounding box ˆbj corresponding to ˆmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The confidence ˆαj of the resulting pseudo-label is given by its corresponding cluster score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The set of pseudo-labels PC(t) = � (ˆbj, ˆmj, ˆαj) | dj ∈ DC(t) � thus contains accurate annotations even for targets that the model is unable to detect at certain orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' a) Rotation-Invariant Loss: To update the model using rotation-invariant pseudo-labels in a robust and efficient man- ner, we propose a novel uncertainty-aware, multi-task loss function given by L = � ˆc∈C � (ˆbj, ˆmj,ˆαj)∈PC(t) ˆαj(Lc(ˆc, ˜c) + Lb(ˆbj,˜bj) + Lm( ˆmj, ˜mj)) + Lrpn, (3) where ˜c, ˜bj, and ˜mj are the object class, bounding box, and segmentation mask predicted by the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Lc, Lb, and Lm are the classification and bounding box regression losses defined in [73] and the pixel-wise binary cross entropy mask loss described in [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' and Lrpn is the region proposal network loss from [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 3, the instance head losses are weighted by their corresponding cluster scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This strategy ensures that instances with low cluster scores that might correspond to incorrect pseudo-labels have little impact on the update of the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 4 indicates, a new set of pseudo- labels is generated at each SSL iteration using the updated model from the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Algorithm 4 Self-Supervised Detection Model Update Input: Image sequence I(t), t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , T Output: Updated detection model DPANet 1: repeat 2: for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , T do 3: SC(t) = AUGMENTEDPROPOSALS(I(t)) 4: DC(t) = CLUSTERREGRESSION(SC(t)) 5: PC(t) = PSEUDOLABELS(DC(t)) 6: end for 7: Fine-tune the DPANet model using the pseudo-labels � PC(t) �T t=1 according to the loss function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 3 8: until Convergence criterion is met B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Multi-View Passenger and Baggage Tracking Our multi-camera tracking framework comprises two main steps: i) a single-camera, multiple-target tracking-by-detection algorithm, and ii) a multi-camera trajectory association mech- anism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our single-camera tracker uses the detections generated by our SSL framework and a Single-Camera Trajectory Asso- ciation (SCA) method to keep track of the identities of individ- ual passengers and baggage items within the field of view of each camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our MCTA strategy then projects the trajectories of passengers observed in cameras with overlapping fields of 6 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 view onto a common image plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' These trajectories are then compared using the Fr´echet distance and associated using a recursive graph-based mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 1) Single-camera Tracking: We use the Tracktor algorithm [8] as our baseline single-camera tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The output of the algorithm at each image frame is a set T C(t) = {ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , ωnC t }, where ωj = [bj, lj], with lj corresponding to a unique identi- fier label for each passenger and baggage item in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' These labels remain the same throughout the video sequence and hence perform temporal association among detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The tracklet for the k-th object is thus given by the set of detections over the entire video sequence whose temporal identifier is lj = k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', τk = ∪T t=1 � ωj | ωj ∈ T C(t), lj = k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Temporary occlusions between passengers may lead to the fragmentation of trajectories within the field of view of a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Tracktor’s simple re-identification strategy is unable to accommodate the longer occlusions, appearance variations, and somewhat erratic motion patterns commonly observed in airport checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Thus, we incorporate an SCA mechanism to resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our method associates new tracklets with recently terminated tracklets such that the Euclidean distance between the centroids of the last detection of the previous tracklet and the first detection of the new tracklet is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, let τm and τn be two distinct tracklets, and bti m, btf n be the first detection of τm and the last detection of τn, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Defining δe = ||bti m − btf n ||2, the association cost between τm and τn is given by Csc(τm, τn) = � δe if 0 < ti − tf ⩽ tth, δe < δmax ∞ otherwise, (4) where tth, δmax are the maximum temporal offset and max- imum distance to consider two tracklets for association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then compute the optimal tracklet assignment using the Hun- garian algorithm based on the costs Csc(τm, τn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 2) Multi-Camera Tracklet Association: Since passengers may temporarily leave and later re-enter the fields of view of individual cameras, their corresponding trajectories may be fragmented into multiple segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To associate tracklets across camera views, we consider the fact that two tracklets corresponding to the same target include temporally over- lapping detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Let the camera whose partial tracklets we wish to complete be our primary camera, and let the auxiliary camera be the one whose tracklets will be used to complement the tracklets observed by the primary camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Further, let Tp and Ta be the sets of tracklets in the primary and auxiliary cameras, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 5 shows, we use the homography Hp,a to project detections from the auxiliary camera onto the primary camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, due to projective distortions, the corresponding bounding boxes in the two cameras may not necessarily overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' we compute the optimal association cost using the Fr´echet distance [74] between the centroids of the detections in each tracklet as follows Cmc(τa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' τp) = � f(˜τp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ˜τa) if ˜τa ̸= ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ˜τp ̸= ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' f < fmax ∞ otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' (5) Algorithm 5 Multi-Camera Tracklet Association Algorithm Input: Set of tracklets from the primary camera Tp and the auxiliary camera Ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' homography Hp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='a mapping the aux- iliary camera image plane to that of the primary camera Output: Updated set of primary tracklet labels 1: Project the detections of tracklets in Ta onto the image plane of the primary camera using Hp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='a 2: Compute the association costs Cmc(τa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' τp) ∀τp ∈ Tp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ∀τa ∈ Ta according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 5 3: Initialize the graph Gmc = (V, E), E = ∅, V = {τ|τ ∈ Tp ∪ Ta} 4: while minτp∈Tp,τa∈Ta (Cmc(τa, τp)) < ∞ do 5: Associate tracklet segments using the Hungarian algo- rithm based on the costs Cmc 6: Update the costs of the tracklets τ ∈ Ta and τ ′ ∈ Tp for which τ ∩ τa /∈ ∅ and τ ′ ∩ τp /∈ ∅ to Cmc(τ, τp) = Cmc(τa, τ ′) = ∞ 7: E = E ∪ (τa, τp) 8: end while 9: for each τp ∈ Tp do 10: Np = DFS(τp, Gmc) 11: Update the labels of tracklets in Np using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 6 12: E = E − {(τi, τj)|(τi, τj) ∈ Np} 13: end for where ˜τp and ˜τa are the temporally overlapping segments of tracklets τp ∈ Tp and τa ∈ Ta, f(˜τp, ˜τa) is the Fr´echet distance of the centroids of the corresponding detections, and fmax is the maximum distance threshold that allows tracklet pairs to be considered for association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We use the Hungarian algorithm again to determine optimal tracklet associations according to the costs Cmc(τa, τp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' How- ever, since the trajectory of a passenger that re-enters the field of view of a camera multiple times consists of a sequence of tracklets, we iteratively update the association costs until no further associations are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We keep track of indirectly associated tracklets by constructing the reachability graph Gmc = (V, E), which contains one edge for each pair of associated tracklets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then set the temporal identifiers of all the tracklets in Tp associated with a common tracklet τa to the first identifier among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, the temporal label of a tracklet τ is given by lτ = min (τi,τj)∈Np (lτi), (6) where lτi is the temporal label of tracklet τi, and Np is the set of tracklets that can be reached from tracklet τp on Gmc, which we obtain through Depth-First Search (DFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' RESULTS AND DISCUSSION In this section, we first discuss the datasets that we used to evaluate our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then present an assessment of the proposed SSL approach in terms of passenger and baggage detection, followed by an evaluation of the single-camera tracking and multi-view tracklet association algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our evaluation is based on the Multi-Object Detection (MOD) and Tracking (MOT) metrics [59], [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Additional results are presented in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Document checking station and divestiture area at the Kostas Research Institute simulated airport checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Datasets The video datasets used in this work were recorded at the Kostas Research Institute (KRI) video analytics laboratory at Northeastern University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 5, the laboratory is configured to emulate a realistic airport checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' It is equipped with 14 standard IP surveillance cameras (Bosch NDN-832-V03P) with 1920 × 1080 resolution and focal lengths between 3 mm and 9 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The cameras are installed approximately three meters from the floor with partially over- lapping fields of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 6 shows a panoramic perspective of the fields of view of the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Several actors traverse the checkpoint with baggage items while performing a variety of activities commonly observed in real airports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 These activities range from simple scenarios in which just a few passengers pass through the checkpoint in sequential order to crowded scenes in which multiple passen- gers divest and retrieve their items in a more erratic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We collected two separate video datasets: CLASP1, which includes relatively simple scenarios, and CLASP2, which is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 7 shows sample frames of videos from the two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Of the 14 cameras in the laboratory, most passenger interactions take place on cameras 9 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Camera 9 monitors the divestiture area and camera 11 observes the baggage retrieval area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Passengers place their belongings into bins or directly on the conveyor belt in the divestiture area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Then, after passing through the metal detector, they collect their belongings in the baggage retrieval area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As Table I shows, a total of 146 passengers carrying 126 baggage items leave and re-enter the fields of view of the cameras several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We manually annotate the videos with uniquely identified axis-aligned bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Given the 1The datasets are available upon request at alert-coe@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Northeastern University’s Institutional Review Board (IRB) and the Com- pliance Assurance Program Office (CAPO) within the DHS Science and Technology Directorate have reviewed the referenced human subjects research protocol and related research documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' No compliance issues or con- cerns related to the use of human subjects in this protocol have been identified through the review, and the DHS policy requirements for human subjects research protocol review has been met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE I DATASETS USED TO EVALUATE OUR ALGORITHMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' FOR EACH VIDEO SEQUENCE, THE TABLE SHOWS THE NUMBER OF PASSENGERS, BAGGAGE ITEMS, VIDEO FRAMES, ANNOTATED FRAMES, AND THE TOTAL NUMBER OF ANNOTATED BOUNDING BOXES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dataset Video Pass- Bag- Video Annotated Bounding seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' engers gage frames frames/rate [fps] boxes CLASP1 A 12 10 6,030 288 (1) 995 B 12 10 6,180 564 (2) 1,720 C 8 9 6,030 491 (2) 853 D 12 8 6,030 523 (2) 1,197 E 9 9 4,719 1,648 (10) 4,254 CLASP2 F 20 20 12,910 179 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='01) 737 G 38 31 10,390 1,346 (3) 4,826 H 35 29 11,200 198 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='01) 900 Total – 146 126 63,489 5,237 15,482 large number of video frames available in the datasets, the an- notation rate for the video sequences varies between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='01 and 10 frames per second (fps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We randomly partition each dataset into a training set containing 80% of the video frames and a test set with the remaining 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For a fair comparison, the Supervised Learning (SL) and SSL models are trained using only the frames from the training set, but the SSL models are fully self-supervised and do not use any manual annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, disregarding every video sequence that includes annotated frames would substantially limit the amount of data available for the computation of tracking performance measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence, to assess tracking performance, we consider all the annotations listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The only method that uses the training set annotations is the SL approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Although this evaluation strategy favors that method, it also more accurately reflects the generalization performance of the SSL approaches to unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Due to space limitations, the results presented in this section were obtained using the aggregated CLASP1 and CLASP2 test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dataset-specific results are given in the Supplementary Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Self-Supervised Learning Detection Performance During training, we freeze the network weights up to the region proposal network layer so that the pre-trained backbone features are effectively used in the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We use an initial learning rate of 5e−3, mini-batch size per image N = 256, r = 20 different orientations, and a cluster confidence threshold λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Similar to the baseline model, we use stochastic gradient descent with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9, weight decay of 1e−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' At each SSL iteration, we fine-tune the model for 20k iterations, reducing the learning rate by a factor of 10 at every 5k iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In our evaluation, we use an IoU threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5, and a non-maximum suppression threshold ηnms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 for all the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The detection threshold for region proposal generation is ηdet = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 8 shows the Multi-Object Detection Accuracy (MODA) of our model as a function of the number of SSL iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To illustrate the impact of the cluster confidence score, we also evaluate a model in which the samples are not weighed by their scores (SSL-wo-α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Instead, this model uses a hard threshold λ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 to discard noisy detections during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The figure also shows the performance of the Multiple-Inference (MI) 8 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Panoramic overview of the camera views at the Kostas Research Institute simulated airport checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Sample images from the datasets collected at the simulated airport checkpoint (left: CLASP2 and right: CLASP1 in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The images show the divestiture area (right: camera 9) and item retrieval area (left: camera 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 0 2 4 6 8 Iteration 40 60 80 100 MODA MI-wo- MI SSL-wo- SSL 0 2 4 6 8 Iteration 40 60 80 100 MODA MI-wo- MI SSL-wo- SSL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' MODA measures for person (left) and baggage (right) classes during SSL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' strategy used to generate the pseudo-labels, which reflects the quality of the pseudo-labels before SSL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, in the MI model, the pseudo-labels themselves are used as model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As the figure indicates, the SSL models gradually approach the performance of the MI strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The incorporation of cluster confidences not only increases the speed of convergence of the models but also leads to noticeable performance gains, particularly for baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 9 shows the precision-recall curves for passenger and baggage detection using four detector models: pre-trained PANet (baseline), PANet trained using SL, SSL-wo-α, and SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Even though the SSL models are trained without manual annotations, they perform on par with the SL model for pas- sengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For baggage items, the maximum average precision for the baseline model is less than half of the performance of the SSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 14, the performance 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 1 Recall 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 1 Precision Base (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='77) SSL-wo- (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='94) SSL (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='95) SL (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='95) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 1 Recall 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 1 Precision Base (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='35) SSL-wo- (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='74) SSL (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='82) SL (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='93) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Precision-recall curves for person (left) and baggage (right) detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The legend shows the average precision of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' difference between the SL and SSL models is due to two main issues: i) appearance similarities among bags and certain garments/items placed inside security bins, and ii) baggage items that can only be partially observed before being placed on the conveyor belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Table VII demonstrates the benefits of incorporating clus- ter uncertainties in the SSL loss function (column α) and of the proposed cluster regression technique (column reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The method that incorporates both cluster uncertainty and regression is equivalent to the approach identified as SSL in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 8 and 9 whereas the method that does not include cluster confidences corresponds to SSL-wo-α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The results in the table correspond to the point that maximizes the F1 score of the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 9 at the best performing SSL iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The top-performing method in Table VII and in the remainder of this section is highlighted in boldface, the second-best is underlined, and ties are broken according to the MODA/MOTA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In comparison with the baseline model, our SSL algorithm substantially increases the recall (Rcll) and precision (Prcn) for passenger detection, which is a result of improvements in true positive (TP), false positive (FP), and false negative (FN) detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The cluster confidence scores substantially reduce the contribution of low-confidence pseudo-labels, especially for baggage items, leading to a noticeable increase in the number of true positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Cluster regression corrects pseudo- 2019 19278440SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Sample results showing failure cases for baggage detection using the SSL model in the CLASP2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The magenta arrows indicate bag-like object detections that are not annotated (false positives), the red arrows indicate annotated baggage items the model fails to detect (false negatives), the green bounding boxes show passenger detections, and the red bounding boxes represent the manual annotations for both classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE II PASSENGER AND BAGGAGE DETECTION EVALUATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Model Method ↑Rcll ↑Prcn ↑TP ↓FP ↓FN ↑MODA α reg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' person bag person bag person bag person bag person bag person bag Baseline \x17 \x17 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 1560 426 228 85 552 724 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 SSL \x17 \x17 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 1989 858 155 194 123 291 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 SSL ✓ \x17 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 1985 863 134 144 127 286 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 SSL \x17 ✓ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 1985 844 73 71 127 305 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 SSL ✓ ✓ 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 2025 903 79 83 87 246 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 SL \x17 \x17 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 2022 1048 70 83 90 101 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 (a) (c) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Qualitative detection results on the CLASP2 dataset using (a) Baseline, (b) SSL, and (c) SL models (the SL model only predicts bounding boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' label errors caused by inaccurate bounding boxes generated from poor segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As a result, the reduction in false positives for both classes is even more pronounced when cluster regression is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Overall, our SSL framework shows a relative MODA score improvement of 46% for passengers and 144% for baggage items with respect to the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 11 shows qualitative results for the models under con- sideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In comparison with the SL model, the SSL models not only improve the accuracy of the predicted bounding boxes but also generate improved segmentation masks since they are trained using instance segmentation pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Single-Camera Tracking We compare the performance of our single-camera tracking algorithm using the proposed SSL detectors with the pre- trained baseline detector and the SL detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We also evaluate the impact of our SCA algorithm, described in Section III-B1, where we use tth = 3 seconds and δmax = 200 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To preserve the entirely self-supervised nature of our pipeline, we refrain from fine-tuning the re-identification module of the baseline tracker, which is pre-trained on the MOT17 [59] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To dissociate the evaluation of the tracking method from our MCTA approach, we use a modified version of the annotations in Table I where a passenger that re-enters the field of view of a camera receives a new identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Thus, the number of unique ground truth passenger identifiers (column GT in Table III) is much higher than those listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We evaluate our system’s ability to maintain consistent passenger identifiers across multiple perspectives in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As Table III shows, the SSL-wo-α and SSL approaches out- perform the tracker using the baseline detector by a large mar- gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The notable improvements in identity-based F1 (IDF1), recall (IDR), and precision (IDP) [76] as well as in standard recall and precision are primarily a result of the reduction in false positives and false negatives generated by the SSL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Self-supervision also improves the tracking-specific metrics of mostly tracked (MT), mostly lost (ML), identity switches (IDs), and fragmented (FM) trajectories [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As a result, our method produces substantial gains in MOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Again, both SSL models perform on par with the SL model for person tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For baggage items, we see similar performance improvements, but the challenges illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 14 again preclude the SSL models from reaching the performance of the SL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Finally, our SCA algorithm leads to further performance gains, particularly in terms of IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Multi-Camera Tracklet Association We evaluate the performance of our MCTA algorithm using the same experimental procedure described in the previous section, with the exception that passengers are now assigned unique identifiers as they leave and re-enter the fields of backpack 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='89 packpack 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='94 suitcase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='78 suitcase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='70 person 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='98rson 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='65 04/16/2019:09:03:46:476bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 person 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='98 person 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='91 person 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='97 person 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 04/16/2019 09:03:46:476bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 bag 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 person 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 person 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 person 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='98 person 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='00 04/16/2019 09:03:46:47610 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 TABLE III SINGLE-CAMERA TRACKING EVALUATION FOR PERSON AND BAGGAGE CLASSES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Class Model α SCA GT ↑IDF1 ↑IDR ↑IDP ↑Rcll ↑Prcn ↓FP ↓FN ↑MT ↓ML ↓IDs ↓FM ↑MOTA ↑MOTP Person Baseline \x17 \x17 391 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 401 822 169 28 48 97 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 SL \x17 \x17 255 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 387 339 226 10 103 69 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 #1357 C11 (a) (b) C05 C11 #2232 C11 #0627 C09 #1652 C09 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Sample results showing (a) cross-camera passenger association between cameras 5 and 11 using MCTA, and (b) tracking and association between passengers and baggage items where the top and bottom rows show image sequences from cameras 9 and 11 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We associate passenger tracklets in cameras 9 and 11 by leveraging the associations between cameras 2 and 5 (passengers flow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 6: C9→C2→C5→C11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Baggage items are associated using temporally constrained distance-based matching when each item receives a unique identifier PiBj, representing the j-th item from the i-th passenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE IV MCTA EVALUATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' THE COLUMN LABELED DIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' INDICATES WHETHER WE EMPLOY THE HAUSDORFF (dh) OR FR´ECHET (df ) DISTANCE TO EVALUATE TRACKLET SIMILARITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' SL SSL-wo-α SSL MCTA ↑IDF1 ↑IDR ↑IDP ↓IDs ↑MOTA \x17 ✓ \x17 \x17 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 157 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 \x17 \x17 ✓ \x17 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 170 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 ✓ \x17 \x17 \x17 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 170 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 dh \x17 ✓ \x17 ✓ 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 115 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 \x17 \x17 ✓ ✓ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 140 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 ✓ \x17 \x17 ✓ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 134 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 df \x17 ✓ \x17 ✓ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 115 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 \x17 \x17 ✓ ✓ 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 132 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 ✓ \x17 \x17 ✓ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 122 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 view of the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Based on the overall flow of passengers through our simulated checkpoint, cameras 9 and 11 are the primary cameras for our tracklet association method (Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Cameras 2 and 5, the cameras immediately below them in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 6, are the respective auxiliary cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For a fair comparison among the detectors, we generate tracklets in the auxiliary cameras using the corresponding SL or SSL model used in the primary cameras (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', trained using only frames from the primary camera).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To provide a set of reference performance measures, we first evaluate our tracking algorithms in the ab- sence of a MCTA mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We then assess the performance of our association method when tracklet similarity is computed using the Fr´echet distance and the more traditional Hausdorff distance [64] with fmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='25 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Table VIII shows that tracklet association improves the IDF1 measure by up to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This is mainly a consequence of the dramatic reduction in the number of identity switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Using the Fr´echet distance to determine tracklet similarity pro- vides consistent performance improvements in all the metrics under consideration, especially for the SSL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The more modest gains in MOTA (up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0%) demonstrate the need for measures that focus specifically on the impact of identity switches on tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 12(a) illustrates the tracklet association procedure be- tween cameras 5 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As the passengers with identities P2 and P3, whose trajectories are represented in green and yellow, move from the field of view of camera 5 to camera 11, their tracklets are projected from the former camera to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The projected trajectories (red for P2 and pink for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P3) are successfully associated with the tracklets from camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P1B1B2P1 transfers P1B1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='Pl carrying P1B1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P2 carrying P2B2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P2 enters into C9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P1B1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P1 enters into C9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P2B2P3B3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P2B2P4B4 left C9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P2 left C9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P4 transfers P4B4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P4 carrying P4B4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P3B3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P3 transfers P3B3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P4B4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P3 carrying P3B3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='p4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P4 enters into C9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P3 enters into C9SIDDIQUE AND MEDEIROS: TRACKING PASSENGERS AND BAGGAGE ITEMS USING MULTIPLE OVERHEAD CAMERAS AT SECURITY CHECKPOINTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='11 based on the Fr´echet distances among their temporally ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='overlapping segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In the instant shown in the figure, passenger P2 is re-entering the field of view of camera 5, and the corresponding tracklet is also correctly associated with that passenger’s tracklet in camera 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence, the passenger’s identity is successfully handed off between the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 12(b) demonstrates a potential application of the proposed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Baggage items are associated with passengers when they are divested in camera 9 and their identifiers can be verified at retrieval time, which is observed in camera 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' CONCLUSION We propose a multistage tracking-by-detection framework to overcome performance limitations of object detection and tracking algorithms in overhead camera videos for which limited training data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our framework is composed of an SSL mechanism to fine-tune object detection models to specific camera views without the need for manual annotations and an MCTA method that only requires the homographies among neighboring cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our experiments show that the proposed framework can accurately detect and track pas- sengers and baggage items across camera views in airport checkpoint scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our framework is flexible and scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' It requires no training data, incurs no detection computational overhead at inference time, and is independent of the number of cameras in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our framework also allows seamless integration of ad- ditional data augmentation strategies and of manually an- notated data when it is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our experiments show that these strategies further improve the selectivity of our detector, particularly for baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For simplicity, our association methods are performed offline, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', after all 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', “End-to-end semi-supervised object detection with soft teacher,” In IEEE/CVF Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' on Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Supplementary Materials: Tracking Passengers and Baggage Items using Multiple Overhead Cameras at Security Checkpoints Abubakar Siddique, Student Member, IEEE, Henry Medeiros, Senior Member, IEEE Abstract—This document supplements our main paper with additional experimental results on the CLASP1 and CLASP2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We extend our Self-Supervised Learning (SSL) ap- proach into a Semi-Supervised Learning (Semi-SL) mechanism to further improve target detection performance, especially for baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We also investigate the impact of additional data augmentation strategies, rotation resolution, and the computa- tional requirements of our proposed technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' These additional evaluation results show that our algorithm outperforms the baseline as well as state-of-the-art supervised and semi-supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Index Terms—Detection, Tracking, Association, Homography, Tracklet, Multi-camera, Surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' SINGLE CAMERA DETECTIONS This section presents a breakdown on the performance of our SSL detector for individual cameras in the CLASP1 and CLASP2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' It also evaluates the performance impact of additional data augmentation strategies, number of rotation an- gles used for data augmentation, and incorporation of labeled data in a semi-supervised approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Self-Supervised Learning Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 13 shows a detailed breakdown of the performance of our SSL detection model for individual camera views in the CLASP1 and CLASP2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The high recall, precision, and MODA values indicate that our SSL approach detects most passengers correctly in these video sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The average precision (AP) for passenger detection is slightly higher for camera 11 in both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The main factor contributing to this performance difference is that in camera 9, passengers are only partially visible most of the time, whereas camera 11 has a better view of the region where the passengers stand next to the conveyor belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' On the other hand, this also contributes to the lower baggage detection performance in Manuscript received August 22, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' accepted November 11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Date of publication December 14, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' †This material is based upon work supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Award Number 2013-ST-061-E0001-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Department of Homeland Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ‡Abubakar Siddique is with the Department of Electrical and Computer Engineering, Marquette University, Milwaukee, USA, e-mail: abubakar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='siddique@marquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='edu §Henry Medeiros is with the Department of Agricultural and Bi- ological Engineering, University of Florida, Gainesville, USA, e-mail: hmedeiros@ufl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='edu ¶Digital Object Identifier (DOI): 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1109/TSMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3225252 camera 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, in camera 11, partially observed baggage items being carried by passengers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 14) are much more common than in camera 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As with passenger detection, we observed similar baggage detection improvements in the camera-specific performance comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This performance could be further improved by using additional unlabelled video frames available in the CLASP1 and CLASP2 datasets to train the SSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Additional Data Augmentation Strategies We investigate the impact of other data augmentation strate- gies during SSL training, including color jittering and motion blur along with multiple rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For color jittering, we increase/decrease image brightness, contrast, saturation, and hue by a factor sampled uniformly from the range [0, maxjit], where maxjit is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 for brightness, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 for contrast, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 for saturation, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='05 for hue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' To emulate motion blur, we use Gaussian blur with kernel size uniformly sampled from the set {5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' , 9} and standard deviation sampled from the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We observe that applying color jittering and mo- tion blur on the pseudo-label augmentation further improves MODA scores by up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8% for passengers and baggage items, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For a fair comparison, we reduced the number of rotation angles used for augmentation such that the total number of augmented images remains the same in both scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Maintaining the original number of rotations would further increase performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE V PERFORMANCE IMPACT OF ADDITIONAL DATA AUGMENTATION STRATEGIES IN THE SSL ITERATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dataset Method ↑AP ↑ F1 ↑MODA Rot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' C-Jit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Mot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='-Blur person bag person bag person bag CLASP1 ✓ \x17 \x17 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 ✓ ✓ ✓ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 CLASP2 ✓ \x17 \x17 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 ✓ ✓ ✓ 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Impact of Rotation Resolution Table VI shows the impact of rotation resolution r on the generation of pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' One SSL iteration with r = 20 improves the MODA scores by up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1% for passengers and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6% for baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The inference time for a single frame increases linearly with the number of rotations, contributing to longer SSL training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' If training time is a concern, r = 10 offers a reasonable speed vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' performance trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' We 13 14 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Passenger and baggage detection performance in cameras 9 and 11 for the CLASP1 (CL1) and CLASP2 (CL2) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Here, P stands for passenger and B for baggage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A description of the methods under consideration is given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='B of the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Additional illustrative failure cases for baggage detection using the SSL model in the CLASP1 dataset (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 10 in the main paper for failures in the more challenging CLASP2 dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The magenta arrows indicate bag-like object detections that are not annotated (false positives), the red arrows indicate annotated baggage items the model fails to detect (false negatives), the green bounding boxes show passenger detections, and the red bounding boxes represent manual annotations for both classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' use r = 20 for all the SSL models to demonstrate the potential performance of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As Table VI indicates, further increasing the value of r would likely lead to minor additional performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE VI PERFORMANCE IMPACT OF THE NUMBER OF ROTATION ANGLES USED IN THE SSL ITERATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dataset r ↓Infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Time ↑ F1 ↑MODA (secs) person bag person bag CLASP1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 CLASP2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 20 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Semi-Supervised Learning As Table VII indicates, the performance of our SSL algo- rithm is limited by the initial accuracy of the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Thus, we extend our method to a semi-supervised approach where we use a certain amount of manual annotations to initialize our model before initiating SSL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For the labeled frames, we employ the same data augmentation pro- cedure used to generate augmented labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 15 shows that training the SSL model using 10% of the manual labels leads to a performance comparable to the SL model, outperforming SoftTeacher [77], a state-of-the-art Semi-SL technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Our method is particularly effective when small amounts of anno- tations are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For example, using only 1% of the manual labels, our Semi-SL approach outperforms SoftTeacher by 104% and is only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6% behind the SL method (Table VII) for baggage items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Furthermore, we observe a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7% MODA improvement over the SL method when we use all the manual annotations during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' SINGLE-CAMERA TRACKING Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 16 shows the Single-Camera Tracking (SCT) perfor- mance of our algorithm for passengers and baggage items in the individual cameras of the CLASP1 and CLASP2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='P-baseline ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='B-baseline ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 822 412 63 95 90 93 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 SSL \x17 ✓ 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 833 354 46 21 79 151 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 SSL ✓ ✓ 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 863 397 47 28 49 108 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 SL \x17 \x17 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 862 472 47 29 50 33 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Semi-SL model performance on CLASP2 using a semi-supervised extension of our proposed SSL method versus SoftTeacher (ST) [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Here, P and B stand for the passenger and baggage categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The SSL model uses no labeled data and the SL model is trained with 100% of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For passenger tracking, the SSL methods outperform the SL approach in terms of IDF1, IDP, and IDR in all the scenarios under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In both datasets, the SL approach shows slightly higher MT results for camera 9, largely due to the partial passenger detection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Since CLASP1 has lower object density, we observe more consistent performance among different methods for both cameras in that dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' While all the methods perform better on the CLASP1 dataset, the benefits of SSL training compared to the baseline detector are particularly evident in the MT results on the CLASP2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Regarding baggage items, although the SSL models lead to a moderate increase in the number of IDs, these switches are offset by substantial gains in MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' As a matter of fact, the SL model shows a much more significant degradation in IDs for the more complex CLASP2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This is particularly evident for camera 9, and it explains the lower IDP obtained by the SL method in that dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The most evident performance gains for baggage tracking are observed in camera 11 on the CLASP2 dataset because the of the difficulty of partially visible baggage items using the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' MULTI-CAMERA TRACKLET ASSOCIATION Regarding our Multi-camera Tracklet Association (MCTA) method, Table VIII shows that the Fr´echet distance metric is particularly useful in crowded scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Although we obtain comparable results using the Hausdorff distance on the easier CLASP1 dataset, we achieve noticeable improvements in all the evaluation criteria on CLASP2 using the Fr´echet distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The single-camera trackers in the auxiliary cameras are trained using frames from the primary cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence, in crowded scenarios they sometimes fails to keep alive trajectories of targets that are temporarily outside the field of view of the primary camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' This is the main reason behind the overall lower tracking performance on the CLASP2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Training camera-specific detectors using our SSL framework would mitigate this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE VIII MCTA EVALUATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' THE COLUMN LABELED DIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' INDICATES WHETHER WE EMPLOY THE HAUSDORFF (dh) OR FR´ECHET (df ) DISTANCE TO EVALUATE TRACKLET SIMILARITY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' SL SSL-wo-α SSL MCTA ↑IDF1 ↑IDR ↑IDP ↓IDs ↑MOTA CLASP1 \x17 ✓ \x17 \x17 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 45 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 \x17 \x17 ✓ \x17 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 48 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 ✓ \x17 \x17 \x17 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 42 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 dh \x17 ✓ \x17 ✓ 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 21 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 \x17 \x17 ✓ ✓ 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 30 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 ✓ \x17 \x17 ✓ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 25 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 df \x17 ✓ \x17 ✓ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 22 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 \x17 \x17 ✓ ✓ 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 27 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 ✓ \x17 \x17 ✓ 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 23 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 CLASP2 \x17 ✓ \x17 \x17 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 112 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 \x17 \x17 ✓ \x17 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 122 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='9 ✓ \x17 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='2 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' COMPUTATIONAL COMPLEXITY In this section, we analyze the theoretical computational complexity of our SSL strategy and measure the computation time and memory utilization of each step of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' All our experiments were performed on a workstation equipped with two RTX-2090Ti GPUs and an Intel® Xeon® Silver 4112 CPU @2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Self-Supervised Learning The computational complexity of our approach increases linearly with the number of rotation angles used for augmenta- tion in the pseudo-label generation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' That is, for a baseline detection algorithm with computational complexity Θ(f(I(t)), 100 80 60 40 P-SL P-ST P-SSL P-Semi-SL 20 B-SL B-ST B-SSL B-Semi-SL 0 1% 5% 10% 50% 100% ↑AP100 80 60 40 P-SL P-ST P-SSL P-Semi-SL 20 B-SL B-ST B-SSL B-Semi-SL 0 1% 5% 10% 50% 100% TMODA16 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Y, DECEMBER 2022 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Comparison of SCT performance of person and baggage classes in individual cameras of the CLASP1 and CLASP2 datasets using the Baseline, SSL-wo-α, SSL, and SL detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' the complexity of our approach is Θ(r·f(I(t)), where r is the number of rotation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' For example, for r = 20, the run- time is 20 times that of a single iteration without augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, these operations are parallelizable as long as the hardware resources support the simultaneous processing of multiple frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' With our unoptimized implementation, the total time to complete one SSL iteration is approximately six hours for both model training and pseudo-label genera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, we have observed that hardware resources are severely underutilized, which indicates substantial room for reduction in overall computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Inference Performance Table IX shows the computation time of the proposed tracking-by-detection algorithm, employing a PANet detector with a ResNet-50 backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' The SCT uses the detector re- sults and a ResNet-50-based Re-Identification (Re-ID) model trained on MOT17 to re-label tracklets lost due to short-term occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Hence, the computation time and memory utiliza- tion for the SCT are similar to those for the detector model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Since we are processing single images individually instead of image batches, the inference time for the detector and the SCT are far from optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Preliminary experiments indicate that processing batches of 10 images simultaneously leads to an approximate six-fold reduction in detector inference time without exceeding the memory capacity of the GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Reusing the backbone features from the detector in the Re-ID model should also lead to a dramatic reduction in SCT time, since The execution time of the proposed MCTA algorithm depends on the average length of the overlapping tracklet segments in each camera pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In the CLASP1 dataset, which contains fewer and shorter tracklets, the algorithm can be executed in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' In CLASP2, it can run at approximately TABLE IX COMPUTATION TIME OF THE PROPOSED TRACKING-BY-DETECTION FRAMEWORK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Data Model Infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Time (ms) Memory (MB) CLASP1 Detector 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 1,850 SCT 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='8 1,748 MCTA 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 CLASP2 Detector 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 1,850 SCT 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 1,750 MCTA 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 feature generation is the most computationally demanding element of the tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' 12 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' However, the current implementation of the proposed system uses the full life-span of a tracklet to compute the Fr´echet association distance in the MCTA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' It is pos- sible to substantially reduce computation time by limiting the length of single-camera tracklets compared by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Table X shows that if we limit the length of the tracklets to 240 frames (or eight seconds), it is possible to achieve real-time performance for both datasets without degrading the accuracy of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' TABLE X COMPUTATION TIME OF THE PROPOSED MCTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Metric Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Size Infer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content=' Time (ms) Memory (MB) MOTA CLASP1 Hausdorff – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 Fr´echet – 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 240 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 CLASP2 Hausdorff – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='64 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='61 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='4 Fr´echet – 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='5 240 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfXvei/content/2301.00190v1.pdf'} +page_content='60 ' 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Lynch,6 and Alessandro Cunsolo6 +1CNR-IOM & INSIDE@ILL c/o Operative Group in Grenoble (OGG), +F-38042 and Institut Laue Langevin, Grenoble, France +2Dipartimento di Economia e Diritto, Universit`a di Macerata, +Via Crescimbeni 20, 62100 Macerata, Italy +3Dipartimento di Fisica e Astronomia, Universit`a di Firenze, +via G. Sansone 1, I-50019 Sesto Fiorentino, Italy +4Consiglio Nazionale delle Ricerche, +Istituto di Fisica Applicata ”Nello Carrara”, +via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy +5Argonne National Laboratory, Advanced Photon Source, +P.O. Box 5000 Upton, 11973 NY, USA +6Department of Physics, University of Wisconsin at Madison, +1150 University Avenue, Madison, WI, USA +1 +arXiv:2301.01385v1 [cond-mat.mes-hall] 3 Jan 2023 + +Abstract +Understanding how molecules engage in collective motions in a liquid where a network of bonds +exists has both fundamental and applied relevance. On the one hand, it can elucidate the “ordering” +role of long-range correlations in an otherwise strongly dissipative system; on the other hand, it +can inspire new avenues to control such order to implement sound manipulation. Water represents +an ideal investigation case to unfold these general aspects and, across the decades, it has been +the focus of thorough scrutiny. Despite this investigative effort, the spectrum of terahertz density +fluctuations of water largely remains a puzzle for Condensed Matter physicists. To unravel it, we +compare previous scattering measurements of water spectra with new ones on ice. Thanks to the +unique asset of Bayesian inference, we draw a more detailed portrayal of the phonon response of +ice. The comparison with the one of liquid water challenges the current understanding of density +fluctuations in water, or more in general, of any networked liquid. +I. +INTRODUCTION +It is a matter of common experience that liquids and solids oppose a different resistance +to mechanical attempts to change their macroscopic shape. A possible way to test such a +different rigidity is to induce a macroscopic mechanical perturbation generating the propaga- +tion of a density wave. An alternative pathway followed in inelastic scattering measurements +consists in stimulating the propagation of density fluctuations at mesoscopic scales by ir- +radiating a material with a beam of particle waves, such as neutrons or photons. When +dealing with a liquid system, such a density wave propagates with a strong damping and at +macroscopic scales, uniquely in a direction parallel to the triggering force. In a structurally +ordered solid such as a crystal, applied stresses propagate for a longer time and in directions +both parallel and orthogonal to the applied force, in the form of longitudinal (LA) or trans- +verse acoustic (TA) phonons, respectively. However, this clear-cut distinction becomes more +elusive over distances and time lapses respectively approaching the size of a single atom first +neighbors cage and the period of such an atom’s “in cage” bounces. +Indeed, abundant experimental evidence endorses the conclusion that density waves in +liquids bear strong evidence for terahertz viscoelasticity [1], i.e., they combine liquid-like +∗ Correspondence email address: defrance@ill.fr +2 + +and solid-like aspects [2–5]. For instance, the transition between the viscous and the elastic +regime in liquid water at about 280 K occurs at a few nm−1, according to the combined +results of Refs. [6] and [7]. Relevant points of the broad picture emerging from the intensive +scrutiny of terahertz viscoelasticity in fluids can be summarized as follows: +1) The first spectacular manifestation of mesoscale viscoelasticity is the increase of sound +velocity when probing distances and times roughly as small as those featuring first-neighbor +atomic interactions. This trend reflects the enhanced rigidity of the system (elastic regime) +at these so-called mesoscopic scales, typically much shorter than those covered by dissipative +- diffusion or relaxation - processes in liquids. +2) The transition to such an elastic regime is paralleled by an increased ability of the +liquid to support the propagation of transverse acoustic waves. +3) As opposed to a long-standing belief, this viscoelasticity persists in thermodynamic +domains extending well above the critical point [8, 9], although their actual delimitation +raised some controversy [10–13]. +4) Aside from its unsettled dependence on thermodynamic conditions, high-frequency +viscoelasticity also has a still poorly understood link with the nature of microscopic interac- +tion; for instance, it is unclear why it is substantially more pronounced in specific relatively +low-viscosity systems as water [3, 14, 15], carbon dioxide [16], deuterium [17], and noble +gases [18–20], while it is often barely detectable in liquid metals [21–25]. +Far from being a mere academic endeavor, gaining insight on these topics could deepen +our understanding of subjects as fundamental as the very nature of liquid aggregation. Also, +it will likewise disclose new avenues in the emerging domain of sound propagation design +and engineering, a field whose enormous practical interest has been recognized [26], yet not +fully explored. +On the experimental side, terahertz phonon propagation in condensed materials can be +directly probed by inelastic scattering methods, such as Inelastic X-Ray (IXS [27, 28]) and +Neutron Scattering (INS, [29, 30]). Conceptually, IXS or INS spectroscopic probes resemble +3 + +large microscopes “pointed on the dynamics”, which can be zoomed in to focus on dynamic +events occurring over various scales. This goal is ordinarily achieved by suitable tuning of +the two main variables of the scattering event, namely energy and momentum exchanged +between the probe wave particles (here assumed to be photons), respectively referred to as +E = ℏω, and p = ℏQ. Here, ℏ is the reduced Planck constant, while Q (ω) represents +the exchanged wave-vector (frequency), i.e., the difference between before-scattering and +after-scattering photon wave-vector (frequency). For small enough values of Q = |Q| and +E, the target system appears as a continuum whose dynamic response is probed as an +average over many microscopic events. Conversely, at extremely large (Q, E)’s, distances +and times spanned become so small that the only event observed is the free recoil of the +single struck atom after the collision with the photon and before subsequent first-neighbor +interactions. Clearly, all dynamic processes characterizing the interatomic dynamics of the +sample can be captured, somewhere between these two limits, by adequately tuning the +spectroscopic probe. Not surprisingly, the developments of the scattering techniques directly +mapping this broad dynamic domain, INS and IXS, have dramatically improved the current +understanding of high-frequency viscoelasticity phenomena in an impressively varied family +of disordered systems. Although water is one of the disordered systems most thoroughly +studied across various decades, its deep scrutiny has often evidenced seemingly anomalous +and poorly understood behaviors, albeit sometimes later recognized as “normal effects” +that water shares with many other liquids. Notable examples include the high propagation +speed of the terahertz sound mode, once called “fast sound” [6, 7], and the onset of a +shear mode propagation [31–35]. Although similar phenomena are being reported for an +increasing number of disordered systems, the case of water found a straightforward and +broadly accepted explanation in terms of hydrogen bond (HB) dynamics. According to this +interpretative scheme, [36, 37] high frequency transverse and longitudinal acoustic waves in +water respectively couple with the bending of two HBs linking triplets of oxygen atoms, or to +the stretching of the HBs connecting two adjacent oxygens. Despite the remarkable insight +achieved over the decades, further experimental effort is still needed to clarify a few more +subtle aspects of water dynamics. These include the microscopic mechanism leading shear +acoustic modes of water to exhibit a nearly flat Q dependence at the crossover between +quasi-macroscopic and mesoscopic distances, while gradually becoming predominant over +the longitudinal acoustic modes. +4 + +From a merely experimental perspective, a natural pathway to understand better the +collective movements of water molecules is to observe how they compare with those in the +solid. Indeed, the study of the acoustic response of (polycrystalline) ice presents significant +simplifications: 1) both atomic diffusion and HB network relaxations are frozen, thus yielding +no visible contribution to sound propagation; 2) the lifetime of acoustic modes is much longer, +and the related spectral features correspondingly sharper. Nonetheless, as illustrated in the +remainder of this paper, the phonon response of ice is far from being trivially interpreted and +characterized. In this paper, we discuss an experimental attempt to elucidate similarities +and distinctive behaviors of the terahertz dynamics of ice and water by comparing their +IXS spectra. Specifically, we jointly investigate collective excitations in water and phonon +modes in polycrystalline hexagonal ice (Ih). On a rigorous ground, inelastic excitations in +the spectrum of a polycrystal should not be referred to as phonons owing to their projection +along multiple crystallographic directions. However, in the following we will resort to this +broadly used nomenclature for consistency with existing literature. +The use of a Bayesian inference-based modeling of IXS lineshapes reveals an unprece- +dentedly complex phonon behavior of ice, while rectifying and complementing the current +understanding of terahertz acoustic excitations in liquid water. +II. +THE MEASUREMENT +Measurements were executed at the Sector 30 beamline [38, 39] of the Advanced Photon +Source at Argonne National Laboratory. The instrument was operated using the ≈ 23.7 +keV harmonic of the undulator source, which corresponds to the Si(12 12 12) backscattering +reflection from both the monochromator and energy analyzers. Spectral acquisitions covered +the 3 ÷ 21 nm−1 Q range with a 2 nm−1 step. The instrumental resolution function was +measured through the IXS signal from a Plexiglas sample at the Q of its first sharp diffrac- +tion maximum, i.e., about 10 nm−1. The resulting spectrum had a 0.8 meV broad (half +width at half maximum, HWHM) nearly Lorentzian profile, sufficient to properly resolve +the relevant spectral features discussed in the remainder of this paper. All measurements +were executed at a temperature of 225 K. The polycrystalline nature of the sample makes +its global orientation irrelevant. As a consequence, the measurement probed the constant- +Q phonon response of ice as an average over the various crystallographic directions. Fig. +5 + +1 illustrates the emergence of phonon excitations in the IXS spectra of ice by comparing +raw and best-fitting model spectra for selected Q values; the semilogarithmic plot enables a +full appreciation of relevant lineshape features across various intensity decades. Model line- +shapes, obtained as discussed in Sec. I of Supplemental Materials, consist of a δ-function +describing the elastic scattering and individual DHO terms accounting for inelastic excita- +tions in the spectrum. The most plausible number of DHO components was determined a +posteriori, based on the outcome of the Bayesian inferential analysis of measured data, as +described in our previous works [40–44]. Individual DHO components of the best-fitting +total spectral distribution are also included in Fig. 1 for reference. The assignment of each +spectral contribution is summarized in the caption of Fig. 1 and illustrated in Table I. +TABLE I. Polycrystalline Ice modes assignment +Mode Label Assignment +Symbol color +1 +TA +Transverse Acoustic +blue +2 +OL +Lower Optical mode +grey +3 +OH +Higher Optical mode +orange +4 +LA +Longitudinal acoustic mode of center of mass magenta +5 +HFM highest frequency steeply dispersive mode +wine +Notice that the same color code is maintained consistent throughout all graphs included +in this work. The IXS spectra in Fig. 1 are reported in the region where they have been +modeled. This region, varying in extent and symmetry with respect to the central peak +position, essentially coincided with the energy range covered by the corresponding energy +resolution measurement. +One can readily notice that spectral shapes displayed in Fig. 1 are dominated by multiple +inelastic features, which, in some cases, partially overlap giving rise to inelastic peaks visibly +broader than the resolution function, as particularly evident in the Q = 21 nm−1 spectrum. +Overall, Fig. 1 provides a comprehensive rendering of the complexity of the phonon response +of polycrystalline ice down to mesoscopic distances. The Q-evolution of various phonon +modes will be discussed in further detail in the next Section, here we only stress that these +modes display quite distinctive Q dependencies revealing their either acoustical or optical +origin. +6 + +FIG. 1. Inelastic X-Ray Scattering (IXS) spectra of polycrystalline ice and their model +lineshapes. IXS spectra measured at representative Q’s and T = 225 K discussed in this work +(open circles) compared with corresponding best-fit lineshapes (red lines through data) along with +their inelastic DHO model components (see text): the low frequency transverse mode (blue line), +the two intermediate frequency optic-like modes (grey and orange lines), the longitudinal acoustic +mode (magenta) and a higher frequency mode (wine line). The green line represents the instru- +mental energy resolution. Error bars are estimated as the square root of the scattering intensity +counts. +III. +DISCUSSION OF RESULTS +The good consistency between the model and measured lineshapes clearly emerges from +the Fig. 1. Selected IXS spectra are compared with their best-fitting model lineshapes +discussed in the Sec. I of Supplemental Materials. Overall, phonon spectra presented in +7 + +10000 +Q=3nm +100 +20 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20 +10000 +I(Q,E) (arbit. units) +Q= 11nm +Q= 17 nm +100 +-20 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20 +30 +Q = 19 nm += 21 nm +100- +-20 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20 +E (meV)this paper are qualitatively similar to those reported in the IXS measurement described in +Ref. [2]. However, the latter was performed on a polycrystalline sample seemingly more +ordered than the present one, as suggested by the weaker elastic contribution, yet with +a resolution significantly coarser, as to be expected considering the still pioneering stage +of high-resolution IXS. Once assessed the ability of the model to reproduce the detail of +the spectral shape, we can now focus on best-fit values of phonon frequencies, whose Q +dependence and assignment are included in Fig. 2. +As apparent from the dispersion curves displayed in Fig. 2, the low-energy (< 25 meV) +phonons of ice probed by this measurement show some rather peculiar features, previously +either undetected [2, 48] or only partially reported by studies in the literature [33, 45]. +Specifically: +1) The longitudinal acoustic phonon of ice, i.e., the one characterized by the sharpest +(linear) low Q growth in Fig. 2a, actually splits into two branches, consistent with what was +reported in a previous IXS work by Cimatoribus and collaborators [33], and previous lattice +calculations [49]. Most importantly, this trend might suggest reconsidering the surprising +line broadening of the longitudinal acoustic mode of both water and polycrystalline ice +reported in Ref. 50. The resolution limitation of such a pioneering IXS measurement is +the most likely explanation for the mode-splitting appearing as a line-broadening. At low +enough Q’s, the two split branches attain similar values following a parallel Q-dependence +while they spread apart at larger Q’s. To estimate the low wavevector propagation speed +of these two modes, say c1 and c2, we looked at the slope of the three lowest Q points of +the corresponding phonon branches (Fig. 2). We found that the ratio of the sound speeds +c1/c2 = 1.09 ± 0.07, a value consistent within the error to that of +� +MCM/MO = 1.06 where +MCM and MO are the masses of the water molecule and the oxygen atom, respectively. This +finding might suggest that, in the linear dispersion regime, the modes reported in Fig. 2 as +LA and HFM, might be connected to collective vibrations involving either the whole H2O +molecule or the oxygen atoms only. This is the main rationale behind the identification of +the lower frequency component of the doublet to the ordinary longitudinal acoustic (LA) +mode. +2) From moderate to high Q values, the LA branch acquires a negative Q slope, bending +downward to some minimum at about 17 nm−1, in agreement with what was previously +observed by the molecular dynamic simulation by Criado and collaborators [45]. Conversely, +8 + +FIG. 2. Phonon branches of polycrystalline ice and corresponding sound dispersions +in liquid water. Panel a: The plot displays the following dispersion branches: the transverse +acoustic (TA) mode (blue squares); the longitudinal acoustic (LA) mode (magenta dots); the +high-frequency mode (HFM, wine dots); the lower and higher energy optical modes OL and OH +(grey and orange lozenges, respectively. Error bars in the dispersion curves are estimated through +the standard deviations of the corresponding posterior distribution functions. The lines refer to +literature results and are colored in black or in red for ice and liquid water respectively. Specifically, +they represent the longitudinal mode of liquid water at 263 K and 2 kbar [34] (solid red line); the +transverse mode of room temperature D2O [31] (dashed red line) and the LA mode of polycrystalline +ice Ref. [45] (black dash-dotted line). The horizontal red arrow points to the optical frequency +measured by optical spectroscopy [46]. +Panel b: The phonon Density of States measured by +neutron scattering on single hexagonal ice ([47], blue line). Horizontal arrows are guides to the eye +showing the rough correspondence between DoS features and dispersion curves extrema. +9 + +b +a +28 +HFM +24- +20- +6 +(me) +0 +12 +16 +4 +8 +20 +0 +0,002 +Q (nm"l) +DoS (meV-1)at about the same Q value, the HFM reaches its maximum at about 25 meV. Noticeably +the plot suggests that the HFM branch is the one most closely resembling the longitudinal +acoustic dispersion of liquid water, which is also reported in Fig. 2a, as measured by IXS at +263 K and 2 kbar [34]. +4) Noticeably, two additional weakly dispersing modes show up in the spectrum inside +the 7-12 meV, Q > 10 nm−1 window: a lower and a higher optical mode here labeled +as OL and OH, respectively. We assign an optic origin to these branches owing to their +mild Q dependence and seemingly non-vanishing low Q trend, which distinguishes them +from the transverse acoustic mode, sitting at an even lower frequency. The existence of +multiple phonon branches at similar energies was reported in a computational study by +Criado and collaborators on polycrystalline ice [45] and by previous works in the single +crystal at ambient [48], or higher pressures [51]. Also, present results are consistent with +Density of State measurements by incoherent INS [52, 53] and optical spectroscopy [46]. In +Fig. 2a, a horizontal arrow points to the Q = 0 (energy axis) value corresponding to one of +the optical mode energies reported by the latter work. Revisited a posteriori in the light of +current results, the optical-like behavior of medium Q phonons of ice is also consistent with +the findings of coherent scattering measurements by INS [31, 48] and IXS [2, 54]. +Fig. 2a suggests that a proper discernment of two distinct modes, LA and HFM, may +be challenging at some Q values. However, the inference of a mode doublet with such a +narrow energy separation is pondered against a precise estimation of a joint probability +distribution. The involved energies are so close that the modes partially overlap, giving +origin to a single peak yet still significantly broader than the instrumental energy resolution +and somewhat flat atop as expected for an excitation composed by multiple spectral modes +of similar amplitude. According to the probability rating of the Bayesian algorithm, the +most plausible number of DHO modes participating to such an excitation is two. It would +be tempting to compare the inelastic modes of the considered polycrystal to its single-crystal +counterpart thoroughly studied in the literature [48, 55]. However, this comparison would +be problematic since, as mentioned, excitations of our sample are averaged over the various +crystallographic directions as opposed to the single-crystal case. For instance, the HFM +dispersion curve might be contaminated by higher energy optical modes especially in the +apex region. In this perspective, the single-crystal phonon Density of States (DoS), being a +wavevector-averaged (non-directional) property, provides a better quantity to compare with. +10 + +For reference, we included this spectral function in Fig. 2b as measured by neutron scattering +on single crystal Ih [47]. The horizontal arrows serve as guides to the eye, highlighting the +overall consistency between relevant DoS features and the extremes of dispersion curves +derived in the current measurement. +Looking at the sound dispersions drawn in Fig. 2a, one may notice the disappearance of +the transverse acoustic (TA) branch for Q > 14 nm−1. We tend to rule out any physical +rationale behind this trend, instead blaming the tendency of several inelastic features to +cram into a narrow energy window, thus challenging the algorithm’s ability to distinguish +them, especially those having a faint spectral fingerprint. In fact at Q =21 nm−1 there might +be a hint for the presence of the transverse mode (see Fig. S1 and S2 of the SM) but, in +our opinion, not with enough experimental evidence. Very likely at this Q this mode is still +overwhelmed by the near more intense optic modes which makes the detection uncertain. +A. +Comparing polycrystalline ice and liquid water +Fig. 3 compares, in a more restricted intensity and energy window, the phonon lineshapes +of ice with the low energy mode we measured in a previous work on liquid water [56]. This +comparison becomes especially informative when considered in combination with results in +Fig. 2. +Before digging further into this aspect, we recall here that the emergence of the low energy +mode in the spectrum of water has been the focus of intensive experimental [31–35], and +computational [35, 57] studies, which, despite some controversy [58, 59], endorsed the now +broadly accepted assignment to a transverse acoustic excitation. However, results in Fig. 2 +and Fig. 3 call for a reconsideration of this assignment, which, ironically enough, was origi- +nally inspired right by the comparison with the ice spectrum [2, 7]. Indeed, Fig. 2 suggests +that the transverse mode of water does coincide with the TA branch of ice at the lowest Q’s +only. Conversely, upon Q-increase, it seems to mimic the lower energy optical phonon of ice, +while acquiring, in turn, a weaker Q dependence. This conclusion is further endorsed by Fig. +3, which shows that the low-energy spectral feature of water closely resembles its transverse +acoustic counterpart in ice for Q ≤ 9 nm−1 only. Conversely, at larger Q’s, it rather parallels +the optical phonons dominating the ice spectrum, even though the latter, owing to the lack +of damping, have visibly higher inelastic shift. These evidences urge us to infer that col- +11 + +lective modes in water correspondingly acquire a dominating optic-like character in this Q +region. On a more general perspective, the above arguments might seem to question the very +nature of a collective mode in a fluid, as they imply that such a mode can actually combine +together independent phonon vibrations. Moreover, the damping enhancement stemming +from diffusive and relaxation processes can make even fuzzier the physical interpretation of +the inelastic mode of a liquid. However, the scenario is not as bewildering for water, at +least when its spectrum is compared to that of its solid-state, ordered, counterpart. In fact, +although vibrational modes of different nature seem to participate in the low energy mode +of water, they dominate complementary Q windows: optical modes become preponderant +upon Q increase while (transverse) acoustic ones follow just the opposite trend. From this +perspective, one is authorized to conclude that, upon Q increase, the low frequency bump +in the water spectrum changes its dominant character from primarily acoustic to mainly +optical. +We believe that this finding is of high relevance since, thus far, no experimental or +computational evidence has ever challenged the consensual assignment of such a spectral +feature to an acoustic phonon-like (transverse) mode. One of the most visible signs of this +acoustic-to-optic transformation is the gradually vanishing group velocity vg = ∂Ω(Q)/∂Q +(with Ω(Q) being the phonon frequency), which reveals an underlying mode localization. +This conclusion rests on the assumption that similar vibration modes are shared by water +and ice, although only in ice they can be properly resolved due to their sharp profile and +relatively tiny energy offset. We finally notice that, even in ice, in some Q window, low energy +modes tend to overcrowd the quasielastic region of the spectrum, thus challenging the ability +of the Bayesian algorithm to properly identify their presence. This “mode overcrowding” +combined with the overwhelming intensity of adjacent optical modes, is likely, as mentioned, +the reason of an apparent disappearance at a certain Q value of the TA phonon (see Fig. +2a). +Let us now comment on the comparison between current measurements on ice and those +on liquid water reported in Ref. [56] and also included in Fig. 4 after arbitrary vertical shift. +Both the HFM mode of ice and its counterpart in water weaken upon Q-increase, down to +almost vanish at high Q’s. In ice the two optical branches, OL and OH, essentially take it over +in the inelastic wings. Notice that the naked eye’s perception of this mode in the 21 nm−1 +spectrum of ice largely owes to the semilogarithmic representation. This strong suppression +12 + +FIG. 3. Comparison of low and intermediate frequency phonon modes in polycrys- +talline ice and in pure water. Selected spectra measured in this work at the indicated Qs are +here displayed in expanded energy transfer and intensity windows. Experimental spectra in ice +(black circles) are compared with best-fitting model profiles (red line), the transverse low energy +component (blue line), and the two optical-like ones (grey and orange lines), whenever present. +The thick olive green line represents the low energy mode in pure water at 300 K as obtained +through the lineshape analysis discussed in Ref. [56]. +13 + +2000 +300 +nm +1500 +200 +1000- +100 +500. +units +nm +arbitrary +200 +200 +100 +100 +0=13nm +100. +200 +50 +100 +-10 +0 +10 +-10 +0 +10 +E(meV)-20 +-10 +0 +10 +20 +30 +10 +100 +1000 + + + + + Q = 11 nm +-1 +a) +-20 +-10 +0 +10 +20 +10 +100 +1000 +10000 + +b) +Q = 21 nm +-1 +I(Q,E) (arbitrary units) +E (meV) +FIG. 4. High-frequency dynamics in polycrystalline ice and in water. IXS spectra (black +circles) measured at two selected Q values are reported along with corresponding best-fitting curves +(red lines) and both the HFM (wine line) and the LA (magenta line) energy model (DHO) com- +ponents. With the orange and grey lines the optical-like modes are reported as in the previous +figures. The corresponding IXS spectra of pure water measured in Ref. [56] are also shown (blue +circles) after arbitrary vertical shift. The two vertical arrows point at the position of the transverse +mode energy in pure water. +14 + +of the HFM is also witnessed by the rapid increase of the damping parameter with increasing +Q (Fig. S4 in the Supplemental Materials). In water, such an effect is even more pronounced, +due to the enhanced damping which makes this mode barely distinguishable (if at all) +from the large inelastic wings of the dominating transverse mode, whose position is roughly +indicated by the two vertical arrows (see also Fig. 2 of Ref. [56]). Although the naked eye +can barely discern the HFM mode in the 21 nm−1 spectrum of Fig. 4, the Bayesian algorithm +endorsed without ambiguity its presence. One can quantify the statistical significance of this +excitation through the high probability rating of a lineshape model explicitly containing it +(66 %). Nonetheless, the posterior of the related excitation frequency (see Fig. S2 of the +Supplemental Information) is quite broad and weak, thus revealing the uncertain location of +this otherwise genuine phonon excitation. Of course, the significance of a poorly discernible +spectral feature should be pondered against all diagnostic factors available, mainly two in +the present case: the probabilistic support of the Bayesian inference and the continuity +with the trend followed by neighboring Q-points. Both indicators persuaded us about the +presence of a loose HFM feature at about 23 meV in the Q = 21 nm−1 spectrum. +B. +The key role of Bayesian analysis +At this stage, it is crucial to understand how the reported result fit into the broader con- +text of available literature data, and, in particular, why the outcome of this study brings to +the surface a more complex phonon behavior than previously observed by individual works. +Aside from some improvement in the statistical accuracy in the currently measured spectra, +the use of Bayesian analysis here provides the critical asset of a minimally biased modelling +of measured spectra. In this approach, the model itself is entrusted to establish on a prob- +abilistic basis the most plausible number of phonon modes in the measured spectrum. This +provides a natural protection against two somewhat opposite risks: the confirmation bias, +holding off the discovery of new spectral excitations, and the model over parametrization, as +would be the inclusion of too many independent excitations in the lineshape model. Bayesian +inference endows researchers with these antidotes also thanks to the intrinsic implementa- +tion of the Occam razor principle, or ”lex parsimoniae”, prescribing that, among equally +plausible competing models, the simplest, i.e., the one with a smaller number of free pa- +rameters, is always to be privileged [60]. Obviously, this aspect can become a game changer +15 + +when facing inherent ambiguities of a measured spectral line, perhaps exacerbated by a less +than adequate statistical accuracy. As an example we can consider the observation of an +anomalously wide spectral line, to be alternatively interpreted as a genuine line broadening, +secondary to an increased excitation lifetime, or the emergence of a new line, indicating the +onset of a mode-splitting. Of course, Bayesian analysis only identifies the probabilistically +most grounded hypothesis based on the measurement obtained, yet it cannot substitute in- +vestigators in difficult assessment of its physical plausibility. Further details on the Bayesian +inference method, model choice and results can be found in the Supplemental Materials , +specifically in Figs. S1, S2, S3 and S5. +IV. +CONCLUSION +We have presented an Inelastic X-Ray Scattering study of the terahertz dynamic response +of water, in which new measurements in the solid phase are compared to our previous +results in the liquid. In this work, a Bayesian inference based analysis of measured spectra +represented a pivotal tool to unveil the full complexity of the phonon response of ice, thereby +improving, via direct comparison, our understanding of the water’s one. +In summary, our data show that, on approaching microscopic scales, high energy modes +becomes increasingly impeded. This hindrance is reflected, on one side, by the rapid increase +of the longitudinal acoustic damping of the highest frequency mode, and, on the other, by a +relative attenuation compared to optic modes which gradually become dominant in adjacent +spectral windows. Indeed, by observing how measured spectra and dispersion curves derived +from them compare in liquid and solid phases, we are urged to conclude that optic terahertz +modes dominate the spectrum of density fluctuations of both water and ice at sufficiently +short distances. This finding is especially noteworthy as clearly at variance with our long- +lasting assumption of a dominating acoustic character of collective modes of water. Despite +this evidence, our results also stimulate the scientific community with new interpretative +challenges. Likewise, the most urgent one is whether the observed behavior is distinctive or +shared with other networked fluids. A dedicated experimental and computational effort is +planned to elucidate this aspect hopefully improving our understanding of the high-frequency +response of disordered systems. +16 + +ACKNOWLEDGMENTS +This research was funded by Ministero dell’Istruzione dell’Universit`a e della Ricerca Ital- +iano (Grant No. PRIN2017-2017Z55KCW). +[1] R. Christensen, Theory of viscoelasticity: an introduction (Elsevier, 2012). +[2] G. Ruocco, F. Sette, U. Bergmann, M. Krisch, C. Masciovecchio, V. Mazzacurati, G. Signorelli, +and R. 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Cun- +solo, Interpreting the terahertz spectrum of complex materials: the unique contribution of the +Bayesian analysis, Materials 12, 2914 (2019). +22 + +Supplemental Materials: Ice phonon spectra and Bayes +inference: a gateway to a new understanding of terahertz sound +propagation in water +I. +EXPERIMENTAL DETAILS AND BAYESIAN APPROACH +To adequately reproduce the highly structured spectrum of density fluctuations of ice, i.e., +the dynamic structure factor S(Q, E), we considered minimally invasive hypothesis on its +shape, which was assumed to include the superposition of an unknown number of elementary +excitations. Namely: +S(Q, E) = Ae(Q)δ(E) + [n(E) + 1] E +kBT +� +k +� +j=1 +Aj(Q)DHOj(Q, E) +� +, +(S1) +here δ(E) is the Dirac delta function of E describing the elastic response and having integral +Ae and n(E) = (eE/kBT −1)−1 is the Bose statistics factor accounting for the detailed balance +condition. The k phonon excitations contributing to the spectrum are approximated by +Damped Harmonic Oscillator (DHOj(Q, E)) multiplied by intensity factors Aj(Q) where: +DHOj(Q, E) = 2 +π +Ω2 +j(Q)Γj(Q) +(E2 − Ω2 +j(Q))2 + 4[EΓj(Q)]2, +(S2) +and where Ωj(Q) and Γj(Q) are the undamped energy and the damping coefficient of the jth +DHO excitation. We stress again that the number k of the DHOj(Q, E) excitations and their +shape coefficients are equally treated as adjustable parameters of the model. To describe +accurately the measured spectrum, the model function in Eq. (S1) must be convoluted with +the instrument resolution function R(Q, E) and a spectral background should also be added +to such a convolution. Explicitly: +˜S(Q, E) = R(Q, E) ⊗ S(Q, E) + B(E), +(S3) +where B(E) is a typically mildly E-dependent background intensity. A Bayesian inferential +procedure can be applied to determine, based on the measurements and on our prior knowl- +edge of the physical problem under investigation, the most plausible shape, intensity and +background parameters entering in Eq. (S3) through Eqs. (S1) and (S2). +1 + +Best-fitting model lineshapes were determined using a Bayesian inferential analysis im- +plemented through a Markov Chain Monte Carlo (MCMC) [61] routine with reversible jump +(RJ) [62] steps. This approach can be used to probabilistically infer, based on a given mea- +surement, the joint posterior probability distribution, or, in short, the posterior, of each +parameter of the model. Notice that in the present case a prior uniform distribution for +the number k of inelastic modes has been deliberately chosen. In this respect the Bayesian +procedure can be used to probabilistically “rate” a guess on the number of excitations in +the sample scattering signal. In general, the knowledge of the entire posterior distribution +of each parameter also authorizes the interpretation of the maximum of such a posterior as +the optimal (most plausible, or best-fitting) value of such a parameter. This assignment is +possible provided the posterior, albeit not necessarily symmetric, is sharply peaked, well- +shaped and unimodal. The shape itself of the posterior carries important information on the +precision and the likelihood of a given best-fitting value. Foundations and working principle +of the performed Bayesian analysis are discussed in great detail in Ref. [40, 42, 63, 64]. +II. +COMPARING MEASURED AND MODEL SPECTRAL SHAPES +The posterior distributions delivered by the MCMC-RJ algorithm (see main text) for +energy shifts and damping coefficients of the Damped Harmonic Oscillator (DHO) model +components are displayed in Figs. S1 and S2, respectively, as an example at Q= 21 nm−1. +The plots demonstrate how the Bayesian approach can draw the entire probability distribu- +tion for all model parameters, rather than providing best-fitting values only, as in a more +traditional χ2-minimization approach. From Fig. S1, it readily appears that the majority +of posteriors drawn for the DHO energy shifts are very sharp, unimodal, and symmetric +about their maxima, their mode thus yielding a straightforward and unambiguous estimate +of the best-fitting parameter value. However, as illustrated in Fig. S2, the scenario is not +always as rosy for the damping coefficients, i.e., the width of the DHO model profiles; this +partially stems from the circumstance that is usually more challenging to determine reliably +the widths than the corresponding peak positions. In general, the width of the posterior +distribution provides a direct measure of the uncertainty affecting the estimate of an optimal +parameter value. This uncertainty can be secondary to the poor statistical accuracy of data, +or can reveal a trend physically significant. For instance, a large posterior width for a mode +2 + +frequency can indicate a gradual approach to a mode overdamping regime. +To provide a meaningful example, we can consider the Q increasing damping of the HFM, +illustrated in Fig. S4. As discussed in the main manuscript in further detail, such a high- +energy phonon gradually damps off as Q increases. This trend is also mirrored by the shape +of the posterior distribution of both frequency and damping of such mode, as clearly emerges +from considering the posteriors drawn for the highest energy phonon mode in Figs. S1 and +S2. These relatively unstructured posteriors hamper a precise determination of the optimal +parameter value. +In the specific case, this difficulty is exacerbated by the finite energy +window of the available experimental data, which in some cases prevented the measurement +to fully capture the position of the tails of the highest energy phonon mode. +FIG. S1. Posterior distribution functions for the phonon mode frequencies in polycrys- +talline ice. +The plot displays the posterior distribution functions conditional on the experimental +data y for the phonon mode frequency (phonon peak position) in the spectrum of polycrystalline +ice at 225 K and wavevector transfer Q = 21 nm−1 as obtained by the MCMC-RJ algorithm (see +main text). Color code is as in the main text. +3 + +5 +4 +3 +['u)d +2 +0 +0 +5 +10 +15 +20 +25 +30 +Qi (meV) j-1...5FIG. S2. +Posterior distribution function for the collective excitations dampings in +pure polycrystalline ice. In the four panels we report the posterior distribution drawn based on +experimental data y for the damping coefficient of the five phonon modes identified in the spectrum +of pure ice at Q = 21 nm−1. +4 + +3 +3 +32 +P(TTAl +0 +0 +1 +2 +3 +4 +5 +0 +2 +3 +4 +5 +ITA (meV) +FOL,OH +(meV) +3 +3 +0 +0 +1 +2 +3 +4 +5 +0 +5 +10 +TLA +(meV) +FHFM (meV)FIG. S3. Posterior distribution function for the excitation frequencies modes in ice at +T = 225 K. In the left column are reported the posterior distribution functions for the excitation +frequencies modes revealed on ice at four selected Q values. The blue, grey, orange, magenta and +wine colors indicate the distributions for the transverse mode, the low frequency optical mode, the +high frequency optical mode, the longitudinal center of mass acoustic mode and the oxygen high +frequency mode, respectively. In the right column the corresponding traceplots are reported with +the same color code. +5 + +20 +5 +t-uu 6 = 0 +10 +0 +0 +0. +5 +10 +15 +20 +25 +30 +0 +2 +4 +6 +8 +X104 +5 +Q = 11 nm-1 +20 +10 +0 +(meV) +0 +P(2;ly) +0 +5 +10 +15 +20 +25 +30 +0 +2 +4 +6 +8 +×104 +30 +5 +Q = 15 nm-1 +5 +20 +10 +5 +10 +15 +20 +25 +30 +0 +0 +0 +2 +4 +6 +8 +X103 +10 +30 +Q = 19 nm-1 +20 +5 +10 +0 +25 +30 +0 +0 +5 +10 +15 +20 +0 +2 +4 +6 +8 +2, (meV) +sweep +×104FIG. S4. Damping of the highest energy mode in pure ice at T = 225 K. Damping of +the HFM associated to the collective excitations of oxygen atoms as a function of the wavevector +transfer Q. +In Fig. S3 we give a graphical demonstration of how, depending on the value of the +wavevector Q, the number of modes that it is possible to recognize in the IXS spectra +changes as well as the greater or lesser “easiness” with which they are solved. According +to their dispersion curve and intensity, some modes may come out from the resolution, be +clearly visible, succumb to more intense ones, or, finally, move out the investigated energy +window. As in Fig. S1, in the left column of the figure we report the fairly well-shaped +posteriors of the excitation frequencies of the different modes. In the right column, we show +the corresponding traceplots for the revealed mode frequencies. These traceplots are one of +the possible graphical (and diagnostic) methods MCMC users adopt to decide with sound +theoretical support when the algorithm reaches convergence. The appearance of traceplots +provides a clear visualization of the efficiency of the Markov Chain in exploring the parameter +6 + +8. +6. + (meV) +4 +HFM +2 +0 +2 +6 +8 +4 +10 +12 +14 +16 +18 +20 +22 +Q (nm-lspace. Bayesian statisticians usually call the correct look of a traceplot a hairy caterpillar. +To provide an example of the uniqueness and the soundness of the model description +within the Bayes inference framework, in Fig. S5, we report the posterior probability distri- +bution of the number of excitations, conditional to the measurements obtained at different +Q’s. This distribution is represented by the histograms drawn by the Bayesian algorithm. +The distribution mode, i.e., the position of the tallest histogram rectangle, identifies the +optimal parameter value. Adopting a model with such an optimal value might appear a +non-unequivocal choice if competing suboptimal model options have comparable probability; +however, such a choice is the one probabilistically most grounded. Notice that suboptimal +solutions having a too small probability are not readily visible in the histograms, for this +reason, all probabilities are also reported in the numerical table here below. +FIG. S5. Posterior distribution function for the number of modes in ice at selected +momentum transfer values. Posterior distribution for the number of inelastic component k +in the spectra of polycrystalline ice at the indicated momentum transfer Q conditional to the +experimental data y. The values of such probabilities are reported also in the Table here below, in +fact too small probabilities are not detectable in the histograms. +7 + +Q=3nm-1 +0.5 +Q=5nm-1 +0.5 +0 +2 +6 +1 +3 +4 +5 +1 +2 +3 +5 +6 +0.5 +1-=0 +1-u6=0 +0.5 +2 +3 +5 +[ey)d +1 +2 +3 +4 +5 +0.5 +Q=13nm-1 +Q=15nm-1 +0.5 +2 +3 +5 +0 +1 +4 +1 +2 +3 +4 +5 +Q=19nm-1 +0.5 +Q=21nm-1 +0.5 +0 +1 +2 +4 +2 +3 +4 +k +kTABLE S1. +Posterior probabilities for the number of inelastic component in the spectra at the +indicated different Q values conditional to the experimental data y +k +Q (nm−1) +1 +2 +3 +4 +5 +3 +0 +0 +0.9995 +0.0005 +0 +5 +0 +0 +0.6883 +0.3095 +0.0022 +7 +0 +0 +0.6413 +0.3401 +0.0186 +9 +0 +0.9981 +0.0019 +0 +0 +11 +0 +0 +0.9225 +0.0775 +13 +0 +0 +0 +0.2741 +0.7259 +15 +0 +0 +0.9832 +0.0168 +0 +17 +0 +0 +0.9382 +0.0612 +0.0006 +19 +0 +0 +0 +1 +0 +21 +0 +0 +0 +0.3381 +0.6619 +8 + diff --git a/jNAzT4oBgHgl3EQfbfwg/content/tmp_files/load_file.txt b/jNAzT4oBgHgl3EQfbfwg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20e4d6b437eae36b97d367af7239e8097f0b6df0 --- /dev/null +++ b/jNAzT4oBgHgl3EQfbfwg/content/tmp_files/load_file.txt @@ -0,0 +1,806 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf,len=805 +page_content='Ice phonon spectra and Bayes inference: a gateway to a new understanding of terahertz sound propagation in water Alessio De Francesco,1, ∗ Luisa Scaccia,2 Ferdinando Formisano,1 Eleonora Guarini,3 Ubaldo Bafile,4 Ahmet Alatas,5 Scott T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Lynch,6 and Alessandro Cunsolo6 1CNR-IOM & INSIDE@ILL c/o Operative Group in Grenoble (OGG), F-38042 and Institut Laue Langevin, Grenoble, France 2Dipartimento di Economia e Diritto, Universit`a di Macerata, Via Crescimbeni 20, 62100 Macerata, Italy 3Dipartimento di Fisica e Astronomia, Universit`a di Firenze, via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Sansone 1, I-50019 Sesto Fiorentino, Italy 4Consiglio Nazionale delle Ricerche, Istituto di Fisica Applicata ”Nello Carrara”, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy 5Argonne National Laboratory, Advanced Photon Source, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Box 5000 Upton, 11973 NY, USA 6Department of Physics, University of Wisconsin at Madison, 1150 University Avenue, Madison, WI, USA 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='01385v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='mes-hall] 3 Jan 2023 Abstract Understanding how molecules engage in collective motions in a liquid where a network of bonds exists has both fundamental and applied relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' On the one hand, it can elucidate the “ordering” role of long-range correlations in an otherwise strongly dissipative system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' on the other hand, it can inspire new avenues to control such order to implement sound manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Water represents an ideal investigation case to unfold these general aspects and, across the decades, it has been the focus of thorough scrutiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Despite this investigative effort, the spectrum of terahertz density fluctuations of water largely remains a puzzle for Condensed Matter physicists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' To unravel it, we compare previous scattering measurements of water spectra with new ones on ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Thanks to the unique asset of Bayesian inference, we draw a more detailed portrayal of the phonon response of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The comparison with the one of liquid water challenges the current understanding of density fluctuations in water, or more in general, of any networked liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' INTRODUCTION It is a matter of common experience that liquids and solids oppose a different resistance to mechanical attempts to change their macroscopic shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' A possible way to test such a different rigidity is to induce a macroscopic mechanical perturbation generating the propaga- tion of a density wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' An alternative pathway followed in inelastic scattering measurements consists in stimulating the propagation of density fluctuations at mesoscopic scales by ir- radiating a material with a beam of particle waves, such as neutrons or photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' When dealing with a liquid system, such a density wave propagates with a strong damping and at macroscopic scales, uniquely in a direction parallel to the triggering force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In a structurally ordered solid such as a crystal, applied stresses propagate for a longer time and in directions both parallel and orthogonal to the applied force, in the form of longitudinal (LA) or trans- verse acoustic (TA) phonons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, this clear-cut distinction becomes more elusive over distances and time lapses respectively approaching the size of a single atom first neighbors cage and the period of such an atom’s “in cage” bounces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Indeed, abundant experimental evidence endorses the conclusion that density waves in liquids bear strong evidence for terahertz viscoelasticity [1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', they combine liquid-like ∗ Correspondence email address: defrance@ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='fr 2 and solid-like aspects [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' For instance, the transition between the viscous and the elastic regime in liquid water at about 280 K occurs at a few nm−1, according to the combined results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [6] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Relevant points of the broad picture emerging from the intensive scrutiny of terahertz viscoelasticity in fluids can be summarized as follows: 1) The first spectacular manifestation of mesoscale viscoelasticity is the increase of sound velocity when probing distances and times roughly as small as those featuring first-neighbor atomic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This trend reflects the enhanced rigidity of the system (elastic regime) at these so-called mesoscopic scales, typically much shorter than those covered by dissipative diffusion or relaxation - processes in liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2) The transition to such an elastic regime is paralleled by an increased ability of the liquid to support the propagation of transverse acoustic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 3) As opposed to a long-standing belief, this viscoelasticity persists in thermodynamic domains extending well above the critical point [8, 9], although their actual delimitation raised some controversy [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4) Aside from its unsettled dependence on thermodynamic conditions, high-frequency viscoelasticity also has a still poorly understood link with the nature of microscopic interac- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' for instance, it is unclear why it is substantially more pronounced in specific relatively low-viscosity systems as water [3, 14, 15], carbon dioxide [16], deuterium [17], and noble gases [18–20], while it is often barely detectable in liquid metals [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Far from being a mere academic endeavor, gaining insight on these topics could deepen our understanding of subjects as fundamental as the very nature of liquid aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Also, it will likewise disclose new avenues in the emerging domain of sound propagation design and engineering, a field whose enormous practical interest has been recognized [26], yet not fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' On the experimental side, terahertz phonon propagation in condensed materials can be directly probed by inelastic scattering methods, such as Inelastic X-Ray (IXS [27, 28]) and Neutron Scattering (INS, [29, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Conceptually, IXS or INS spectroscopic probes resemble 3 large microscopes “pointed on the dynamics”, which can be zoomed in to focus on dynamic events occurring over various scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This goal is ordinarily achieved by suitable tuning of the two main variables of the scattering event, namely energy and momentum exchanged between the probe wave particles (here assumed to be photons), respectively referred to as E = ℏω, and p = ℏQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Here, ℏ is the reduced Planck constant, while Q (ω) represents the exchanged wave-vector (frequency), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', the difference between before-scattering and after-scattering photon wave-vector (frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' For small enough values of Q = |Q| and E, the target system appears as a continuum whose dynamic response is probed as an average over many microscopic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Conversely, at extremely large (Q, E)’s, distances and times spanned become so small that the only event observed is the free recoil of the single struck atom after the collision with the photon and before subsequent first-neighbor interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Clearly, all dynamic processes characterizing the interatomic dynamics of the sample can be captured, somewhere between these two limits, by adequately tuning the spectroscopic probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Not surprisingly, the developments of the scattering techniques directly mapping this broad dynamic domain, INS and IXS, have dramatically improved the current understanding of high-frequency viscoelasticity phenomena in an impressively varied family of disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Although water is one of the disordered systems most thoroughly studied across various decades, its deep scrutiny has often evidenced seemingly anomalous and poorly understood behaviors, albeit sometimes later recognized as “normal effects” that water shares with many other liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Notable examples include the high propagation speed of the terahertz sound mode, once called “fast sound” [6, 7], and the onset of a shear mode propagation [31–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Although similar phenomena are being reported for an increasing number of disordered systems, the case of water found a straightforward and broadly accepted explanation in terms of hydrogen bond (HB) dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' According to this interpretative scheme, [36, 37] high frequency transverse and longitudinal acoustic waves in water respectively couple with the bending of two HBs linking triplets of oxygen atoms, or to the stretching of the HBs connecting two adjacent oxygens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Despite the remarkable insight achieved over the decades, further experimental effort is still needed to clarify a few more subtle aspects of water dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' These include the microscopic mechanism leading shear acoustic modes of water to exhibit a nearly flat Q dependence at the crossover between quasi-macroscopic and mesoscopic distances, while gradually becoming predominant over the longitudinal acoustic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4 From a merely experimental perspective, a natural pathway to understand better the collective movements of water molecules is to observe how they compare with those in the solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Indeed, the study of the acoustic response of (polycrystalline) ice presents significant simplifications: 1) both atomic diffusion and HB network relaxations are frozen, thus yielding no visible contribution to sound propagation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2) the lifetime of acoustic modes is much longer, and the related spectral features correspondingly sharper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Nonetheless, as illustrated in the remainder of this paper, the phonon response of ice is far from being trivially interpreted and characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In this paper, we discuss an experimental attempt to elucidate similarities and distinctive behaviors of the terahertz dynamics of ice and water by comparing their IXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Specifically, we jointly investigate collective excitations in water and phonon modes in polycrystalline hexagonal ice (Ih).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' On a rigorous ground, inelastic excitations in the spectrum of a polycrystal should not be referred to as phonons owing to their projection along multiple crystallographic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, in the following we will resort to this broadly used nomenclature for consistency with existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The use of a Bayesian inference-based modeling of IXS lineshapes reveals an unprece- dentedly complex phonon behavior of ice, while rectifying and complementing the current understanding of terahertz acoustic excitations in liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' THE MEASUREMENT Measurements were executed at the Sector 30 beamline [38, 39] of the Advanced Photon Source at Argonne National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The instrument was operated using the ≈ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='7 keV harmonic of the undulator source, which corresponds to the Si(12 12 12) backscattering reflection from both the monochromator and energy analyzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Spectral acquisitions covered the 3 ÷ 21 nm−1 Q range with a 2 nm−1 step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The instrumental resolution function was measured through the IXS signal from a Plexiglas sample at the Q of its first sharp diffrac- tion maximum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', about 10 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The resulting spectrum had a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='8 meV broad (half width at half maximum, HWHM) nearly Lorentzian profile, sufficient to properly resolve the relevant spectral features discussed in the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' All measurements were executed at a temperature of 225 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The polycrystalline nature of the sample makes its global orientation irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' As a consequence, the measurement probed the constant- Q phonon response of ice as an average over the various crystallographic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 5 1 illustrates the emergence of phonon excitations in the IXS spectra of ice by comparing raw and best-fitting model spectra for selected Q values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' the semilogarithmic plot enables a full appreciation of relevant lineshape features across various intensity decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Model line- shapes, obtained as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' I of Supplemental Materials, consist of a δ-function describing the elastic scattering and individual DHO terms accounting for inelastic excita- tions in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The most plausible number of DHO components was determined a posteriori, based on the outcome of the Bayesian inferential analysis of measured data, as described in our previous works [40–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Individual DHO components of the best-fitting total spectral distribution are also included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1 for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The assignment of each spectral contribution is summarized in the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1 and illustrated in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Polycrystalline Ice modes assignment Mode Label Assignment Symbol color 1 TA Transverse Acoustic blue 2 OL Lower Optical mode grey 3 OH Higher Optical mode orange 4 LA Longitudinal acoustic mode of center of mass magenta 5 HFM highest frequency steeply dispersive mode wine Notice that the same color code is maintained consistent throughout all graphs included in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The IXS spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1 are reported in the region where they have been modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This region, varying in extent and symmetry with respect to the central peak position, essentially coincided with the energy range covered by the corresponding energy resolution measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' One can readily notice that spectral shapes displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1 are dominated by multiple inelastic features, which, in some cases, partially overlap giving rise to inelastic peaks visibly broader than the resolution function, as particularly evident in the Q = 21 nm−1 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Overall, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1 provides a comprehensive rendering of the complexity of the phonon response of polycrystalline ice down to mesoscopic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The Q-evolution of various phonon modes will be discussed in further detail in the next Section, here we only stress that these modes display quite distinctive Q dependencies revealing their either acoustical or optical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Inelastic X-Ray Scattering (IXS) spectra of polycrystalline ice and their model lineshapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' IXS spectra measured at representative Q’s and T = 225 K discussed in this work (open circles) compared with corresponding best-fit lineshapes (red lines through data) along with their inelastic DHO model components (see text): the low frequency transverse mode (blue line), the two intermediate frequency optic-like modes (grey and orange lines), the longitudinal acoustic mode (magenta) and a higher frequency mode (wine line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The green line represents the instru- mental energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Error bars are estimated as the square root of the scattering intensity counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' DISCUSSION OF RESULTS The good consistency between the model and measured lineshapes clearly emerges from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Selected IXS spectra are compared with their best-fitting model lineshapes discussed in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' I of Supplemental Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Overall, phonon spectra presented in 7 10000 Q=3nm 100 20 10 0 10 20 20 10 0 10 20 10000 I(Q,E) (arbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' units) Q= 11nm Q= 17 nm 100 20 10 0 10 20 20 10 0 10 20 30 Q = 19 nm = 21 nm 100- 20 10 0 10 20 20 10 0 10 20 E (meV)this paper are qualitatively similar to those reported in the IXS measurement described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, the latter was performed on a polycrystalline sample seemingly more ordered than the present one, as suggested by the weaker elastic contribution, yet with a resolution significantly coarser, as to be expected considering the still pioneering stage of high-resolution IXS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Once assessed the ability of the model to reproduce the detail of the spectral shape, we can now focus on best-fit values of phonon frequencies, whose Q dependence and assignment are included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' As apparent from the dispersion curves displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2, the low-energy (< 25 meV) phonons of ice probed by this measurement show some rather peculiar features, previously either undetected [2, 48] or only partially reported by studies in the literature [33, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Specifically: 1) The longitudinal acoustic phonon of ice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', the one characterized by the sharpest (linear) low Q growth in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2a, actually splits into two branches, consistent with what was reported in a previous IXS work by Cimatoribus and collaborators [33], and previous lattice calculations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Most importantly, this trend might suggest reconsidering the surprising line broadening of the longitudinal acoustic mode of both water and polycrystalline ice reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The resolution limitation of such a pioneering IXS measurement is the most likely explanation for the mode-splitting appearing as a line-broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' At low enough Q’s, the two split branches attain similar values following a parallel Q-dependence while they spread apart at larger Q’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' To estimate the low wavevector propagation speed of these two modes, say c1 and c2, we looked at the slope of the three lowest Q points of the corresponding phonon branches (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' We found that the ratio of the sound speeds c1/c2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='07, a value consistent within the error to that of � MCM/MO = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='06 where MCM and MO are the masses of the water molecule and the oxygen atom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This finding might suggest that, in the linear dispersion regime, the modes reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2 as LA and HFM, might be connected to collective vibrations involving either the whole H2O molecule or the oxygen atoms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This is the main rationale behind the identification of the lower frequency component of the doublet to the ordinary longitudinal acoustic (LA) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2) From moderate to high Q values, the LA branch acquires a negative Q slope, bending downward to some minimum at about 17 nm−1, in agreement with what was previously observed by the molecular dynamic simulation by Criado and collaborators [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Conversely, 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Phonon branches of polycrystalline ice and corresponding sound dispersions in liquid water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Panel a: The plot displays the following dispersion branches: the transverse acoustic (TA) mode (blue squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' the longitudinal acoustic (LA) mode (magenta dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' the high-frequency mode (HFM, wine dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' the lower and higher energy optical modes OL and OH (grey and orange lozenges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Error bars in the dispersion curves are estimated through the standard deviations of the corresponding posterior distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The lines refer to literature results and are colored in black or in red for ice and liquid water respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Specifically, they represent the longitudinal mode of liquid water at 263 K and 2 kbar [34] (solid red line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' the transverse mode of room temperature D2O [31] (dashed red line) and the LA mode of polycrystalline ice Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [45] (black dash-dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The horizontal red arrow points to the optical frequency measured by optical spectroscopy [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Panel b: The phonon Density of States measured by neutron scattering on single hexagonal ice ([47], blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Horizontal arrows are guides to the eye showing the rough correspondence between DoS features and dispersion curves extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 9 b a 28 HFM 24- 20- 6 (me) 0 12 16 4 8 20 0 0,002 Q (nm"l) DoS (meV-1)at about the same Q value, the HFM reaches its maximum at about 25 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Noticeably the plot suggests that the HFM branch is the one most closely resembling the longitudinal acoustic dispersion of liquid water, which is also reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2a, as measured by IXS at 263 K and 2 kbar [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4) Noticeably, two additional weakly dispersing modes show up in the spectrum inside the 7-12 meV, Q > 10 nm−1 window: a lower and a higher optical mode here labeled as OL and OH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' We assign an optic origin to these branches owing to their mild Q dependence and seemingly non-vanishing low Q trend, which distinguishes them from the transverse acoustic mode, sitting at an even lower frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The existence of multiple phonon branches at similar energies was reported in a computational study by Criado and collaborators on polycrystalline ice [45] and by previous works in the single crystal at ambient [48], or higher pressures [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Also, present results are consistent with Density of State measurements by incoherent INS [52, 53] and optical spectroscopy [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2a, a horizontal arrow points to the Q = 0 (energy axis) value corresponding to one of the optical mode energies reported by the latter work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Revisited a posteriori in the light of current results, the optical-like behavior of medium Q phonons of ice is also consistent with the findings of coherent scattering measurements by INS [31, 48] and IXS [2, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2a suggests that a proper discernment of two distinct modes, LA and HFM, may be challenging at some Q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, the inference of a mode doublet with such a narrow energy separation is pondered against a precise estimation of a joint probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The involved energies are so close that the modes partially overlap, giving origin to a single peak yet still significantly broader than the instrumental energy resolution and somewhat flat atop as expected for an excitation composed by multiple spectral modes of similar amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' According to the probability rating of the Bayesian algorithm, the most plausible number of DHO modes participating to such an excitation is two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' It would be tempting to compare the inelastic modes of the considered polycrystal to its single-crystal counterpart thoroughly studied in the literature [48, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, this comparison would be problematic since, as mentioned, excitations of our sample are averaged over the various crystallographic directions as opposed to the single-crystal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' For instance, the HFM dispersion curve might be contaminated by higher energy optical modes especially in the apex region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In this perspective, the single-crystal phonon Density of States (DoS), being a wavevector-averaged (non-directional) property, provides a better quantity to compare with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 10 For reference, we included this spectral function in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2b as measured by neutron scattering on single crystal Ih [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The horizontal arrows serve as guides to the eye, highlighting the overall consistency between relevant DoS features and the extremes of dispersion curves derived in the current measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Looking at the sound dispersions drawn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2a, one may notice the disappearance of the transverse acoustic (TA) branch for Q > 14 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' We tend to rule out any physical rationale behind this trend, instead blaming the tendency of several inelastic features to cram into a narrow energy window, thus challenging the algorithm’s ability to distinguish them, especially those having a faint spectral fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In fact at Q =21 nm−1 there might be a hint for the presence of the transverse mode (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1 and S2 of the SM) but, in our opinion, not with enough experimental evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Very likely at this Q this mode is still overwhelmed by the near more intense optic modes which makes the detection uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Comparing polycrystalline ice and liquid water Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 3 compares, in a more restricted intensity and energy window, the phonon lineshapes of ice with the low energy mode we measured in a previous work on liquid water [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This comparison becomes especially informative when considered in combination with results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Before digging further into this aspect, we recall here that the emergence of the low energy mode in the spectrum of water has been the focus of intensive experimental [31–35], and computational [35, 57] studies, which, despite some controversy [58, 59], endorsed the now broadly accepted assignment to a transverse acoustic excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 3 call for a reconsideration of this assignment, which, ironically enough, was origi- nally inspired right by the comparison with the ice spectrum [2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2 suggests that the transverse mode of water does coincide with the TA branch of ice at the lowest Q’s only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Conversely, upon Q-increase, it seems to mimic the lower energy optical phonon of ice, while acquiring, in turn, a weaker Q dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This conclusion is further endorsed by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 3, which shows that the low-energy spectral feature of water closely resembles its transverse acoustic counterpart in ice for Q ≤ 9 nm−1 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Conversely, at larger Q’s, it rather parallels the optical phonons dominating the ice spectrum, even though the latter, owing to the lack of damping, have visibly higher inelastic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' These evidences urge us to infer that col- 11 lective modes in water correspondingly acquire a dominating optic-like character in this Q region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' On a more general perspective, the above arguments might seem to question the very nature of a collective mode in a fluid, as they imply that such a mode can actually combine together independent phonon vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Moreover, the damping enhancement stemming from diffusive and relaxation processes can make even fuzzier the physical interpretation of the inelastic mode of a liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, the scenario is not as bewildering for water, at least when its spectrum is compared to that of its solid-state, ordered, counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In fact, although vibrational modes of different nature seem to participate in the low energy mode of water, they dominate complementary Q windows: optical modes become preponderant upon Q increase while (transverse) acoustic ones follow just the opposite trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' From this perspective, one is authorized to conclude that, upon Q increase, the low frequency bump in the water spectrum changes its dominant character from primarily acoustic to mainly optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' We believe that this finding is of high relevance since, thus far, no experimental or computational evidence has ever challenged the consensual assignment of such a spectral feature to an acoustic phonon-like (transverse) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' One of the most visible signs of this acoustic-to-optic transformation is the gradually vanishing group velocity vg = ∂Ω(Q)/∂Q (with Ω(Q) being the phonon frequency), which reveals an underlying mode localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This conclusion rests on the assumption that similar vibration modes are shared by water and ice, although only in ice they can be properly resolved due to their sharp profile and relatively tiny energy offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' We finally notice that, even in ice, in some Q window, low energy modes tend to overcrowd the quasielastic region of the spectrum, thus challenging the ability of the Bayesian algorithm to properly identify their presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This “mode overcrowding” combined with the overwhelming intensity of adjacent optical modes, is likely, as mentioned, the reason of an apparent disappearance at a certain Q value of the TA phonon (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Let us now comment on the comparison between current measurements on ice and those on liquid water reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [56] and also included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4 after arbitrary vertical shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Both the HFM mode of ice and its counterpart in water weaken upon Q-increase, down to almost vanish at high Q’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In ice the two optical branches, OL and OH, essentially take it over in the inelastic wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Notice that the naked eye’s perception of this mode in the 21 nm−1 spectrum of ice largely owes to the semilogarithmic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This strong suppression 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Comparison of low and intermediate frequency phonon modes in polycrys- talline ice and in pure water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Selected spectra measured in this work at the indicated Qs are here displayed in expanded energy transfer and intensity windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Experimental spectra in ice (black circles) are compared with best-fitting model profiles (red line), the transverse low energy component (blue line), and the two optical-like ones (grey and orange lines), whenever present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The thick olive green line represents the low energy mode in pure water at 300 K as obtained through the lineshape analysis discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 13 2000 300 nm 1500 200 1000- 100 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' units nm arbitrary 200 200 100 100 0=13nm 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 200 50 100 10 0 10 10 0 10 E(meV)-20 10 0 10 20 30 10 100 1000 Q = 11 nm 1 a) 20 10 0 10 20 10 100 1000 10000 b) Q = 21 nm 1 I(Q,E) (arbitrary units) E (meV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' High-frequency dynamics in polycrystalline ice and in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' IXS spectra (black circles) measured at two selected Q values are reported along with corresponding best-fitting curves (red lines) and both the HFM (wine line) and the LA (magenta line) energy model (DHO) com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' With the orange and grey lines the optical-like modes are reported as in the previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The corresponding IXS spectra of pure water measured in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [56] are also shown (blue circles) after arbitrary vertical shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The two vertical arrows point at the position of the transverse mode energy in pure water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 14 of the HFM is also witnessed by the rapid increase of the damping parameter with increasing Q (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S4 in the Supplemental Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In water, such an effect is even more pronounced, due to the enhanced damping which makes this mode barely distinguishable (if at all) from the large inelastic wings of the dominating transverse mode, whose position is roughly indicated by the two vertical arrows (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Although the naked eye can barely discern the HFM mode in the 21 nm−1 spectrum of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4, the Bayesian algorithm endorsed without ambiguity its presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' One can quantify the statistical significance of this excitation through the high probability rating of a lineshape model explicitly containing it (66 %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Nonetheless, the posterior of the related excitation frequency (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S2 of the Supplemental Information) is quite broad and weak, thus revealing the uncertain location of this otherwise genuine phonon excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Of course, the significance of a poorly discernible spectral feature should be pondered against all diagnostic factors available, mainly two in the present case: the probabilistic support of the Bayesian inference and the continuity with the trend followed by neighboring Q-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Both indicators persuaded us about the presence of a loose HFM feature at about 23 meV in the Q = 21 nm−1 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The key role of Bayesian analysis At this stage, it is crucial to understand how the reported result fit into the broader con- text of available literature data, and, in particular, why the outcome of this study brings to the surface a more complex phonon behavior than previously observed by individual works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Aside from some improvement in the statistical accuracy in the currently measured spectra, the use of Bayesian analysis here provides the critical asset of a minimally biased modelling of measured spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In this approach, the model itself is entrusted to establish on a prob- abilistic basis the most plausible number of phonon modes in the measured spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This provides a natural protection against two somewhat opposite risks: the confirmation bias, holding off the discovery of new spectral excitations, and the model over parametrization, as would be the inclusion of too many independent excitations in the lineshape model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Bayesian inference endows researchers with these antidotes also thanks to the intrinsic implementa- tion of the Occam razor principle, or ”lex parsimoniae”, prescribing that, among equally plausible competing models, the simplest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', the one with a smaller number of free pa- rameters, is always to be privileged [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Obviously, this aspect can become a game changer 15 when facing inherent ambiguities of a measured spectral line, perhaps exacerbated by a less than adequate statistical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' As an example we can consider the observation of an anomalously wide spectral line, to be alternatively interpreted as a genuine line broadening, secondary to an increased excitation lifetime, or the emergence of a new line, indicating the onset of a mode-splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Of course, Bayesian analysis only identifies the probabilistically most grounded hypothesis based on the measurement obtained, yet it cannot substitute in- vestigators in difficult assessment of its physical plausibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Further details on the Bayesian inference method, model choice and results can be found in the Supplemental Materials , specifically in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1, S2, S3 and S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' CONCLUSION We have presented an Inelastic X-Ray Scattering study of the terahertz dynamic response of water, in which new measurements in the solid phase are compared to our previous results in the liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In this work, a Bayesian inference based analysis of measured spectra represented a pivotal tool to unveil the full complexity of the phonon response of ice, thereby improving, via direct comparison, our understanding of the water’s one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In summary, our data show that, on approaching microscopic scales, high energy modes becomes increasingly impeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This hindrance is reflected, on one side, by the rapid increase of the longitudinal acoustic damping of the highest frequency mode, and, on the other, by a relative attenuation compared to optic modes which gradually become dominant in adjacent spectral windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Indeed, by observing how measured spectra and dispersion curves derived from them compare in liquid and solid phases, we are urged to conclude that optic terahertz modes dominate the spectrum of density fluctuations of both water and ice at sufficiently short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This finding is especially noteworthy as clearly at variance with our long- lasting assumption of a dominating acoustic character of collective modes of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Despite this evidence, our results also stimulate the scientific community with new interpretative challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Likewise, the most urgent one is whether the observed behavior is distinctive or shared with other networked fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' A dedicated experimental and computational effort is planned to elucidate this aspect hopefully improving our understanding of the high-frequency response of disordered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 16 ACKNOWLEDGMENTS This research was funded by Ministero dell’Istruzione dell’Universit`a e della Ricerca Ital- iano (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' PRIN2017-2017Z55KCW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Christensen, Theory of viscoelasticity: an introduction (Elsevier, 2012).' metadata={'source': 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scatter- ing spectroscopy, in Inelastic X-Ray Scattering and X-Ray Powder Diffraction Applications, edited by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Cunsolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Franco, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Yokaichiya (IntechOpen, 2020) Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [64] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' De Francesco, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Scaccia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Maccarini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Formisano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Guarini, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Bafile, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Cun- solo, Interpreting the terahertz spectrum of complex materials: the unique contribution of the Bayesian analysis, Materials 12, 2914 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 22 Supplemental Materials: Ice phonon spectra and Bayes inference: a gateway to a new understanding of terahertz sound propagation in water I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' EXPERIMENTAL DETAILS AND BAYESIAN APPROACH To adequately reproduce the highly structured spectrum of density fluctuations of ice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', the dynamic structure factor S(Q, E), we considered minimally invasive hypothesis on its shape, which was assumed to include the superposition of an unknown number of elementary excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Namely: S(Q, E) = Ae(Q)δ(E) + [n(E) + 1] E kBT � k � j=1 Aj(Q)DHOj(Q, E) � , (S1) here δ(E) is the Dirac delta function of E describing the elastic response and having integral Ae and n(E) = (eE/kBT −1)−1 is the Bose statistics factor accounting for the detailed balance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The k phonon excitations contributing to the spectrum are approximated by Damped Harmonic Oscillator (DHOj(Q, E)) multiplied by intensity factors Aj(Q) where: DHOj(Q, E) = 2 π Ω2 j(Q)Γj(Q) (E2 − Ω2 j(Q))2 + 4[EΓj(Q)]2, (S2) and where Ωj(Q) and Γj(Q) are the undamped energy and the damping coefficient of the jth DHO excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' We stress again that the number k of the DHOj(Q, E) excitations and their shape coefficients are equally treated as adjustable parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' To describe accurately the measured spectrum, the model function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' (S1) must be convoluted with the instrument resolution function R(Q, E) and a spectral background should also be added to such a convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Explicitly: ˜S(Q, E) = R(Q, E) ⊗ S(Q, E) + B(E), (S3) where B(E) is a typically mildly E-dependent background intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' A Bayesian inferential procedure can be applied to determine, based on the measurements and on our prior knowl- edge of the physical problem under investigation, the most plausible shape, intensity and background parameters entering in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' (S3) through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' (S1) and (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 1 Best-fitting model lineshapes were determined using a Bayesian inferential analysis im- plemented through a Markov Chain Monte Carlo (MCMC) [61] routine with reversible jump (RJ) [62] steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This approach can be used to probabilistically infer, based on a given mea- surement, the joint posterior probability distribution, or, in short, the posterior, of each parameter of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Notice that in the present case a prior uniform distribution for the number k of inelastic modes has been deliberately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In this respect the Bayesian procedure can be used to probabilistically “rate” a guess on the number of excitations in the sample scattering signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In general, the knowledge of the entire posterior distribution of each parameter also authorizes the interpretation of the maximum of such a posterior as the optimal (most plausible, or best-fitting) value of such a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This assignment is possible provided the posterior, albeit not necessarily symmetric, is sharply peaked, well- shaped and unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The shape itself of the posterior carries important information on the precision and the likelihood of a given best-fitting value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Foundations and working principle of the performed Bayesian analysis are discussed in great detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' [40, 42, 63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' COMPARING MEASURED AND MODEL SPECTRAL SHAPES The posterior distributions delivered by the MCMC-RJ algorithm (see main text) for energy shifts and damping coefficients of the Damped Harmonic Oscillator (DHO) model components are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1 and S2, respectively, as an example at Q= 21 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The plots demonstrate how the Bayesian approach can draw the entire probability distribu- tion for all model parameters, rather than providing best-fitting values only, as in a more traditional χ2-minimization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1, it readily appears that the majority of posteriors drawn for the DHO energy shifts are very sharp, unimodal, and symmetric about their maxima, their mode thus yielding a straightforward and unambiguous estimate of the best-fitting parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' However, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S2, the scenario is not always as rosy for the damping coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', the width of the DHO model profiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' this partially stems from the circumstance that is usually more challenging to determine reliably the widths than the corresponding peak positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In general, the width of the posterior distribution provides a direct measure of the uncertainty affecting the estimate of an optimal parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This uncertainty can be secondary to the poor statistical accuracy of data, or can reveal a trend physically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' For instance, a large posterior width for a mode 2 frequency can indicate a gradual approach to a mode overdamping regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' To provide a meaningful example, we can consider the Q increasing damping of the HFM, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' As discussed in the main manuscript in further detail, such a high- energy phonon gradually damps off as Q increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This trend is also mirrored by the shape of the posterior distribution of both frequency and damping of such mode, as clearly emerges from considering the posteriors drawn for the highest energy phonon mode in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' These relatively unstructured posteriors hamper a precise determination of the optimal parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In the specific case, this difficulty is exacerbated by the finite energy window of the available experimental data, which in some cases prevented the measurement to fully capture the position of the tails of the highest energy phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Posterior distribution functions for the phonon mode frequencies in polycrys- talline ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The plot displays the posterior distribution functions conditional on the experimental data y for the phonon mode frequency (phonon peak position) in the spectrum of polycrystalline ice at 225 K and wavevector transfer Q = 21 nm−1 as obtained by the MCMC-RJ algorithm (see main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Color code is as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=" 3 5 4 3 ['u)d 2 0 0 5 10 15 20 25 30 Qi (meV) j-1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Posterior distribution function for the collective excitations dampings in pure polycrystalline ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In the four panels we report the posterior distribution drawn based on experimental data y for the damping coefficient of the five phonon modes identified in the spectrum of pure ice at Q = 21 nm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 4 3 3 32 P(TTAl 0 0 1 2 3 4 5 0 2 3 4 5 ITA (meV) FOL,OH (meV) 3 3 0 0 1 2 3 4 5 0 5 10 TLA (meV) FHFM (meV)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Posterior distribution function for the excitation frequencies modes in ice at T = 225 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In the left column are reported the posterior distribution functions for the excitation frequencies modes revealed on ice at four selected Q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The blue, grey, orange, magenta and wine colors indicate the distributions for the transverse mode, the low frequency optical mode, the high frequency optical mode, the longitudinal center of mass acoustic mode and the oxygen high frequency mode, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In the right column the corresponding traceplots are reported with the same color code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 5 20 5 t-uu 6 = 0 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 5 10 15 20 25 30 0 2 4 6 8 X104 5 Q = 11 nm-1 20 10 0 (meV) 0 P(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='ly) 0 5 10 15 20 25 30 0 2 4 6 8 ×104 30 5 Q = 15 nm-1 5 20 10 5 10 15 20 25 30 0 0 0 2 4 6 8 X103 10 30 Q = 19 nm-1 20 5 10 0 25 30 0 0 5 10 15 20 0 2 4 6 8 2, (meV) sweep ×104FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Damping of the highest energy mode in pure ice at T = 225 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Damping of the HFM associated to the collective excitations of oxygen atoms as a function of the wavevector transfer Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S3 we give a graphical demonstration of how, depending on the value of the wavevector Q, the number of modes that it is possible to recognize in the IXS spectra changes as well as the greater or lesser “easiness” with which they are solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' According to their dispersion curve and intensity, some modes may come out from the resolution, be clearly visible, succumb to more intense ones, or, finally, move out the investigated energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S1, in the left column of the figure we report the fairly well-shaped posteriors of the excitation frequencies of the different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' In the right column, we show the corresponding traceplots for the revealed mode frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' These traceplots are one of the possible graphical (and diagnostic) methods MCMC users adopt to decide with sound theoretical support when the algorithm reaches convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The appearance of traceplots provides a clear visualization of the efficiency of the Markov Chain in exploring the parameter 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' (meV) 4 HFM 2 0 2 6 8 4 10 12 14 16 18 20 22 Q (nm-lspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Bayesian statisticians usually call the correct look of a traceplot a hairy caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' To provide an example of the uniqueness and the soundness of the model description within the Bayes inference framework, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S5, we report the posterior probability distri- bution of the number of excitations, conditional to the measurements obtained at different Q’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' This distribution is represented by the histograms drawn by the Bayesian algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The distribution mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=', the position of the tallest histogram rectangle, identifies the optimal parameter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Adopting a model with such an optimal value might appear a non-unequivocal choice if competing suboptimal model options have comparable probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' however, such a choice is the one probabilistically most grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Notice that suboptimal solutions having a too small probability are not readily visible in the histograms, for this reason, all probabilities are also reported in the numerical table here below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Posterior distribution function for the number of modes in ice at selected momentum transfer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Posterior distribution for the number of inelastic component k in the spectra of polycrystalline ice at the indicated momentum transfer Q conditional to the experimental data y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' The values of such probabilities are reported also in the Table here below, in fact too small probabilities are not detectable in the histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' 7 Q=3nm-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 Q=5nm-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 0 2 6 1 3 4 5 1 2 3 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 1-=0 1-u6=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 2 3 5 [ey)d 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 Q=13nm-1 Q=15nm-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 2 3 5 0 1 4 1 2 3 4 5 Q=19nm-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 Q=21nm-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='5 0 1 2 4 2 3 4 k kTABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content=' Posterior probabilities for the number of inelastic component in the spectra at the indicated different Q values conditional to the experimental data y k Q (nm−1) 1 2 3 4 5 3 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='9995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='0005 0 5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='6883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='3095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='0022 7 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='6413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} +page_content='3401 0.' metadata={'source': 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+page_content='6619 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAzT4oBgHgl3EQfbfwg/content/2301.01385v1.pdf'} diff --git a/lNAyT4oBgHgl3EQfYfcF/content/tmp_files/2301.00202v1.pdf.txt b/lNAyT4oBgHgl3EQfYfcF/content/tmp_files/2301.00202v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9957d2f457ee208f1585a689c0206b30a1e9a659 --- /dev/null +++ b/lNAyT4oBgHgl3EQfYfcF/content/tmp_files/2301.00202v1.pdf.txt @@ -0,0 +1,2147 @@ +arXiv:2301.00202v1 [nucl-th] 31 Dec 2022 +A transport model description of Time-Dependent Generator Coordinate under +Gaussian overlap approximation +Fangyuan Wang,1 Yingxun Zhang,1, ∗ and Zhipan Li2 +1Department of Nuclear Physics, China Institute of Atomic Energy, Beijing 102413, People’s Republic of China +2School of Physical Science and Technology, Southwest University, Chongqing 400715, People’s Republic of China +(Dated: January 3, 2023) +In this work, we derived a transport equation based on a generalized equation of time-dependent +generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). The +transport equation is obtained by using quantum-mechanics phase space distributions under a +“quasi-particle” picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierar- +chy. The theoretical advantage of this transport equation is that time evolution of s-body phase +space density distribution is coupled with s + 1-body phase space density distributions, and thus, +non-adiabatic effects and dynamical fluctuations could be involved by more collective degrees and +entanglement of phase space trajectories. In future, we will perform the numerical calculations for +fission nuclei after obtaining collective inertia and potential energy surface (PES). +I. +INTRODUCTION +After the discovery of nuclear fission by Hahn and +Strassmann [1] in 1939, nuclear fission has become one +of the most challenging topics in physics since it is a key +ingredient for modeling nucleosynthesis [2], energy pro- +duction [3], medicine [4], and nuclear safeguard [5]. Even +with recent progress in experimental techniques, mea- +surements of nuclear fission are not possible for all fis- +sion nuclei. Thus, theoretical simulations are mandatory +for fully understanding the fission dynamics and comple- +menting the missing data [6–12]. +One kind of models is useful macroscopic model by +taking into account shell effects, collective variables, and +correlations between collective degree and single parti- +cles motions, such as Brownian shape motion +[13–15] +and Langevin model [16–18]. With four or five collective +degrees, it could depict fission dynamics process appro- +priately and reproduce fission yields distribution well. +Another kind of models is microscopic model, which +based on nucleonic Hamiltonian and solve fission dy- +namics with Schr¨odinger or Dirac equation in time +domain. +For example, time-dependent density func- +tional theory, such as time-dependent superfluid lo- +cal density approximation (TDSLDA) [19, 20], Con- +strained and time-dependent Hartree-Fock calculations +with dynamical Bardeen–Cooper–Schrieffer pairing cor- +relations (CHF+BCS) [21, 22], time dependent Hartree- +Fock-Bogliubov (TDHFB)/TDHF+BCS [23], the adia- +batic time-dependent HFB (ATDHFB) [24] and time- +dependent covariant density functional theory (TD- +CDFT) [25] describe fission process with full quantum +microscopic approaches. At tremendous costs of compu- +tations, these models successfully describe the fission dy- +namics and predict the most probable fission yields. The +great understanding of fission dynamics from microscopic +model are obtained [19, 22]. However, describing fission +∗ zhyx@ciae.ac.cn +yields distribution with these models is still a theoreti- +cal challenge due to the lacking of fluctuation in initial +state and fission process. The efforts on this direction is +to describe quantum fluctuation by a sampling of initial +conditions followed by TDDFT[26] but without quantum +interference. +An alternative method to include the correlation is +to represent a many-body wave function of the system +with a mixture of states with different shapes. It stim- +ulates the description of fission dynamics with time- +dependent generator coordinate method under Gaussian +overlap approximation (TDGCM+GOA) [27–29]. In the +TDGCM+GOA, fission is assumed as adiabatic process +since the typical time for the motion of individual nucle- +ons inside the fission nucleus (roughly 10−22 s) is roughly +ten times smaller than the time scale of the system’s col- +lective deformation (10−21 s) [29]. Thus, the fission dy- +namics are approximately described in terms of a few +shape coordinates. +Currently, most of the TDGCM+GOA calculations +were performed by using only two degrees of freedom, +usually (q20, q30) or (β2, β3), under adiabatic assump- +tion [30–32]. While, the semiphenomenological and fully +microscopic approaches illustrate that at least four or +five collective variables play a role in the dynamics of +fission [19, 33–37]. Regnier et al. [38] have started some +trials on the rigorous three degrees of freedom calculation +of the PES for 240Pu in the collective space (q20, q30, q40), +and their work is still in progress. In Ref. [39], Zhao et +al. did the calculations with the dynamical pairing de- +gree of freedom as the third degree of freedom besides +(β2, β3), and their results also demonstrate the impor- +tance of including more degree of freedom. Furthermore, +one should note that an ad hoc Gaussian smoothing have +to be used at the end of the TDGCM+GOA calculations, +to account for the fluctuations in particle number of the +fragments due to both pairing effects and the finite num- +ber of particles in the neck region for points along the +fission line. Thus, one would expect to develop a micro- +scopic method that can include more collective degrees + +2 +and the correlations to account for dynamical fluctua- +tion and non-adiabatic effects, and reasonably describing +fission dynamics and distribution of fission yields. +In this work, we derive a transport equation based on +a generalized N-dimensional TDGCM+GOA equation to +describe fission dynamics, in which non-adiabatic effects +are introduced by more collective degrees, fluctuations +are introduced by initial state fluctuation and entan- +glement of phase space trajectories. +The paper is or- +ganized as follows: in Sec.II, the transport equation is +obtained by using quantum-mechanics phase space dis- +tributions under a “quasi-particle” picture and strategy +of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hi- +erarchy. One of the advantages of this hierarchy is that +correlations from high-order degree can be involved in +the evolution of the one-body phase-space density dis- +tribution. In Sec.III, a numerical recipes for solving the +transport equation is provided. Sec.IV is the summary +and outlook. +II. +THEORY FRAMWORK +A. +Overview of TDGCM for fission +For convenience, we briefly review the TDGCM +GOA +theory which describes induced fission as a slow adiabatic +process determined by a small number of collective de- +grees of freedom. Under the Griffin-Hill-Wheeler ansatz, +the many-body state of fissioning system at any time +reads +|Ψ(t)⟩ = +� +q∈E +dq|φ(q)⟩f(q, t). +(1) +The set {|φ(q)⟩} is a family of the generator states +which are the solutions of a constrained Hartree-Fock- +Bogoliubov equation. +f(q, t) is the complex-valued +weights of the quantum mixture of states. The gener- +ator coordinate q = {q1, ..., qN}, and each of these qi is a +collective variable chosen based on the physics of fission. +The time-dependent Schr¨odinger equation for the +many-body state of fission system |Ψ(t)⟩, +( ˆH − iℏ d +dt)|Ψ(t)⟩ = 0, +(2) +can yield an equation of the unknown weight func- +tion f(q, t), i.e., the Hill-Wheeler equation with time- +dependent form, +� +dq′⟨φq| +� +ˆH − iℏ d +dt +� +|φq′⟩f(q′, t) = 0. +(3) +Here, ˆH is the Hamiltonian acting on the full many-body +system. Principally, Eq. (3) can be solved numerically, +but it needs a tremendous amount of computations. To +overcome these difficulties, a popular approach named +as Gaussian overlap approximation (GOA) is used. The +simplest formulation of GOA assumes that the overlap +between two generator states ⟨φq|φq′⟩ has a Gaussian +shape, +N(q, q′) = ⟨φq|φq′⟩ ≡ exp +� +−1 +2(q − q′)tG(¯q)(q − q′) +� +. +(4) +N(q, q′) = ⟨φq|φq′⟩ is peaked functions for q = q′, and +¯q = (q + q′)/2. By changing a new collective coordinate +α by the relation +α(q) = +� +a∈Cq +0 +G1/2(a)da, +(5) +in terms of which the overlap matrix becomes +N(α, α′) = exp +� +−1 +2(α − α′)2 +� +, +(6) +G (q) is the metric of new coordinates of α(q), and G (q) +is the determinant of G. +Within this approximation, the time-dependent Hill- +Wheeler equation is reduced to a local, time-dependent +Schr¨odinger-like equation as +iℏ∂g(q, t) +∂t += ˆHcoll(q)g(q, t). +(7) +g(q, t) is related to the weight function f(q, t) as g = +N 1/2f, and contains all the information about the fis- +sion dynamics of system [29]. The collective Hamiltonian +ˆHcoll(q) is a local operator acting on g(q, t), +ˆHcoll(q) = +(8) +� +−ℏ2 +2 +� +kl +1 +� +G (q) +∂ +∂qk +� +G (q)Bkl (q) ∂ +∂ql ++ V (q) +� +. +The potential V (q), +V (q) = ⟨q| ˆH|q⟩ − ǫ0(q), +(9) +with the zero-point energy-correction +ǫ0 = 1 +2Gij(q) +∂2h +∂qi∂q′j +���� +q=q′. +(10) +The symmetric collective inertial tensor B(q) ≡ Bij(q), +Bkl(q) = +1 +2ℏ2 Gkm(q) +�∂2h(q, q′) +∂qm∂q′n − ∂2h(q, q′) +∂qm∂qn ++ +� +i +mn +� ∂h(q, q′) +∂qi +����� +q=q′ +Gnl(q), +(11) +the expression in braces is the Christoffel symbol of the +second kind. h(q, q′) is +h(q, q′) = +� +φq| ˆH|φq′ +� +⟨φq|φq′⟩ +. +(12) + +3 +They are usually calculated from the nuclear Hamilto- +nian ˆH and the generator states |φq⟩ with HFB [30] or +RMF+BCS [40]. +The number of collective degree of freedom are usually +selected as N = 2, and shape coordinates q are the mul- +tipole moments Q20 and Q30 in Ref. [30–32], or β2 and +β3 as in Refs. [39, 40], and G (q) = 1 Ref. [31]. In this +case, the equation of TDGCM+GOA is +iℏ ∂ +∂tg (q1, q2; t) = +(13) +� +−ℏ2 +2 +� +kl +∂ +∂qk +Bkl (q1, q2) ∂ +∂ql ++ V (q1, q2) +� +g (q1, q2; t) . +This equation has been solved by the software package +FELIX-1.0 [31] or FELIX-2.0 [32] with finite element +method. +B. +A transport equation for N-dimensional +TDGCM+GOA +The TDGCM has achieved great progress on describ- +ing the fission dynamics [30–32, 39–41], but previous cal- +culations in Refs. [33–36, 38] also showed it is necessary +to include more degrees to describe nonadiabatic effects +which may arise from the coupling between collective and +intrinsic degrees of freedom, and invovle dynamical fluc- +tuations to describe fission products distributions. Now, +the question is that can we effectively involve more collec- +tive degrees of freedom into the equation with two degrees +that we are currently using? +Principally, nuclear shape can be described by an ex- +pansion in spherical harmonics, i.e., +R(θ, φ, t) = R0 + +1 + +N +� +λ=0 +λ +� +µ=−λ +α∗ +λµ(t)Yλµ(θ, φ) + + . +(14) +The number of shape coordinates (or collective degree +of freedom) N depends on the choice of collective co- +ordinates or generator coordinates and stage of fission. +In the stage of fissionning system from the quasista- +tionary initial state to the outer fission barrier, evolu- +tion is slow and fission process can be described by a +small number of collective degree, i.e., a small N, with +adiabatic approximation[42]. In the stage of saddle-to- +scission, the nucleus quickly elongates toward scission +and non-adiabatic effects have to be considered and N +may vary with the stage of fission process. Thus, the col- +lective wave function is presented with N degrees, i.e., +g(q1, q2, ..., qN), for fissioning system, and Eq.(7) will be- +come a generalized N-dimensional TDGCM+GOA equa- +tion since the degrees are not limited to a few. +In this work, +we interpret the wave function of +g(q1, q2, ..., qN) for fissioning system as a wave function +for ‘N-body quasiparticles in 1-Dimension space’ sys- +tem (NB1D), and a convention gN(q1, q2, ..., qN) is used +in following to represent the wave function with parti- +cle from 1 to N. Then, we derive a transport equation +which can effectively couple one more collective degree +of freedom to the degrees currently used. Firstly, we per- +form Wigner transformation [43] on NB1D wave function +gN (q1, · · · , qN) to get their quantum mechanically phase +space density fN as, +fN (q1, · · · , qN; p1, · · · , pN) +(15) += +1 +(πℏ)N +� +· · · +� +dy1 · · · dyNg∗ +N(q1 − y1, · · · , qN − yN) +gN(q1 + y1, · · · , qN + yN) +× exp[−2i(p1 · y1 + · · · + pN · yN)/ℏ]. +Here pi is the conjugated momentum of qi for quasi- +particle i. After a trivial deviation, the corresponding +transport equation reads, +∂fN +∂t += − +� +kl +¯B(b) +kl (q)pk +∂fN +∂ql +(16) ++ +� +λ=1,3,··· +( ℏ +2i)λ1+···+λN−1 +1 +λ1! · · · λN! +×∂λ1+···+λNVN (q) +∂qλ1 +1 · · · ∂qλN +N +∂λ1+···+λN fN +∂pλ1 +1 · · · ∂pλN +N += − +� +kl +¯B(b) +kl (q)pk +∂fN +∂ql ++ +� +l +∂VN +∂ql +∂fN +∂pl ++ +� +λ=3,··· +( ℏ +2i)λ1+···+λN−1 +1 +λ1! · · · λN! +×∂λ1+···+λNVN (q) +∂qλ1 +1 · · · ∂qλN +N +∂λ1+···+λN fN +∂pλ1 +1 · · · ∂pλN +N +. +Here, λ = �N +i=1 λi and ¯B(b) +kl (q) is an effective collective +inertia, which is defined as +¯B(b) +kl (q) = +� +Bkl(q − y∗ +(1b)) + Bkl(q − y∗ +(2b)) +� +/2. +(17) +y∗ +(1b) and y∗ +(2b) are corrections on q, and their origins can +be found in appendix A. VN(q) is N-body potential. +Principally, VN(q) = V (q1, q2, · · · , qN). If the N-body +potential is calculated from the two-body interaction, i.e., +V (q1, q2, · · · , qN) = +� +i≤j +V (qi, qj), +(18) +the transport equation is simplified as, +∂fN +∂t += − +� +kl +¯B(b) +kl (q)pk +∂fN +∂ql ++ 1 +2 +� +k,m̸=k +∂Vkm +∂qk +∂fN +∂pk +.(19) +Vkm is two-body potential between quasi-particle k and +m, which can be obtained by HFB/RMF+BCS as in +Refs.[30, 40]. +When the N-body potential is obtained +from multi-dimensional PES by HFB/RMF+BCS, trans- +port equation of Eq.(16) should be used. + +4 +A standard procedure to solve the N-body transport +equation is to use BBGKY hierarchy, in which one-body +degrees of freedom (DOF) is coupled to two-body DOF +that are themselves coupled to three-body DOFs and +so forth. +As an example, we present the time evolu- +tion of fs under the condition of V (q1, q2, · · · , qN) = +� +i≤j V (qi, qj). The s-body phase space density distri- +bution fs is defined as, +fs(q1, · · · , qs, p1, · · · , ps) +(20) += +1 +ΩN−s +� +fN(q1, · · · , qN, p1, · · · , pN)dΓs+1 · · · dΓN, +dΓi = dqidpi, +here, Ω is volume in phase space. Thus, +∂fs(q1, · · · , qs, p1, · · · , ps) +∂t +(21) += +1 +ΩN−s +� ∂fN(q1, · · · , qN, p1, · · · , pN) +∂t +dΓs+1 · · · dΓN += +1 +ΩN−s +� � +− +� +kl +¯B(b) +kl (q)pk +∂fN +∂ql ++ +� +1≤kqsci +fs(qs, ps, t)dpsdqs +(44) += +1 +2Ntest +2 +� +k=1 +Ntest +� +i=1 +Θ(qki − ¯qki,sci(t)). +Thus, the yield of mass of fragment +Y (A) = +� +S +Y (A, S). +(45) +The sum on S runs over the whole scission hypersurface. +IV. +SUMMARY AND OUTLOOK +In this work, we derive a transport equation based +on a generalized N-dimensional TDGCM+GOA equation +to describe fission dynamics. The transport equation is +obtained by using quantum-mechanics phase space dis- +tributions under a “quasi-particle” picture and strategy +of BBGKY hierarchy. The advantages of this transport +equation is that time evolution of s-body phase space +density distribution is coupled with s + 1-body phase +space density distributions. Thus, one can expect that +non-adiabatic effects and dynamical fluctuations could +be introduced by involving more collective degrees and +entanglement of phase space trajectories. +Different than directly solving the TDGCM+GOA +equation with finite elements method, our approach is +realized by using the test particle method. The coordi- +nates and momentum of test particles in initialization of +fissioning system are sampled according to initial phase +space density distribution of system. +The time evolu- +tion of test particles are governed by a Hamiltonian-like +equation coupled with a random scattering between test +particles, which will naturally provide a fluctuation on +the mass of fission fragments. Finally, the numerical re- +sults on this transport equations are still on the way, +since we need to select reasonable collective coordinates +to obtain the results of PES and collective inertia, and + +7 +the high order effects of degree of freedom on collective +inertia should also be investigated in this approach before +presenting numerical results. +ACKNOWLEDGEMENTS +The authors thank Zhuxia Li, Zaochun Gao and +Siyu Zhuo for reading the manuscript and providing +useful feedback. +This work was partly supported by +the National Natural Science Foundation of China Nos. +11875323, 12275359, 11875225, 11705163, 11790320, +11790323, and 11961141003, the National Key R&D Pro- +gram of China under Grant No. +2018 YFA0404404, +the Continuous Basic Scientific Research Project (No. +WDJC-2019-13, BJ20002501), Key Laboratory of Nu- +clear +Data +foundation +(No.JCKY2022201C158) and +the +funding +of +China +Institute +of +Atomic +Energy +YZ222407001301. +The Leading Innovation Project of +the CNNC under Grant No. +LC192209000701, No. +LC202309000201. +Appendix A: Derivation of transport equation +The equation of motion of gN is determined by the +Schr¨odinger-like equation, i.e., +iℏ ∂ +∂tgN (q, t) = +(A1) +� +−ℏ2 +2 +� +kl +∂ +∂qk +Bkl (q) ∂ +∂ql ++ VN (q) +� +gN (q, t) . +By using the Wigner transformation [43], fN(q, p) is ob- +tained. The equation of motion of fN(q, p) reads as, +∂fN(q, p, t) +∂t += +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y/ℏ +(A2) +�∂g∗ +N(q − y) +∂t +gN(q + y) + g∗ +N(q − y)∂gN(q + y) +∂t +� +. +where p is the conjugate momentum of q. +In the derivations of Eq. (A2), the ∂g∗ +N/∂t and ∂gN/∂t +are replaced with the right hand side of Eq. (A1) and we +have, +∂fN(q, p, t) +∂t += +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y/ℏ +(A3) +� � +kl +� +− iℏ +2 +∂ +∂(qk − yk) +� +Bkl (q − y) ∂g∗ +N (q − y) +∂(ql − yl) +� +gN(q + y) ++iℏ +2 g∗ +N(q − y) +∂ +∂(qk + yk) +� +Bkl (q + y) ∂gN (q + y) +∂(ql + yl) +�� ++ i +ℏ (V (q − y) − V (q + y)) g∗ +N(q − y)gN(q + y) +� +. +One should note that the kinetic energy term contains the +collective coordinate q dependence of inertia Bkl, which +is different than the N-body system with fixed particle +mass. +In coordinate space, the kinetic energy terms in +Eq. (A3) are as follows, +Mkl = −iℏ +2 +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y/ℏ +(A4) +� ∂ +∂yk +� +Bkl(q − y)∂g∗ +N(q − y) +∂yl +� +gN(q + y) +−g∗ +N(q − y) ∂ +∂yk +� +Bkl(q + y)∂gN(q + y) +∂yl +�� += Mkl,1 + Mkl,2. +Since the gN and g∗ +N depend on the integration vari- +able yk, one can replace the differentiations with respect +to qk +yk by differentiations with respect to yk in deriva- +tions. By doing one partial integration with respect to +yk, one has +Mkl,1 = +(A5) +−iℏ +2 +1 +(πℏ)N Bkl(q − y)∂g∗ +N(q − y) +∂yl +gN(q + y)e−2ip·y/ℏ +���� ++∞ +−∞ ++iℏ +2 +1 +(πℏ)N +� +dy1 · · · dyNBkl(q − y)∂g∗ +N(q − y) +∂yl +× ∂ +∂yk +� +gN(q + y)e−2ip·y/ℏ� +The first term in Eq. (A5) vanishes due to the boundary +condition at infinity, and second term becomes +Mkl,1 = iℏ +2 +1 +(πℏ)N +� +dy1 · · · dyN +(A6) +×Bkl(q − y)∂g∗ +N(q − y) +∂yl +∂ +∂yk +� +gN(q + y)e−2ip·y/ℏ� += iℏ +2 +1 +(πℏ)N +� +dy1 · · · dyNs +×Bkl(q − y)∂g∗ +N(q − y) +∂yl +�∂gN(q + y) +∂yk +e−2ip·y/ℏ ++gN(q + y)e−2ip·y/ℏ(−2ipk +ℏ ) +� += iℏ +2 Bkl(q − y∗ +(1a)) +1 +(πℏ)N +� +dy1 · · · dyN +(A7) +∂g∗ +N(q − y) +∂yl +∂gN(q + y) +∂yk +e−2ip·y/ℏ ++iℏ +2 Bkl(q − y∗ +(1b)) +1 +(πℏ)N +� +dy1 · · · dyN +∂g∗ +N(q − y) +∂yl +gN(q + y)e−2ip·y/ℏ(−2ipk +ℏ ). +In above derivations, the Bkl(q − y) is moved out by +assuming the following relationship, i.e., +� +Bkl(q − y)O(q, ∂/∂q)dy1 · · · dyN = +(A8) +Bkl(q − y∗ +(I)) +� +O(q, ∂/∂q)dy1 · · · dyN. + +8 +Similarly, the Mkl,2 becomes +Mkl,2 = −iℏ +2 Bkl(q + y∗ +(2a)) +1 +(πℏ)N +� +dy1 · · · dyN(A9) +∂g∗ +N(q − y) +∂yk +∂gN(q + y) +∂yl +e−2ip·y/ℏ +−iℏ +2 Bkl(q + y∗ +(2b)) +1 +(πℏ)N +� +dy1 · · · dyN +g∗ +N(q − y)∂gN(q + y) +∂yl +e−2ip·y/ℏ(−2ipk +ℏ ) +By defining ¯B(a) +kl , ¯B(b) +kl , δB(a) +kl and δB(b) +kl as, +¯B(a) +kl = +� +Bkl(q − y∗ +(1a)) + Bkl(q + y∗ +(2a)) +� +/2, +¯B(b) +kl = +� +Bkl(q − y∗ +(1b)) + Bkl(q + y∗ +(2b)) +� +/2, +δB(a) +kl = +� +Bkl(q − y∗ +(1a)) − Bkl(q + y∗ +(2a)) +� +, +δB(b) +kl = +� +Bkl(q − y∗ +(2a)) − Bkl(q + y∗ +(2b)) +� +, +and assuming δB(a) +kl ≈ 0 and δB(b) +kl ≈ 0, Mkl becomes +Mkl = Mkl,1 + Mkl,2 +(A10) += −iℏ +2 +¯B(a) +kl +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y/ℏ +�∂g∗ +N(q − y) +∂ql +∂gN(q + y) +∂qk +− ∂g∗ +N(q − y) +∂qk +∂gN(q + y) +∂ql +� +−iℏ +2 +¯B(b) +kl +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y/ℏ +�∂g∗ +N(q − y) +∂ql +g(q + y) − g∗ +N(q − y)∂gN(q + y) +∂ql +� +(−2ipk +ℏ ). +In Eq.(A10), an identical relationship between differen- +tial with respect to yk(yl) and with respect to qk(ql) is +used. +Due to the symmetric property of ¯B(a) +kl and summation +of � +kl in Eq.(A3), the contributions from first term in +Eq.(A10) vanishes. Thus, the kinetic part can be written +as, +Mkl = −pk ¯B(b) +kl +∂fN +∂ql +(A11) +For the potential part in Eq. (A3), a Taylor series with +respect to q is performed with potential fields VN (q + y) +and VN (q − y). +N = +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y/ℏ +(A12) +× i +ℏ [V (q − y) − V (q + y)] g∗ +N(q − y)gN(q + y) += +1 +(πℏ)N +� +dy1 · · · dyNe−2ip·y +×2i +ℏ +� +λ +1 +λ1! · · · λN! +∂λ1+···λN VN (q) +∂qλ1 +1 · · · ∂qλN +N +yλ1 +1 · · · yλN +N g∗ +N (q − y) gN (q + y) += 2i +ℏ +� +λ +( ℏ +2i)λ1+···+λN +1 +λ1! · · · λN! +×∂λ1+···+λNVN (q) +∂qλ1 +1 · · · ∂qλN +N +∂λ1+···+λN fN +∂pλ1 +1 · · · ∂pλN +N +Finally, Eq. (A3) could be written as +∂fN(q, p, t) +∂t += − +� +kl +pk ¯B(b) +kl (q)∂fN +∂ql +(A13) ++ +� +λ +( ℏ +2i)λ1+···+λN−1 +1 +λ1! · · · λN! +×∂λ1+···+λN VN (q) +∂qλ1 +1 · · · ∂qλN +N +∂λ1+···+λN fN +∂pλ1 +1 · · · ∂pλN +N += − +� +kl +pk ¯B(b) +kl (q)∂fN +∂ql ++ +N +� +l +∂V +∂ql +· ∂fN +∂pl ++ +� +λ +( ℏ +2i)λ1+···+λN−1 +1 +λ1! · · · λN! +×∂λ1+···+λN VN (q) +∂qλ1 +1 · · · ∂qλN +N +∂λ1+···+λN fN +∂pλ1 +1 · · · ∂pλN +N +.. +where all λ1, · · · , λN are non-negative integer values and +λ1 + · · · + λN is an odd number. This is a general form +of transport equation for TDGCM+GOA. +When the potential is calculated from the two-body +interaction, i.e., +V (q1, q2, · · · , qN) = +� +i≤j +V (qi, qj), +(A14) +the equation is further simplified as, +∂fN +∂t += − +� +kl +¯B(b) +kl (q)pk +∂fN +∂ql ++ +� +k≤m +∂Vkm +∂qk +∂fN +∂pk +. +(A15) + +9 +Appendix B: Time evolution of s-body phase space +density distribution +The s-body phase space density is defined as, +fs(q1, · · · , qs, p1, · · · , ps) +(B1) += +1 +ΩN−s +� +fN(q1, · · · , qN, p1, · · · , pN)dΓs+1 · · · dΓN, +dΓi = dqidpi. +When the potential is calculated from the two-body in- +teraction, the time-dependent probability distribution +fs is obtained by the similar strategy of the derivation +of the Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) +hierarchy, i.e., +∂fs(q1, · · · , qs, p1, · · · , ps) +∂t += +1 +ΩN−s +� +dΓs+1 · · · dΓN +∂fN(q1, · · · , qN, p1, · · · , pN) +∂t += +1 +ΩN−s +� +dΓs+1 · · · dΓN +� +− +� +kl +¯B(b) +kl (q)pk +∂fN +∂ql ++ +� +k≤m +∂Vkm +∂qk +∂fN +∂pk +� +. +(B2) +Different than the transport equation for fixed-mass +many-particle system, the inertia Bkl(q) depends on the +coordinate and it causes the equation much more com- +plexity. +To avoid the difficulty caused by ¯Bkl(q), we move the +¯Bkl out from the integration by using the equivalent of +the integration, i.e., +� +¯Bkl(q)O(q, p)dΓs+1 · · · dΓN = +(B3) +¯Bkl(qs, q∗ +s+1, · · · , q∗ +N) +� +O(q, p)dΓs+1 · · · dΓN. +Consequently, the derivation will be simplified. +For the first term in r.h.s of Eq. (B2), we perform one +partial integration with respect to ql and the result is +I1 = − +1 +ΩN−s +� � � +kl +¯B(b) +kl (q)pk +∂fN +∂ql +� +dΓs+1 · · · dΓN += − +� +kl +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N) +× +1 +ΩN−s +� � +pk +∂fN +∂ql +� +dΓs+1 · · · dΓN += − +s +� +k=1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N)pk +∂fs +∂ql +− +N +� +k=s+1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N) +× +1 +ΩN−s +� +pk +∂fN +∂ql +dΓs+1 · · · dΓN += − +s +� +k=1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N)pk +∂fs +∂ql +− δI1 +(B4) +In +above +derivation, +we +use +the +term +of +�N +l=s+1 ¯B(b) +kl (q)pkfN|ql→∞ +ql→−∞ += +0. +This is because +the finite values of fN, which means fN(q) = 0 at +q → ±∞. The δI1 is defined as, +δI1 = +N +� +k=s+1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N) +(B5) +· +1 +ΩN−s +� +pk +∂fN +∂ql +dΓs+1 · · · dΓN += +N +� +k=s+1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N)p∗ +k +∂fs +∂ql +which is connected to the N-body density distribution +fN, and reflects how the momentum field evolve with the +coordinates. +The second term in Eq. (B2) reads, +I3 = +1 +ΩN−s +� � +k≤m +∂Vkm +∂qk +∂fN +∂pk +dΓs+1 · · · dΓN +(B6) += +1 +ΩN−s +� +� +1≤k≤m≤s +∂Vkm +∂qk +∂fN +∂pk +dΓs+1 · · · dΓN ++ (N − s) +1 +ΩN−s +� +l +� +k=1 +�∂Vk,s+1 +∂qk +� �∂fN +∂pk +� +dΓs+1 · · · dΓN += +� +1≤k≤m≤s +∂Vkm +∂qk +∂fs +∂pk ++ N − s +Ω +� +s +� +k=1 +�∂Vk,s+1 +∂qk +� +× +�∂fs+1 +∂pk +� +dΓs+1 + +10 +Finally, the time evolution of fs is, +∂fs +∂t = − +s +� +k=1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N)pk +∂fs +∂ql +(B7) +− +N +� +k=s+1 +s +� +l=1 +¯B(b) +kl (qs, q∗ +s+1, . . . , q∗ +N)p∗ +k +∂fs +∂ql +− +� +1≤k≤m≤s +∂Vkm +∂qk +∂fs +∂pk ++ N − s +Ω +� +s +� +k=1 +�∂Vk,s+1 +∂qk +� +× +�∂fs+1 +∂pk +� +dΓs+1 +[1] O. +Hahn +and +F. +Strassmann, +Naturwissenschaften 27, 11 (1939). +[2] A. +Baran, +M. +Kowal, +P.-G. +Reinhard, +L. +Robledo, +A. +Staszczak, +and +M. +Warda, +Nuclear Physics A 944, 442 (2015). +[3] E. Fermi, E. Amaldi, B. Pontecorvo, F. Rasetti, and +E. Segr¨a, Ricerca. 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Xie, +Progress in Particle and Nuclear Physics 125, 103962 (2022). + diff --git a/lNAyT4oBgHgl3EQfYfcF/content/tmp_files/load_file.txt b/lNAyT4oBgHgl3EQfYfcF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91246661e8be523b6e180a9bbac60992b8e97bf8 --- /dev/null +++ b/lNAyT4oBgHgl3EQfYfcF/content/tmp_files/load_file.txt @@ -0,0 +1,727 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf,len=726 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='00202v1 [nucl-th] 31 Dec 2022 A transport model description of Time-Dependent Generator Coordinate under Gaussian overlap approximation Fangyuan Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='1 Yingxun Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' ∗ and Zhipan Li2 1Department of Nuclear Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' China Institute of Atomic Energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Beijing 102413,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' People’s Republic of China 2School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Southwest University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Chongqing 400715,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' People’s Republic of China (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' 2023) In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' we derived a transport equation based on a generalized equation of time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The transport equation is obtained by using quantum-mechanics phase space distributions under a “quasi-particle” picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierar- chy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The theoretical advantage of this transport equation is that time evolution of s-body phase space density distribution is coupled with s + 1-body phase space density distributions, and thus, non-adiabatic effects and dynamical fluctuations could be involved by more collective degrees and entanglement of phase space trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In future, we will perform the numerical calculations for fission nuclei after obtaining collective inertia and potential energy surface (PES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' INTRODUCTION After the discovery of nuclear fission by Hahn and Strassmann [1] in 1939, nuclear fission has become one of the most challenging topics in physics since it is a key ingredient for modeling nucleosynthesis [2], energy pro- duction [3], medicine [4], and nuclear safeguard [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Even with recent progress in experimental techniques, mea- surements of nuclear fission are not possible for all fis- sion nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Thus, theoretical simulations are mandatory for fully understanding the fission dynamics and comple- menting the missing data [6–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' One kind of models is useful macroscopic model by taking into account shell effects, collective variables, and correlations between collective degree and single parti- cles motions, such as Brownian shape motion [13–15] and Langevin model [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' With four or five collective degrees, it could depict fission dynamics process appro- priately and reproduce fission yields distribution well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Another kind of models is microscopic model, which based on nucleonic Hamiltonian and solve fission dy- namics with Schr¨odinger or Dirac equation in time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' time-dependent density func- tional theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' such as time-dependent superfluid lo- cal density approximation (TDSLDA) [19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' 20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Con- strained and time-dependent Hartree-Fock calculations with dynamical Bardeen–Cooper–Schrieffer pairing cor- relations (CHF+BCS) [21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' 22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' time dependent Hartree- Fock-Bogliubov (TDHFB)/TDHF+BCS [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' the adia- batic time-dependent HFB (ATDHFB) [24] and time- dependent covariant density functional theory (TD- CDFT) [25] describe fission process with full quantum microscopic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' At tremendous costs of compu- tations, these models successfully describe the fission dy- namics and predict the most probable fission yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The great understanding of fission dynamics from microscopic model are obtained [19, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' However, describing fission ∗ zhyx@ciae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='cn yields distribution with these models is still a theoreti- cal challenge due to the lacking of fluctuation in initial state and fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The efforts on this direction is to describe quantum fluctuation by a sampling of initial conditions followed by TDDFT[26] but without quantum interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' An alternative method to include the correlation is to represent a many-body wave function of the system with a mixture of states with different shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' It stim- ulates the description of fission dynamics with time- dependent generator coordinate method under Gaussian overlap approximation (TDGCM+GOA) [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In the TDGCM+GOA, fission is assumed as adiabatic process since the typical time for the motion of individual nucle- ons inside the fission nucleus (roughly 10−22 s) is roughly ten times smaller than the time scale of the system’s col- lective deformation (10−21 s) [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Thus, the fission dy- namics are approximately described in terms of a few shape coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Currently, most of the TDGCM+GOA calculations were performed by using only two degrees of freedom, usually (q20, q30) or (β2, β3), under adiabatic assump- tion [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' While, the semiphenomenological and fully microscopic approaches illustrate that at least four or five collective variables play a role in the dynamics of fission [19, 33–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Regnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [38] have started some trials on the rigorous three degrees of freedom calculation of the PES for 240Pu in the collective space (q20, q30, q40), and their work is still in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [39], Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' did the calculations with the dynamical pairing de- gree of freedom as the third degree of freedom besides (β2, β3), and their results also demonstrate the impor- tance of including more degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Furthermore, one should note that an ad hoc Gaussian smoothing have to be used at the end of the TDGCM+GOA calculations, to account for the fluctuations in particle number of the fragments due to both pairing effects and the finite num- ber of particles in the neck region for points along the fission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Thus, one would expect to develop a micro- scopic method that can include more collective degrees 2 and the correlations to account for dynamical fluctua- tion and non-adiabatic effects, and reasonably describing fission dynamics and distribution of fission yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In this work, we derive a transport equation based on a generalized N-dimensional TDGCM+GOA equation to describe fission dynamics, in which non-adiabatic effects are introduced by more collective degrees, fluctuations are introduced by initial state fluctuation and entan- glement of phase space trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The paper is or- ganized as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='II, the transport equation is obtained by using quantum-mechanics phase space dis- tributions under a “quasi-particle” picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hi- erarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' One of the advantages of this hierarchy is that correlations from high-order degree can be involved in the evolution of the one-body phase-space density dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='III, a numerical recipes for solving the transport equation is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='IV is the summary and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' THEORY FRAMWORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Overview of TDGCM for fission For convenience, we briefly review the TDGCM +GOA theory which describes induced fission as a slow adiabatic process determined by a small number of collective de- grees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Under the Griffin-Hill-Wheeler ansatz, the many-body state of fissioning system at any time reads |Ψ(t)⟩ = � q∈E dq|φ(q)⟩f(q, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (1) The set {|φ(q)⟩} is a family of the generator states which are the solutions of a constrained Hartree-Fock- Bogoliubov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' f(q, t) is the complex-valued weights of the quantum mixture of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The gener- ator coordinate q = {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', qN}, and each of these qi is a collective variable chosen based on the physics of fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The time-dependent Schr¨odinger equation for the many-body state of fission system |Ψ(t)⟩, ( ˆH − iℏ d dt)|Ψ(t)⟩ = 0, (2) can yield an equation of the unknown weight func- tion f(q, t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', the Hill-Wheeler equation with time- dependent form, � dq′⟨φq| � ˆH − iℏ d dt � |φq′⟩f(q′, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (3) Here, ˆH is the Hamiltonian acting on the full many-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Principally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (3) can be solved numerically, but it needs a tremendous amount of computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' To overcome these difficulties, a popular approach named as Gaussian overlap approximation (GOA) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The simplest formulation of GOA assumes that the overlap between two generator states ⟨φq|φq′⟩ has a Gaussian shape, N(q, q′) = ⟨φq|φq′⟩ ≡ exp � −1 2(q − q′)tG(¯q)(q − q′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (4) N(q, q′) = ⟨φq|φq′⟩ is peaked functions for q = q′, and ¯q = (q + q′)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' By changing a new collective coordinate α by the relation α(q) = � a∈Cq 0 G1/2(a)da, (5) in terms of which the overlap matrix becomes N(α, α′) = exp � −1 2(α − α′)2 � , (6) G (q) is the metric of new coordinates of α(q), and G (q) is the determinant of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Within this approximation, the time-dependent Hill- Wheeler equation is reduced to a local, time-dependent Schr¨odinger-like equation as iℏ∂g(q, t) ∂t = ˆHcoll(q)g(q, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (7) g(q, t) is related to the weight function f(q, t) as g = N 1/2f, and contains all the information about the fis- sion dynamics of system [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The collective Hamiltonian ˆHcoll(q) is a local operator acting on g(q, t), ˆHcoll(q) = (8) � −ℏ2 2 � kl 1 � G (q) ∂ ∂qk � G (q)Bkl (q) ∂ ∂ql + V (q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The potential V (q), V (q) = ⟨q| ˆH|q⟩ − ǫ0(q), (9) with the zero-point energy-correction ǫ0 = 1 2Gij(q) ∂2h ∂qi∂q′j ���� q=q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (10) The symmetric collective inertial tensor B(q) ≡ Bij(q), Bkl(q) = 1 2ℏ2 Gkm(q) �∂2h(q, q′) ∂qm∂q′n − ∂2h(q, q′) ∂qm∂qn + � i mn � ∂h(q, q′) ∂qi ����� q=q′ Gnl(q), (11) the expression in braces is the Christoffel symbol of the second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' h(q, q′) is h(q, q′) = � φq| ˆH|φq′ � ⟨φq|φq′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (12) 3 They are usually calculated from the nuclear Hamilto- nian ˆH and the generator states |φq⟩ with HFB [30] or RMF+BCS [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The number of collective degree of freedom are usually selected as N = 2, and shape coordinates q are the mul- tipole moments Q20 and Q30 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [30–32], or β2 and β3 as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [39, 40], and G (q) = 1 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In this case, the equation of TDGCM+GOA is iℏ ∂ ∂tg (q1, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' t) = (13) � −ℏ2 2 � kl ∂ ∂qk Bkl (q1, q2) ∂ ∂ql + V (q1, q2) � g (q1, q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' This equation has been solved by the software package FELIX-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='0 [31] or FELIX-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='0 [32] with finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' A transport equation for N-dimensional TDGCM+GOA The TDGCM has achieved great progress on describ- ing the fission dynamics [30–32, 39–41], but previous cal- culations in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [33–36, 38] also showed it is necessary to include more degrees to describe nonadiabatic effects which may arise from the coupling between collective and intrinsic degrees of freedom, and invovle dynamical fluc- tuations to describe fission products distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Now, the question is that can we effectively involve more collec- tive degrees of freedom into the equation with two degrees that we are currently using?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Principally, nuclear shape can be described by an ex- pansion in spherical harmonics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', R(θ, φ, t) = R0 \uf8eb \uf8ed1 + N � λ=0 λ � µ=−λ α∗ λµ(t)Yλµ(θ, φ) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (14) The number of shape coordinates (or collective degree of freedom) N depends on the choice of collective co- ordinates or generator coordinates and stage of fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In the stage of fissionning system from the quasista- tionary initial state to the outer fission barrier, evolu- tion is slow and fission process can be described by a small number of collective degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', a small N, with adiabatic approximation[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In the stage of saddle-to- scission, the nucleus quickly elongates toward scission and non-adiabatic effects have to be considered and N may vary with the stage of fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Thus, the col- lective wave function is presented with N degrees, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', g(q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', qN), for fissioning system, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (7) will be- come a generalized N-dimensional TDGCM+GOA equa- tion since the degrees are not limited to a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' In this work, we interpret the wave function of g(q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', qN) for fissioning system as a wave function for ‘N-body quasiparticles in 1-Dimension space’ sys- tem (NB1D), and a convention gN(q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', qN) is used in following to represent the wave function with parti- cle from 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Then, we derive a transport equation which can effectively couple one more collective degree of freedom to the degrees currently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Firstly, we per- form Wigner transformation [43] on NB1D wave function gN (q1, · · · , qN) to get their quantum mechanically phase space density fN as, fN (q1, · · · , qN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' p1, · · · , pN) (15) = 1 (πℏ)N � · · � dy1 · · · dyNg∗ N(q1 − y1, · · · , qN − yN) gN(q1 + y1, · · · , qN + yN) × exp[−2i(p1 · y1 + · · · + pN · yN)/ℏ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Here pi is the conjugated momentum of qi for quasi- particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' After a trivial deviation, the corresponding transport equation reads, ∂fN ∂t = − � kl ¯B(b) kl (q)pk ∂fN ∂ql (16) + � λ=1,3,··· ( ℏ 2i)λ1+···+λN−1 1 λ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' · · · λN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' ×∂λ1+···+λNVN (q) ∂qλ1 1 · · · ∂qλN N ∂λ1+···+λN fN ∂pλ1 1 · · · ∂pλN N = − � kl ¯B(b) kl (q)pk ∂fN ∂ql + � l ∂VN ∂ql ∂fN ∂pl + � λ=3,··· ( ℏ 2i)λ1+···+λN−1 1 λ1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' · · · λN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' ×∂λ1+···+λNVN (q) ∂qλ1 1 · · · ∂qλN N ∂λ1+···+λN fN ∂pλ1 1 · · · ∂pλN N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Here, λ = �N i=1 λi and ¯B(b) kl (q) is an effective collective inertia, which is defined as ¯B(b) kl (q) = � Bkl(q − y∗ (1b)) + Bkl(q − y∗ (2b)) � /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (17) y∗ (1b) and y∗ (2b) are corrections on q, and their origins can be found in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' VN(q) is N-body potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Principally, VN(q) = V (q1, q2, · · · , qN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' If the N-body potential is calculated from the two-body interaction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=', V (q1, q2, · · · , qN) = � i≤j V (qi, qj), (18) the transport equation is simplified as, ∂fN ∂t = − � kl ¯B(b) kl (q)pk ∂fN ∂ql + 1 2 � k,m̸=k ∂Vkm ∂qk ∂fN ∂pk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (19) Vkm is two-body potential between quasi-particle k and m, which can be obtained by HFB/RMF+BCS as in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' [30, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' When the N-body potential is obtained from multi-dimensional PES by HFB/RMF+BCS, trans- port equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' (16) should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' 4 A standard procedure to solve the N-body transport equation is to use BBGKY hierarchy, in which one-body degrees of freedom (DOF) is coupled to two-body DOF that are themselves coupled to three-body DOFs and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' As an example, we present the time evolu- tion of fs under the condition of V (q1, q2, · · · , qN) = � i≤j V (qi, qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' The s-body phase space density distri- bution fs is defined as, fs(q1, · · · , qs, p1, · · · , ps) (20) = 1 ΩN−s � fN(q1, · · · , qN, p1, · · · , pN)dΓs+1 · · · dΓN, dΓi = dqidpi, here, Ω is volume in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNAyT4oBgHgl3EQfYfcF/content/2301.00202v1.pdf'} +page_content=' Thus, ∂fs(q1, · · · , qs, p1, · · · , ps) ∂t (21) = 1 ΩN−s � ∂fN(q1, · · · , qN, p1, · · · , pN) ∂t dΓs+1 · · · dΓN = 1 ΩN−s � � − � kl ¯B(b) kl (q)pk ∂fN ∂ql + � 1≤k h+1 where h is the Coxeter number of the reductive group +(see Section 2.1). Under this assumption the pro-p Iwahori I is p-saturated in the sense of +Lazard (see Theorem 2.4.1, [LS22]). The main result of the article is, +Theorem 1.0.1. For p > h + 1, the Iwasawa algebra Λ(I) is naturally isomorphic as +topological rings to A/R (see Theorem 6.0.2). +Here A is the noncommutative power series ring over Zp in a certain number of variables +(depending on G) and R is a two-sided ideal of A coming from the Chevalley relations on +the p-adic group G. +2020 Mathematics Subject Classification. Primary: 11R23, 20G25 Secondary: 16L30. +Key words and phrases. Iwasawa algebra, pro-p Iwahori, reductive groups. +1 + +2 +ARANYA LAHIRI AND JISHNU RAY +The first result in this direction was carried out by Clozel in [Clo11], then taken up in +[Clo17] and used in [Clo18] in context of a proposed ‘p-adic base change map’. There he +gives an explicit ring theoretic presentation of the Iwasawa algebra Λ(IQp) for the pro-p +Iwahori subgroup IQp of SL2(Qp) and of Λ(IL), the pro-p Iwahori subgroup IL of SL2(L) +for some unramified extension L/Qp. Later, the second author of this article generalized +the result to the pro-p Iwahori subgroup of SLn(Qp) (see [Ray20]). He also solved the case +of a general uniform pro-p group in [Ray18a]. +Here we follow the circle of ideas of Clozel and the second authors’ work for SL2 and SLn. +The main technical challenge here is that the pro-p Iwahori subgroup can be equipped +with a p-valuation with respect to which it is p-saturated but in general it is not a uniform +group. Essentially, the main difference being the associated graded algebra in the case of +a uniform pro-p group is a commutative polynomial ring, while that is not the case in our +setup. +Clozel could carry out his calculations by hand because of a relatively small number of +variables and the relations between them being simple and easily calculable. The techniques +used by the second author in [Ray20] were also explicit but involved somewhat tedious +computations involving large matrices. +Clozel’s main interest in the explicit ring theoretic presentation of the Iwasawa algebras +seem to stem from the fact that he could propose a formal candidate for the p-adic base +change map between Λ(IL) → Λ(IQp). But even Clozel himself notes that the right setup for +such a base change map perhaps involves the rigid analytic distribution algebras associated +to the corresponding rigid analytic groups of the pro-p Iwahori subgroups. And as of now, +we are not sure of the significance of such a formal base change map and thus for the purpose +of exposition and ease of calculations we have restricted ourselves to groups defined over +Qp. The same technique should generalize to groups defined over an unramified extension +L/Qp as well, and using the explicit presentation it should not be difficult to define a +‘formal base change map’ between Λ(IL) → Λ(IQp). +1.1. Sketch of ideas for the main theorem. Before starting to write things formally +we give a brief and slightly vague account of the main ideas going into the proof of the +main theorem. +The natural p-valuation on the pro-p Iwahori subgroup I of G makes it a p-saturated group, +which in turn implies that I is homeomorphic to Zd +p as a Qp-analytic manifold. Thus the +Iwasawa algebra Λ(I) is isomorphic as a Zp-module to Zp[[X1, · · · , Xd]]. Now the Chevalley +relations between the Chevalley basis of the Lie algebra of I induces relations between the +variables Xi. We show here that this is a complete set of generator and relations. +A key observation towards the proof is to realize that it is enough to prove an analogous +statement for the mod-p Iwasawa algebra Ω(I) := Λ(I)⊗Zp Fp. Reduction mod-p makes the +Chevalley relations much easier to work with. The proof then boils down to a dimension +counting of associated graded vector space of Ω(I), as executed in Theorem 4.0.3. +We point out some useful context and contrast for the reader of this article: + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS +3 +• By using results of a previous paper by the first author and Sorensen ([LS22]) we +conceptualize the calculations significantly. The key input being a systematic way +of writing down a p-valuation and an ordered basis for the pro-p Iwahori subgroup. +This was missing in literature when the second author published [Ray20], which +caused him to resort to the long calculations. This simplification also allows us +to deal with the Chevalley relations without having to break them into too many +cases. +• We replace the lexicographic ordering on the set of positive roots in [Ray20] by +any additive ordering as constructed by Papi in [Pap94]. In the process perhaps we +shade more light on the seemingly adhoc choice of order in the ordered basis for I +in [Ray20]. +• We also provide a more conceptual proof of Theorem 4.0.3, which is in fact the +technical heart of the article. +• In [WE21] the author also gives a ring theoretic presentation of the mod-p Iwasawa +algebra. But their methods are very different from ours, and we do not know the +connection between these two approaches yet. +1.2. Applications in Iwasawa theory. Let E be an elliptic curve without complex +multiplication, over a number field F with good ordinary reduction at places of F above +p. +Consider the trivializing extension F∞ := F[Ep∞] = ∪n≥1F[Epn], where Epn is the +pn-torsion points of E(F). +This trivializing extension has Galois group G which is a +compact open subgroup of GL(2, Zp) for all primes p; and G = GL(2, Zp) for all but +finitely many primes p. The Pontryagin dual X∞ of the Selmer group +� +Sel(E/F∞) over this +trivializing extension carries several arithmetic information of E and its structure as an +Iwasawa module has deep number theoretic consequences. It is conjectured (see [Coa99, +Conjecture 1.7]) that X∞ is a finitely generated torsion Λ(G)-module. See Theorem 1.10 +and the example following it of (loc.cit) for cases when the conjecture holds. A partial +result giving the explicit structure of this dual Selmer group as an Iwasawa module is +also known. More precisely, X∞ is pseudo-isomorphic to a direct sum of cyclic Iwasawa +modules: +X∞ ∼ ⊕m +i=1Λ(G)/Li, +where Li are left reflexive ideals in the Iwasawa algebra Λ(G) (see the structure theorem +in [CSS03, pg. 74]). The explicit description of these left reflexive ideals is not known. +So a natural question is whether one can use the explicit presentation of the Iwasawa +algebra Λ(G) to explicitly describe these reflexive ideals that are of arithmetic interest. +Several progress has been made in the literature regarding the nonexistence of the two- +sided reflexive ideals and over the mod-p Iwasawa algebra Ω(G). An element x ∈ Ω(G) is +called normal if xΩ(G) = Ω(G)x and the ideal generated by a normal element is a two-sided +reflexive ideal of Ω(G). When G is the first congruence kernel of a split, semi-simple, simply +connected Chevalley group over Zp, under certain hypothesis, the explicit presentation of +its Iwasawa algebra given by the second author in [Ray18b] can be used to show that +every nonzero normal element of Ω(G) must be a unit (see [HRW21, Theorem 5.1]). Hence +there cannot be any two-sided non-trivial reflexive ideal generated by a normal element + +4 +ARANYA LAHIRI AND JISHNU RAY +in the mod-p Iwasawa algebra. Note that this technique of using the explicit presentation +to prove such a result on the normal element (loc.cit) also works for the prime p = 3 +and hence generalizes an earlier result by Ardakov, Wei and Jhang; their techniques were +also entirely different (see [AWZ08, Theorem A]). Such an explicit presentation of Iwasawa +algebra could also be used to reprove known results regarding the center of the mod-p +Iwasawa algebra (see [HRW21, Proposition 7.1]). We must mention that the techniques +in [HRW21] concerns only the first congruence kernel and hence the underlying group is +uniform pro-p. It is a natural question, which we believe should have an affirmative answer, +to generalize the results in (loc.cit) and investigate whether the explicit presentation of the +pro-p Iwahori subgroup presented in this paper could be used to explore one-sided or +two-sided reflexive ideals of the correponding Iwasawa algebra. +Acknowledgement +The authors thank Peter Schneider and Claus Sorensen for several discussions throughout +this project. The second author is supported by the Inspire research grant. +2. Preliminaries +2.1. Iwahori factorization. Let p be a prime. For the sake of clarity we will discuss +everything with the assumption that our ground field is Qp and just note at the beginning +that most of the results in this preliminary section are usually stated in their corresponding +sources in an appropriately generalized fashion for a finite extension L/Qp. +We fix a connected reductive group G which is split over Qp, and choose an Qp-split +maximal torus T ⊂ G. Let (X∗(T), Φ, X∗(T), Φ∨) be the associated root datum. We fix +a system of positive roots Φ+ with simple roots ∆ ⊂ Φ+ and let Φ− = −Φ+. We let ht(·) +be the height function on Φ, and we recall that the Coxeter number of G is h = 1 + ht(θ) +where θ is the highest root. +Let G = G(Qp) with the usual locally profinite topology, and similarly T = T(Qp). Further +we will denote the maximal compact subgroup by T 0 ⊂ T and its Sylow pro-p-subgroup by +T 1. Recall that to each α ∈ Φ one can attach a unipotent root subgroup Uα normalized by +T. We let Uα = Uα(Qp) and note that it comes with a natural isomorphism uα : Qp +∼ +−→ Uα +such that tuα(x)t−1 = uα(α(t)x) for all t ∈ T and x ∈ Qp. This gives a filtration of Uα by +the subgroups Uα,r = uα((pr)) for varying r ∈ Z. +We consider the full Iwahori subgroup J corresponding to Φ+. If G is assumed to be +semisimple and simply connected then J has the geometric interpretation of being the +stabilizer of the maximal facet (or alcove) corresponding to the root datum of the Bruhat- +Tits building B(G) of G. We don’t pursue the geometric aspects of J and instead just note +that the so-called Iwahori factorization says that multiplication defines a homeomorphism, +� +α∈Φ− +Uα,1 × T 0 × +� +α∈Φ+ +Uα,0 +∼ +−→ J. + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS +5 +We concentrate our attention on the Sylow pro-p subgroup of J, namely the pro-p Iwahori +subgroup I. Again multiplication gives a homeomorphism, +(2.1.0) +� +α∈Φ− +Uα,1 × T 1 × +� +α∈Φ+ +Uα,0 +∼ +−→ I. +Here the products over Φ− and Φ+ are ordered in an arbitrarily chosen way, see the proof +of [OS19, Lem. 2.3] and Lemma 2.1, part i, in [OS19]. From Section 2.4, we will fix an +ordering of this product to discuss ordered basis and subsequent developments that depend +on the ordered basis. +2.2. p-saturated groups. To make the article somewhat self-contained we collect here a +few definitions related to p-valuation on a group G which can be any pro-p, torsion free, +compact p-adic Lie group. Let us recall that a p-valuation [Sch11, Chapter 23, pg 169] ω +on the group G is a real valued function, +ω : G \ {1} → ( +1 +p−1, ∞) with the convention ω(1) = ∞, satisfying +(1) ω(g−1h) ≥ min(ω(g), ω(h)), +(2) ω([g, h]) ≥ ω(g) + ω(h), +(3) ω(gp) = ω(g) + 1. +A sequence of elements (g1, · · · , gr) in a group G equipped with a p-valuation ω is called +an ordered basis [Sch11, Def, Chapter 26, pg 182] of (G, ω) if it satisfies +(1) Zd +p → G, (x1, · · · , xd) → gx1 +1 · · · gxd +d is a homeomorphism, +(2) ω(gx1 +1 · · · gxd +d ) = min1≤i≤d(ω(gi) + vp(xi)) for any x1, · · · , xd ∈ Zp. +Finally a p-valuable group (G, ω) equipped with the valuation ω is saturated [Sch11, Def, +chapter 26, pg 187] if any g with ω(g) > +p +p−1 is a p-th power of some element of G. +2.3. Additive ordering on the set of positive roots. We recollect results from [Pap94] +in a form we will need to prove Lemma 5.3.1 later. +Let W be the associated Weyl group to the root system Φ. Let ∆ = {δ1, · · · , δn} be a +set of simple roots and Φ+ cooresponding set of positive roots and Φ− = −Φ+. And let +s1, · · · , sn be the reflections corresponding to simple roots. +Definition 2.3.1. If S := {σ1, · · · , σr} ⊂ Φ+ is a subset of positive roots of Φ we say that +S is associated to w ∈ W if there is a reduced expression w = si1 · · · sir of w such that +σ1 = δi1, σj = si1 · · · sij−1(δij), δij ∈ ∆. +In particular note that, if w0 is the longest element of the Weyl group then Φ+ is associated +to w0. +Definition 2.3.2. An ordered subset (S, <) of Φ+ is said to be compatibly ordered if +the following two conditions hold. +(1) If λ, µ ∈ S with λ < µ, and λ + µ ∈ Φ then λ + µ ∈ S and λ < λ + µ < µ. +(2) If λ + µ ∈ S, λ, µ ∈ Φ+ then λ or µ (or both) belong to S and one of them precedes +λ + µ. + +6 +ARANYA LAHIRI AND JISHNU RAY +The main result of [Pap94] is that +Theorem 2.3.3 (Theorem in pg 662 of [Pap94]). A subset S ⊂ Φ+ can be given a com- +patible ordering if and only if it is associated to some w of W. +Since we already observed that Φ+ is associated to w0, as a corollary of 2.3.3 we get +Corollary 2.3.4. We can impose a compatible ordering on Φ+. +2.4. Ordered basis of pro-p Iwahori. We recall what we need from [LS22, Section 3] +with slight modification and without any proofs. +In [LS22] it was shown that the pro-p Iwahori subgroup I can be equipped with a p- +valuation ω with respect to which I is saturated. For us it will be important to work with +a choice of an ordered basis for (I, ω) and to be able to explicitly write down the valuations +of these basis elements. +Let us fix any ordering on the elements of Φ− with increasing height function on the roots, +the elements of ∆ in an arbitrary way and the elements of Φ+ with respect to an arbitrary +but fixed compatible ordering in the sense of Definition 2.3.2. And finally we combine this +ordering to a totally ordered Φ in a way such that for any β ∈ Φ−, δ ∈ ∆, α ∈ Φ+ we have +β < δ < α. +Theorem 2.4.1. [LS22, Proposition 3.4, Corollary 3.6] With p − 1 > h, an ordered basis +with their respective valuations for (I, ω) is given by +• +uβ(p)β∈Φ− +, +ω(uβ(p)) += +1 + ht(β) +h , +• +δ∨(1 + p)δ∈∆ +, +ω(δ∨(1 + p)) += +1, +• +uα(1)α∈Φ+, +, +ω(uα(1)) += +ht(α) +h . +where we order the elements of the ordered basis corresponding to a total ordering of Φ +chosen as above. +Note that the description given in [LS22] chooses ep as an ordered basis of 1 + (p). Here +to make the calculations more straight forward the natural choice seems to be 1 + p. From +this point onward we change our notation to write hδ instead of δ∨ to match with the +notation used in [Ray20]. +Remark 2.4.2. For future reference we note that for any element g of the ordered basis +just mentioned above we have 1 +h ≤ ω(g) ≤ 1. +2.5. Iwasawa Algebra. Let G be any profinite group and let N (G) the set of open normal +subgroups in G. Then we know that +G = +lim +←− +N∈N (G) +G/N +is the projective limit, as a topological group, of the finite groups G/N equipped witb +the discrete topology. By functoriality of associated group rings, the algebraic group rings +O[G/N] form, for varying N in N (G), a projective system of rings. The projective limit +Λ(G) := O[[G]] := +lim +←− +N∈N (G) +O[G/N] + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS +7 +is called the completed group ring or the Iwasawa algebra of G over O. Equipping Λ(G) +with the projective limit topology of the natural topologies of O[G/N] makes it a complete +pseudo-compact topological ring. +The embedding of G → Λ(G) given by g → δg promotes to an injective embedding of the +group ring O[G] into Λ(G) with dense image. +Now let (G, ω) be a p-saturated group over Zp with a p-valuation ω and a choice of ordered +basis (h1, · · · , hd). In this situation we get a homeomorphism of analytic manifolds +c : Zd +p → G +(x1, · · · , xd) �→ hx1 +1 · · · hxd +d . +We use the standard notation bj = hj − 1 ∈ Λ(G), and for α = (α1, · · · , αd) ∈ Nd we let +bα = bα1 +1 · · · bαd +d . Then any element λ ∈ Λ(G) can be written as λ := � +α cαbα with cα ∈ O +and we can naturally equip it with the valuation, +(2.5.0) +˜ωΛ( +� +α +cαbα) = inf +α (v(cα) + +d +� +i=1 +αiω(hi)). +After pulling back and dualizing we get an isomorphism of topological Zp-modules +c⋆ : Λ(Zd +p) ∼= Λ(G) +δei → δhi. +By [Sch11, 20.1] we know that Λ(Zd +p) ∼= Zp[[X1, · · · , Xd]] the power series ring in d variables, +as a topological ring. +Using 2.4.1 we thus rewrite for I, +Lemma 2.5.1 (Lazard). The following are isomorphic as Zp-modules with an isomorphism +given by, +˜c : Zp[[Uβ, Vα, Wδ : β ∈ Φ−, α ∈ Φ+, δ ∈ ∆]] ∼= Λ(I) +1 + Vα → uα(1), +1 + Wδ → hδ(1 + p), +1 + Uβ → uβ(p). +2.6. The graded structure. For any p-valued group G, the valuation ˜ω induces a natural +filtration of Λ(G) as follows. Define for v ≥ 0, +Λ(G)v +:= +� +λ ∈ Λ(G) +��˜ω(λ) ≥ v +� +, +Λ(G)v+ +:= +� +λ ∈ Λ(G) +��˜ω(λ) > v +� +. +This gives a filtration and the associated graded algebra +grvΛ(G) := Λ(G)v/Λ(G)v+; grΛ(G) = +� +v≥0 +grvΛ(G). +Under this filtration Λ(I) is filtered by 1 +hN. + +8 +ARANYA LAHIRI AND JISHNU RAY +This follows from the fact that, grvΛ(I) is generated by the independent elements +ps � +β∈Φ− U +nβ +β +� +δ∈∆ W nδ +δ +� +α∈Φ+ V nα +α +such that +s + +� +β∈Φ− +h + ht(β) +h +nβ + +� +δ∈∆ +1 +hnδ + +� +α∈Φ+ +ht(α) +h +nα = v. +The filtration on Λ(I) induces a natural filtration and thus a grading on Ω(I) = Λ(I)⊗ZpFp. +We will use this to compute the dimension of grvΩ(I) as an Fp-vector space. We know +that grvΩ(I) is generated by � +β∈Φ− U +nβ +β +� +δ∈∆ W nδ +δ +� +α∈Φ+ V nα +α +with +� +β∈Φ− +h + ht(β) +h +nβ + +� +δ∈∆ +1 +hnδ + +� +α∈Φ+ +ht(α) +h +nα = v. +We don’t change the filtered pieces if we replace v ∈ 1 +hN by m = vh. This change assigns +the following valuations (which we still denote by ˜ω by abuse of notation) to the generators +of the Iwasawa algebra. +˜ω(uβ(p)) += +h + ht(β), +˜ω((hδ(1 + p)) += +h, +˜ω(uα(1)) += +ht(α). +The remarks above recover the following fact which is really due to Lazard. +Theorem 2.6.1. The dimension of grmΩ(I) as an Fp-vector space is equal to the di- +mension of space of homogeneous symmetric polynomials of degree m in the variables +{Uβ, Vα, Wδ : β ∈ Φ−, α ∈ Φ+, δ ∈ ∆} where the variables are assigned the degrees equal to +the corresponding shifted valuations of the ordered basis, deg(Uβ) = h+ht(β), deg(Wδ) = h +and deg(Vα) = ht(α). +3. Relations in the Iwasawa Algebra +3.1. The Chevalley relations. The Chevalley relations on the Chevalley basis of I trans- +late to relations between the non-commutative variables defining the Iwasawa algebra. As +remarked in the introduction the main result of this article is that these are a defining set +of relations for the Iwasawa algebra. +The Chevalley relations are the following (see [Ste16, pg 23]): +(Diag) hα(u)hβ(t) = hβ(t)hα(u). +(Diag-uni) hδ(t)xα(u)hδ(t)−1 = xα(t⟨α,δ⟩u) where ⟨α, δ⟩ ∈ Z. +(Uni-1) If α + β ̸= 0 and α + β /∈ Φ, then we have xα1(t)xα2(u) = xα2(u)xα1(t). +(Uni-2) If α1+α2 ̸= 0 and α1+α2 ∈ Φ, then we have xα(t)xβ(u) = � +i,j>0 xiα+jβ(cijtiuj)xβ(u)xα(t), +where cij’s are unique integers depending on α and β. +(Uni-3) Let α ∈ Φ+. Writing α = � +δi∈∆ niδi, we have hα(t) = � +δi∈∆ hδi(t)ni for some + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS +9 +non-negative integers ni. We obtain +xα(1)x−α(p) = x−α(p)Q � +δi∈∆ +hδi(1 + p)nixα(1)Q, +where Q = (1 + p)−1. +We include a proof of (Uni-3); the other relations are exactly as in [Ste16]. For any root +α ∈ Φ+, we have a homomorphism +φα : SL2(Qp) → ⟨xα(t), x−α(t); t ∈ Qp⟩ +such that +φα(1 + tE12) = xα(t), +φα(1 + tE21) = x−α(t), +φα(tE11 + t−1E22) = hα(t), +where Eij is the 2 × 2 matrix with 1 in the (i, j)-th entry and zero elsewhere. In SL2(Zp) +we have the following matrix relation +(1 + E12)(1 + pE21) = +� +1 + p(1 + p)−1E21 +�� +(1 + p)E11 + (1 + p)−1E22 +�� +1 + (1 + p)−1E12 +� +. +Since φα is a homomorphism we obtain +xα(1)x−α(p) = x−α(p(1 + p)−1)hα(1 + p)xα((1 + p)−1), +which implies +xα(1)x−α(p) = x−α(p)(1+p)−1 � +δi∈∆ +hδi(1 + p)nixα(1)(1+p)−1. +3.2. Relations in the Iwasawa algebra. The corresponding relations in the Iwasawa +algebra are +(R-1) (1 + Wδ1)(1 + Wδ2) = (1 + Wδ2)(1 + Wδ1). +(R-2) (1 + Wδ)(1 + Vα) = (1 + Vα)M(1 + Wδ), α ∈ Φ+, M = (1 + p)⟨α,δ⟩. +(R-3) (1 + Wδ)(1 + Uβ) = (1 + Uβ)M(1 + Wδ), β ∈ Φ−, M = (1 + p)⟨β,δ⟩. +(R-4) VαUβ = UβVα, α ∈ Φ+, β ∈ Φ−, α + β ̸= 0, α + β /∈ Φ. +(R-5) For α1, α2 ∈ Φ+, (1 + Vα1)(1 + Vα2) = +� +i,j>0 +iα1+jα2∈Φ +(1 + Viα1+jα2)cij(1 + Vα2)(1 + Vα1). +(R-6) For β1, β2 ∈ Φ−, (1+Uβ1)(1+Uβ2) = +� +i,j>0 +iβ1+jβ2∈Φ +(1+Uiβ1+jβ2)cijpi+j−1(1+Uβ2)(1+Uβ1). +(R-7) For α ∈ Φ+, β ∈ Φ−, +(1 + Vα)(1 + Uβ) = +� +i,j>0 +iα+jβ∈Φ− +(1 + Uiα+jβ)cijpj−1 +� +i,j>0 +iα+jβ∈Φ+ +(1 + Viα+jβ)cijpj(1 + Uβ)(1 + Vα). +(R-8) With α = � +i niδi, (1 + Vα)(1 + U−α) = (1 + U−α)Q � +δi∈∆ +(1 + Wδi)ni(1 + Vα)Q, where +Q = (1 + p)−1. + +10 +ARANYA LAHIRI AND JISHNU RAY +4. Explicit presentation of the Iwasawa algebra +Let |Φ| + |∆| = d and let us totally order S := {Vα, Wδ, Uβ, α ∈ Φ+, δ ∈ ∆, β ∈ Φ−} with +an order as prescribed in Theorem 2.4.1. Let X := {X1, · · · , Xd}. It is clear that there is +a natural bijection between ψ : X → S sending Xj to the j-th element of S. +Let A be the universal p-adic algebra of non-commutative power series in the variables +X1, · · · , Xd with coefficients in Zp with the ordering as in set S. Let us note that any +monomial of degree n in A can be expressed as aiXi(1) · · · Xi(n) where i is a map i : +{1, · · · , n} → {1, · · · , d}. Thus, writing Xi := Xi(1) · · · Xi(n) we can write +A := +� +f = +� +n +� +i +aiXi +�� ai ∈ Zp +� +. +The topology of A is given by the valuation +vA(f = +� +n +� +i +aiXi) = inf +i (vp(ai) + |i|), +where |i| = i(1) + · · ·+ i(n). The powers of the maximal ideal mA := (p, X1, · · · , Xd) gives +a fundamental system of neighborhoods of 0. +Let R ⊂ A be the closed two-sided ideal generated in A by the relations (R-1)-(R-8) +(after we have translated them as a relation between the variables Xi’s using ψ). Let A +be the reduction modulo p of A. The same proof as [Clo11, lemma 1.3] gives us +Lemma 4.0.1. Let R be the image of R in A. Then R is the closed two-sided ideal in A +generated by the relations (R-1)-(R-8). +Let A := Zp{X1, · · · , Xd} be the non-commutative polynomial ring in the variables X1, · · · , Xd. +The natural inclusion map A → A has dense image. +Lemma 4.0.2. Let us (by abuse of notation) define a natural map ψ : A → Λ(I) mapping +Xi ∈ A to the corresponding i-th element among S := {Vα, Wδ, Uβ, α ∈ Φ+, δ ∈ ∆, β ∈ Φ−} +in the Iwasawa algebra Λ(I). Then ψ extends continuously to a surjective homomorphism +A → Λ(I). +Proof. Using Lemma 2.5.1 and identifying via ψ we can write an element µ ∈ Λ(I) as +µ = � +i aiψ(Xi) and we have +˜ωΛ(µ) += +infi(vp(ai) + �d +j=1 i(j)ω(hj)) (from (2.5.0)), +≥ +infi( vp(ai) +h ++ �d +j=1 i(j) 1 +h) (as ω(hd) ≥ 1 +h, remark 2.4.2), +≥ +1 +hvA(� +i ai(Xi)). +Hence the map is continuous. Surjectivity is almost automatic from the definition of ψ. +□ +Consider the the natural filtration of A by powers of the maximal ideal mA, which we +denote by F nA. We have F nA/F n+1A = grnA. The filtration F n induces a filtration on +B = A/R +F nB = F nA + R, +and hence a gradation +grnB = F nB/F n+1B. + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 11 +Theorem 4.0.3. With the notation as above, for n ≥ 0, we have +dimFp grnB ≤ dimFp grnΩ(I). +The proof of this theorem is the main heart of the paper and occupies the whole of Section +5. +5. Bounding the dimension +5.1. Plan for the proof of Theorem 4.0.3. The theorem is obviously true for n = 0 +as gr0 is Fp and both sides equal 1. For n = 1, gr1B is a quotient of the space F 1A/F 2A +with basis {Vα, Uβ} where α varies over all the simple roots (i.e. ht(α) = 1) and β is the +negative of the highest root (hence h + ht(β) = 1). Hence +dimFp gr1B ≤ |∆| + 1 = dimFp gr1Ω(I). +For the general case, we first reduce all the relations (R-1) − (R-8) modulo appropriate +filtrations (see Proposition 5.2.1 in Section 5.2). Proposition 5.2.1 tells us how the product +of two variables in the wrong order is related to the product of the variables in the right +order (the right order is the one prescribed by Theorem 2.4.1). Next we show that we can +arrange the variables in the wrong order in grnB, by a sequence of transpositions and put +them in the right order, and at each step we reduce the number of inversions (see Lemma +5.3.1 in Section 5.3). This shows that dimFp grnB ≤ dimFp grnΩ(I) since grnΩ(I) consists +of homogeneous symmetric polynomials. +Let us start the proof of Theorem 4.0.3 by analyzing the mod-p reduction of certain bi- +nomial coefficients which will be necessary to study the relations between the variables +modulo filtrations of an appropriate degree inside R. +Lemma 5.1.1. Let 1 ≤ c ≤ p − 1, k ≥ 1, pk ∤ r and 1 ≤ r < pk then we have +�cpk +r +� +≡ 0 (mod p) +Proof. For any positive integer n, write n in base p, i.e. n = �d +i=0 aipd, where 0 ≤ ai ≤ p−1 +and ad ̸= 0. Define the function sp(n) := �d +i=0 ai. Let us recall vp(n!) = n−sp(n) +p−1 . Thus we +have +vp( +�cpk +r +� +) += cpk−sp(cpk) +p−1 +− r−sp(r) +p−1 +− cpk−r−sp(cpk−r) +p−1 += sp(cpk−r)+sp(r)−sp(cpk) +p−1 +From [HLS11, Proposition 2.1] we get +sp(cpk − r) += +sp(c − 1) + (p − 1)k − sp(r − 1) +sp(cpk − r) + sp(r) − sp(cpk) += +sp(c − 1) + (p − 1)k − sp(r − 1) + sp(r) − c += c − 1 + (p − 1)k − sp(r − 1) + sp(r) − c += +(p − 1)k + sp(r) − sp(r − 1) − 1 +Since we assumed r < pk we have sp(r − 1) < (p − 1)k and of course sp(r) − 1 ≥ 0 . Thus +we have + +12 +ARANYA LAHIRI AND JISHNU RAY +vp( +�cpk +r +� +) = (p − 1)k + sp(r) − sp(r − 1) − 1 > (p − 1)k − (p − 1)k = 0 +And since it is an integer we get vp( +�cpk +r +� +) ≥ 1, which is what we needed to show. +□ +5.2. Relations modulo appropriate filtrations. Recall that deg(Wδ) = h, deg(Vα) = +ht(α), deg(Uβ) = h + ht(β). +The variables in relations (R-1) and (R-4) commute modulo any filtration. In this fol- +lowing proposition we compute the other relations modulo appropriate filtrations. We will +see that we are reduced to only three relations (R-5’), (R-7’) and (R-8’) given below +where the variables don’t commute modulo filtration of an appropriate order. +Proposition 5.2.1. We have +(R-2’) WδVα = VαWδ (mod Filht(α)+h+1). +(R-3’) WδUβ = UβWδ (mod Fil2h+ht(β)+1). +(R-5’) Vα1Vα2 = +� +i,j>0 +iα1+jα2∈Φ +cijViα1+jα2 + Vα2Vα1 (mod Filht(α1)+ht(α2)+1). +(R-6’) Uβ1Uβ2 = Uβ2Uβ1 (mod Fil2h+ht(β1)+ht(β2)+1). +(R-7’) VαUβ = c11Uα+β + UβVα (mod Filh+ht(α)+ht(β)+1). +(R-8’) VαU−α = � +δi∈∆ +niWδi + U−αVα (mod Filh+1). +Proof. Consider relation (R-2) +(1 + Wδ)(1 + Vα) = (1 + Vα)M(1 + Wδ), α ∈ Φ+, M = (1 + p)⟨α,δ⟩. +Note that deg(WδVα) = h + ht(α) and so we are interested in studying relation (R-1) +modulo Filht(α)+h+1. +We want to show that relation (R-2) reduces to +(1 + Wδ)(1 + Vα) = (1 + Vα)(1 + Wδ) (mod Filht(α)+h+1). +We will show that by proving +(1 + Vα)M ≡ (1 + Vα) (mod Filht(α)+h+1). +We note that M ≡ 1 (mod p) and +(1 + Vα)M = 1 + MVα + +�M +2 +� +V 2 +α + · · · +Since degree of V m +α += ht(α)m for any m ∈ N we see that V m +α +≡ 0 (mod Filht(α)+h+1) as +long as ht(α)m > ht(α)+h. So we will be done if we can show +�M +m +� +≡ 0 (mod p) for m ≥ 2 +and ht(α)m ≤ ht(α) + h or rewriting, ht(α)(m − 1) ≤ h then we are done. But note that, +if ht(α)(m − 1) ≤ h then certainly m − 1 ≤ h. Thus if we can prove +�M +m +� +≡ 0 (mod p) for + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 13 +m − 1 ≤ h we are done. By assumption, h < p − 1, thus if m − 1 ≤ h < p − 1 then m < p. +So it’s clear in that case +�M +m +� +≡ 0 (mod p). +This proves (R-2’). +Next we want to show that relation (R-3) +(1 + Wδ)(1 + Uβ) = (1 + Uβ)M(1 + Wδ), β ∈ Φ−, M = (1 + p)⟨β,δ⟩, +reduces to +(1 + Wδ)(1 + Uβ) = (1 + Uβ)(1 + Wδ) (mod Fil2h+ht(β)+1). +Just like the last relation we want to show +(1 + Uβ)M ≡ 1 + Uβ (mod Fil2h+ht(β)+1). +When (ht(β) + h)m > 2h + ht(β) then Um +β ≡ 0 (mod Fil2h+ht(β)+1) thus we need to worry +about m ≥ 2 and (ht(β) + h)m ≤ 2h + ht(β) or rewriting (ht(β) + h)(m − 1) ≤ h. But +pretty much by the same argument as above we are reduced to the case m − 1 ≤ h and +the above calculation goes through which proves (R-3’). +Next consider relation (R-6). We want to show the relations, for β1, β2 ∈ Φ−, +(1 + Uβ1)(1 + Uβ2) = +� +i,j>0 +iβ1+jβ2∈Φ +(1 + Uiβ1+jβ2)cijpi+j−1(1 + Uβ2)(1 + Uβ1), +reduces to +(1 + Uβ1)(1 + Uβ2) = (1 + Uβ2)(1 + Uβ1) (mod Fil2h+ht(β1)+ht(β2)+1). +We need to analyze (1 + Uiβ1+jβ2)cijpi+j−1 carefully. As usual +deg(Um +iβ1+jβ2) = m(h + ht(iβ1 + jβ2)). +Thus for any m such that m(h + ht(iβ1 + jβ2)) > 2h + ht(β1) + ht(β2), we have +deg(Um +iβ1+jβ2) ≡ 0 (mod Fil2h+ht(β1)+ht(β2) + 1). +From now on we safely make the assumption, +m(h + ht(iβ1 + jβ2)) +≤ +2h + ht(β1) + ht(β2) +gives m +≤ +2h+ht(β1)+ht(β2) +h+ht(iβ1+jβ2) +≤ +2h − 2 (as h + ht(iβ1 + jβ2) ≥ 1) +≤ +2(p − 1) − 2 = 2p − 4 (as p > 1 + h). +For any prime p ≥ 3 and any k ≥ 2 we have pk > 2p − 4. Thus we have +�cijpi+j−1 +m +� +≡ 0 (mod p), m ≤ 2p − 4 and i + j − 1 ≥ 2. + +14 +ARANYA LAHIRI AND JISHNU RAY +We are left to deal with the case i + j = 2 or i = 1, j = 1. Note that by Lemma 5.1.1, +�cijp +m +� +≡ 0 (mod p) for m < p. Thus we can make the assumption m ≥ p. But in this case +we have +m(h + ht(β1 + β2)) +≥ +p(h + ht(β1 + β2)) +> +(1 + h)(h + ht(β1 + β2)) = h + ht(β1 + β2) + h(h + ht(β1 + β2)) +≥ +h + ht(β1 + β2) + h (as h + ht(β1 + β2) ≥ 1) += +2h + ht(β1 + β2). +Hence this proves (R-6’). +Next consider relation (R-5). We want to show the relation, α1, α2 ∈ Φ+, +(1 + Vα1)(1 + Vα2) = +� +i,j>0 +iα1+jα2∈Φ +(1 + Viα1+jα2)cij(1 + Vα2)(1 + Vα1), +reduces to +Vα1Vα2 = +� +i,j>0 +iα1+jα2∈Φ +cijViα1+jα2 + Vα2Vα1 (mod Filht(α1)+ht(α2)+1). +This follows from the observations that +(1 + Viα1+jα2)cij ≡ 1 + cijViα1+jα2 (mod Filht(α1)+ht(α2)+1) +by degree calculation and that deg(Viα1+jα2) + deg(Vαi) > ht(α1) + ht(α2) for i ∈ {1, 2}. +This shows (R-5’). +Next consider the relation (R-7). For α ∈ Φ+, β ∈ Φ−, +(1 + Vα)(1 + Uβ) = +� +i,j>0 +iα+jβ∈Φ− +(1 + Uiα+jβ)cijpj−1 +� +i,j>0 +iα+jβ∈Φ+ +(1 + Viα+jβ)cijpj(1 + Uβ)(1 + Vα). +Let us first deal with the case iα + jβ ∈ Φ−. For deg Um +iα+jβ = m(h + ht(iα + jβ)) > +h + ht(β) + ht(α), we have Um +iα+jβ = 0 (mod Filh+ht(β)+ht(α)+1). Hence we can restrict to +the case m(h + ht(iα + jβ)) ≤ h + ht(β) + ht(α). +m(h + ht(iα + jβ)) +≤ +h + ht(β) + ht(α) +gives m +≤ +h+ht(β)+ht(α) +h+ht(iα+jβ) +≤ +2h − 2 (as h + ht(iα + jβ) ≥ 1, ht(β) ≤ −1, ht(α) ≤ h − 1) +≤ +2(p − 1) − 2 = 2p − 4 (as p > 1 + h). +By Lemma 5.1.1, similar to the argument we gave while deducing (R-6’), we see that if +j − 1 ≥ 2, then with this restriction on m, none of the corresponding terms will contribute +modulo the filtration as the coefficients become trivial modulo p. +When j − 1 = 1 or j = 2 then we potentially will have contribution from m = p term. In +that case we have, deg(Uiα+2β)p = p(h + ht(iα + 2β)) > (h + 1)(h + ht(iα + 2β)). + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 15 +Note that as h + ht(iα + 2β) > 1, (h + 1)(h + ht(iα + 2β)) ≥ 2(h + 1). So, of course, +(h + 1)(h + ht(iα + 2β)) > h + ht(α) + ht(β) if ht(α) + ht(β) < 1. So we assume +ht(α) + ht(β) ≥ 1 +ht(α) + ht(β) +≥ +1 +hence +iht(α) +≥ +i(1 − ht(β)) +which gives +iht(α) + 2ht(β) +≥ +i(1 − ht(β)) + 2ht(β) = i + (2 − i)ht(β). +So if 2 − i ≤ 0 then the right hand side is definitely > 0, but this is not possible as iα + 2β +was assumed to be a negative root. Thus we see that only value i can take is 1. But in +that case as p > 1 + h, +p(h + ht(α) + 2ht(β)) > (h + 1)(h + ht(α) + 2ht(β)) += (h + ht(α) + ht(β)) + h +� +h + ht(α) + 2ht(β) +� ++ ht(β) += (h + ht(α) + ht(β)) + h +� +h + ht(α + 2β) +� ++ ht(β) +> (h + ht(α) + ht(β)) + (h + ht(β)) (as h + ht(α + 2β) > 1) +> h + ht(α) + ht(β) (as h + ht(β) > 1). +When j = 1 then m(h+ ht(iα + β)) ≥ h + ht(α) + ht(β) for all m, with equality only when +m = 1, i = 1. Thus the only contribution we can possibly have is (1 + c11Uα+β). +Now to look at the contribution coming from positive roots part of the equation (R-7), +i.e. from the terms +� +i,j>0 +iα+jβ∈Φ+ +(1 + Viα+jβ)cijpj. +By the same logic as above (i.e. bounding m by 2p − 4, then showing that the binomial +coefficients become trivial modulo p) this time also j ≥ 2 gets ruled out immediately. The +calculation is similar, hence omitted. +When j = 1 we are again concerned with the m = p term only. +In this case we have +deg(Viα+β)p = p +� +ht(iα + β) +� +> (h + 1) +� +iht(α) + ht(β) +� += iht(α) + ht(β) + h +� +ht(iα + β) +� +≥ h + htα + ht(β) (as i > 0, ht(iα + β) > 0). +Thus relation (R-7) reduce to +(1+Vα)(1+Uβ) = (1+c11Uα+β)(1+Uβ)(1+Vα) (mod Filh+ht(β)+ht(α)+1), when α+β ∈ Φ−. +Now, deg(Uα+βVα) = h + 2ht(α) + ht(β) > h + ht(β) + ht(α) and +deg(Uα+βUβ) = 2h + ht(α) + 2ht(β) += h + ht(β) + ht(α) + (h + ht(β)) +> h + ht(β) + ht(α) (as h + ht(β) > 1). +hence, after further simplifying we obtain + +16 +ARANYA LAHIRI AND JISHNU RAY +VαUβ = c11Uα+β + UβVα (mod Filh+ht(β)+ht(α)+1) +which is exactly relation (R-7’). +□ +Now consider relation (R-8). First, we will show that (1+U−α)Q = (1+U−α) (mod Filh+1). +If (h + ht(−α))m > h then Um +−α = 0 (mod Filh+1). So we need to worry about the case +m ≥ 2 and (h + ht(−α))m ≤ h. But then again m ≤ h < p − 1 and so +�Q +m +� += 0 (mod p) +since Q = 1 (mod p). +Next, we will show that (1 + Vα)Q = (1 + Vα) (mod Filh+1). +If mht(α) > h, the the +degree of V m +α is strictly larger than h. If mht(α) ≤ h, then m ≤ h < p − 1 and so again +�Q +m +� += 0 (mod p). Hence relation (R-8) reduces to +(1 + Vα)(1 + U−α) = (1 + U−α) +� +δi∈∆ +(1 + Wδi)ni(1 + Vα) (mod Filh+1). +Since deg(Wδi) = h and ni ≥ 0, further simplification yields +VαU−α = +� +δi∈∆ +niWδi + U−αVα (mod Filh+1). +5.3. Computing inversions modulo filtrations. Recall that {X1, · · · , Xd} was an or- +dered set of variables generating the noncommutative p-adic power series ring A. We set +Xi := Xi1 · · · Xiw to mean the monomial Xi(1) · · ·Xi(w). Then we can change Xi to a well- +ordered monomial (b → ib increasing) by a sequence of transpositions. Consider a move +(b, b + 1) → (b + 1, b) and assume ib > ib+1. We write +Xi = XfXbXb+1Xe, +and we look at Xi (mod Fildeg(Xi)+1). Suppose (Xb, Xb+1) is any pair of variables among +{Uβ, Wδi, Vα, β ∈ Φ−, δi ∈ ∆, α ∈ Φ+} but in the wrong ordering (for example, let +(Xb, Xb+1) = (Wδ, Vα) from relation (R-2’)). From Proposition 5.2.1, we see that variables +in the wrong ordering commute module filtration of an appropriate degree except relations +(R-5’), (R-7’) and (R-8’). Hence, except for these three relations, +XbXb+1 = Xb+1Xb (mod Fildeg(Xb)+deg(Xb+1)+1). +Therefore, we can reduce the number of inversions and change the wrong ordered variables +into right order, modulo suitable filtration. For the three relations (R-5’), (R-7’) and +(R-8’), we have to compute the number of inversions explicitly. +Lemma 5.3.1. Suppose (Xb, Xb+1) be either of the three pairs (Vα1, Vα2), (Vα, Uβ) or +(Vα, U−α) in the wrong ordering coming from left hand side of the relations (R-5’), (R-7’) +and (R-8’) satisfying equations of the form +XbXb+1 = Xb+1Xb + linear terms in Xi (mod Fildeg(Xb)+deg(Xb+1)+1). +Then the number of inversions in XfXb+1XbXe and XfXiXe are both strictly less than the +number of inversions of Xi. That is, the two moves XbXb+1 → Xb+1Xb, and XbXb+1 → Xi +reduces the number of inversions modulo Fildeg(Xb)+deg(Xb+1)+1. + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 17 +Proof. First let us compute the number of inversions of Xi. Let ξ and µ be denote indices +̸= b, b + 1 in the product Xi. +Consider relation (R-5’). The number of inversions in Xi was originally as follows: +inv = +� +ξ<µ +iξ>iµ +1 + +� +µ>b+1 +iµ∈[I(Vα2 ),I(Vα1 )) +1 + +� +2 +� +µ>b+1 +iµI(Vα1 ) +1 +� ++ 1, +where I(∗) := index(∗) is the index of the variable ∗. +As I(Vα2) < I(Vα1), the transition Vα1Vα2 → Vα2Vα1 clearly reduces the number of inver- +sions, and by changing Vα1Vα2 to cijViα1+jα2 we get +inv′ = +� +ξ<µ +iξ>iµ +1 + +� +µ>b+1 +iµI(Viα1+jα2 ) +1 +Because of the choice of an additive ordering on the set of positive roots in Section 2.3, +(5.3.1) +I(Vα2) < I(Viα1+jα2) < I(Vα1). +Hence the second term of inv′ is +� +µ>b+1 +iµb+1 +iµb+1 +iµ∈[I(Vα2 ),I(Vα1 )) +1 + +� +µ>b+1 +iµb+1 +iµ∈[I(Vα2 ),I(Vα1 )) +1 + +� +2 +� +µ>b+1 +iµI(Viα1+jα2 ) +1 ≤ +� +ξI(Vα2 ) +1 = +� +ξI(Vα1 ) +1 ≤ +� +ξI(Vα1 ) +1 +� +. +Hence inv′ < inv. for the relation (R-5’). +The proofs for the relations (R-7’) and (R-8’) are similar and hence omitted, the key +necessary things corresponding to (5.3.1) being I(Uβ) < I(Uα+β) < I(Vα) for the relation +(R-7’) and I(U−α) < I(Wδi) < I(Vα) for the relation (R-8’). +□ +6. Proof of the main theorem +In this section we recall and prove the main theorem in two steps. Both the proofs are +exactly same proof as that of Clozel (see [Clo11, pg. 554, Theorems 1.2 and 1.4]), we recall +them here for the purpose of completion. +Theorem 6.0.1. The mod-p Iwasawa algebra Ω(I) is naturally isomorphic to A/R. +Proof. The natural map ψ : A → Λ(I) sends Mn +A to Mn +Λ(I), where MΛ(I) is the maximal +ideal of Λ(I). The reduction mod-p of ψ respects the natural filtration on both sides and +thus induce a natural map gr•ψ : gr•A → gr•Λ(I). The induced map on the associated +graded is surjective since ψ is surjective. Thus it is an isomorphism by Theorem 4.0.3. +This then imply the theorem since the filtration on B is complete. +□ +Theorem 6.0.2. The Iwasawa algebra Λ(I) is naturally isomorphic to A/R. + +18 +ARANYA LAHIRI AND JISHNU RAY +We replicate the proof which is also the same argument as Clozel’s corresponding statement. +Proof. The reduction of ψ : A/R → Λ(I) is ψ. Recall that R is the image of R in A. +Assume f ∈ A satisfies ψ(f) = 0. We then have f ∈ R by Theorem 6.0.1. So f = r1 +pf1, +r1 ∈ R, f1 ∈ A. Then by applying ψ to both sides we get ψ(f1) = 0. Inductively, we +obtain an expression f = rn + pnfn with rn ∈ R and fn ∈ A. Since pnfn tends to 0 in A +and R is closed, we see that f ∈ R. +□ +References +[AB06] +K Ardakov and K Brown. Ring-theoretic properties of Iwasawa algebras: a survey. Documenta +Mathematica, Extra Volume: John H. Coates’ Sixtieth Birthday (2006):7–33, 2006. +[AWZ08] Konstantin Ardakov, Feng Wei, and James J Zhang. Reflexive ideals in Iwasawa algebras. Ad- +vances in Mathematics, 218(3):865–901, 2008. +[Clo11] +Laurent Clozel. Presentation of an Iwasawa algebra: the case of Γ1SL(2, Zp). Doc. Math., 16:545– +559, 2011. +[Clo17] +L. Clozel. Globally analytic p-adic representations of the pro-p-Iwahori subgroup of GL(2) and +base change, I: Iwasawa algebras and a base change map. Bull. Iranian Math. Soc., 43(4):55–76, +2017. +[Clo18] +Laurent Clozel. Globally analytic p-adic representations of the pro-p Iwahori subgroup of GL(2) +and base change, II: A Steinberg tensor product theorem. In Cohomology of arithmetic groups, +volume 245 of Springer Proc. Math. Stat., pages 1–33. Springer, Cham, 2018. +[Coa99] +John Coates. Fragments of the GL2 Iwasawa theory of elliptic curves without complex multipli- +cation. In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in +Math., pages 1–50. Springer, Berlin, 1999. +[CSS03] +John Coates, Peter Schneider, and Ramdorai Sujatha. Modules over Iwasawa algebras. J. Inst. +Math. Jussieu, 2(1):73–108, 2003. +[HLS11] +Kevin Hare, Shanta Laishram, and Thomas Stoll. Stolarsky’s conjecture and the sum of digits +of polynomial values. Proceedings of the American Mathematical Society, 139(1):39–49, 2011. +[HRW21] Dong Han, Jishnu Ray, and Feng Wei. Normal elements in the Iwasawa algebras of Chevalley +groups. manuscripta mathematica, 165(3):415–451, 2021. +[LS22] +Aranya Lahiri and Claus Sorensen. Rigid vectors in p-adic principal series representations. to +appear in Israel Journal of Mathematics, arXiv: 2205.02952., 2022. +[OS19] +Rachel Ollivier and Peter Schneider. The modular pro-p Iwahori-Hecke Ext-algebra. In Repre- +sentations of reductive groups, volume 101 of Proc. Sympos. Pure Math., pages 255–308. Amer. +Math. Soc., Providence, RI, 2019. +[Pap94] +Paolo Papi. A characterization of a special ordering in a root system. Proceedings of the American +Mathematical Society, 120(3):661–665, 1994. +[Ray18a] Jishnu Ray. Explicit ring-theoretic presentation of Iwasawa algebras. Comptes Rendus Mathe- +matique, 356(11):1075–1080, 2018. +[Ray18b] Jishnu Ray. Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, +simply connected Chevalley group over Zp. Journal of Algebra, 511:405–419, 2018. +[Ray20] +Jishnu Ray. Explicit presentation of an Iwasawa algebra: the case of pro-p Iwahori subgroup of +SLn(Zp). Forum Math., 32(2):319–338, 2020. +[Sch11] +Peter Schneider. p-adic Lie groups, volume 344 of Grundlehren der Mathematischen Wis- +senschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 2011. +[Ste16] +Robert Steinberg. Lectures on Chevalley groups, volume 66 of University Lecture Series. Ameri- +can Mathematical Society, Providence, RI, 2016. Notes prepared by John Faulkner and Robert +Wilson, Revised and corrected edition of the 1968 original [ MR0466335], With a foreword by +Robert R. Snapp. + +PRESENTATION OF AN IWASAWA ALGEBRA: +THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 19 +[WE21] +Carl Wang-Erickson. Higher Yoneda product structures and Iwasawa algebras modulo p. +arXiv:2101.06295, 2021. +(Lahiri) University of California San Diego +Email address: arlahiri@ucsd.edu +(Ray) Harish Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) +211 019 India +Email address: jishnuray@hri.res.in; jishnuray1992@gmail.com + diff --git a/o9AyT4oBgHgl3EQflvh2/content/tmp_files/load_file.txt b/o9AyT4oBgHgl3EQflvh2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..808ca00b9c4c68fa9ccc5f5cf98ea9a5cdfb763e --- /dev/null +++ b/o9AyT4oBgHgl3EQflvh2/content/tmp_files/load_file.txt @@ -0,0 +1,596 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf,len=595 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='00458v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='RT] 1 Jan 2023 PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS ARANYA LAHIRI AND JISHNU RAY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In this article we generalize results of Clozel and Ray (for SL2 and SLn respectively) to give explicit ring-theoretic presentation in terms of a complete set of gen- erators and relations of the Iwasawa algebra of the pro-p Iwahori subgroup of a connected, split, reductive group G over Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Introduction In this article we are interested in the Iwasawa algebra of the pro-p Iwahori subgroup of the Qp-valued points G := G(Qp), of a connected reductive group G defined and split over Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let’s recall that for a pro-p group G and a finite extension L/Qp with ring of integers O, the projective limit over the set of open normal subgroups N (G) of G, Λ(G) := O[[G]] := lim ←− N∈N (G) O[G/N], equipped with the appropriate projective limit topology, is called the completed group ring or the Iwasawa algebra of G over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' It is a complete pseudo-compact topological ring that plays an important role in the study of representations of p-adic groups and number theory via it’s connection to the p-adic Local Langlands program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The reader may find in the excellent survey article [AB06] a collection of several ring theoretic properties of Iwasawa algebras of a p-adic group and several open problems on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In this article we give a ring theoretic presentation of Λ(I), where I is the pro-p Iwahori subgroup of a connected reductive group G satisfying the assumptions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We work under the assumption that p > h+1 where h is the Coxeter number of the reductive group (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Under this assumption the pro-p Iwahori I is p-saturated in the sense of Lazard (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1, [LS22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The main result of the article is, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For p > h + 1, the Iwasawa algebra Λ(I) is naturally isomorphic as topological rings to A/R (see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Here A is the noncommutative power series ring over Zp in a certain number of variables (depending on G) and R is a two-sided ideal of A coming from the Chevalley relations on the p-adic group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Primary: 11R23, 20G25 Secondary: 16L30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Iwasawa algebra, pro-p Iwahori, reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 1 2 ARANYA LAHIRI AND JISHNU RAY The first result in this direction was carried out by Clozel in [Clo11], then taken up in [Clo17] and used in [Clo18] in context of a proposed ‘p-adic base change map’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' There he gives an explicit ring theoretic presentation of the Iwasawa algebra Λ(IQp) for the pro-p Iwahori subgroup IQp of SL2(Qp) and of Λ(IL), the pro-p Iwahori subgroup IL of SL2(L) for some unramified extension L/Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Later, the second author of this article generalized the result to the pro-p Iwahori subgroup of SLn(Qp) (see [Ray20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' He also solved the case of a general uniform pro-p group in [Ray18a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Here we follow the circle of ideas of Clozel and the second authors’ work for SL2 and SLn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The main technical challenge here is that the pro-p Iwahori subgroup can be equipped with a p-valuation with respect to which it is p-saturated but in general it is not a uniform group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Essentially, the main difference being the associated graded algebra in the case of a uniform pro-p group is a commutative polynomial ring, while that is not the case in our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Clozel could carry out his calculations by hand because of a relatively small number of variables and the relations between them being simple and easily calculable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The techniques used by the second author in [Ray20] were also explicit but involved somewhat tedious computations involving large matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Clozel’s main interest in the explicit ring theoretic presentation of the Iwasawa algebras seem to stem from the fact that he could propose a formal candidate for the p-adic base change map between Λ(IL) → Λ(IQp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But even Clozel himself notes that the right setup for such a base change map perhaps involves the rigid analytic distribution algebras associated to the corresponding rigid analytic groups of the pro-p Iwahori subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' And as of now, we are not sure of the significance of such a formal base change map and thus for the purpose of exposition and ease of calculations we have restricted ourselves to groups defined over Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The same technique should generalize to groups defined over an unramified extension L/Qp as well, and using the explicit presentation it should not be difficult to define a ‘formal base change map’ between Λ(IL) → Λ(IQp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Sketch of ideas for the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Before starting to write things formally we give a brief and slightly vague account of the main ideas going into the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The natural p-valuation on the pro-p Iwahori subgroup I of G makes it a p-saturated group, which in turn implies that I is homeomorphic to Zd p as a Qp-analytic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus the Iwasawa algebra Λ(I) is isomorphic as a Zp-module to Zp[[X1, · · · , Xd]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Now the Chevalley relations between the Chevalley basis of the Lie algebra of I induces relations between the variables Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We show here that this is a complete set of generator and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' A key observation towards the proof is to realize that it is enough to prove an analogous statement for the mod-p Iwasawa algebra Ω(I) := Λ(I)⊗Zp Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Reduction mod-p makes the Chevalley relations much easier to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The proof then boils down to a dimension counting of associated graded vector space of Ω(I), as executed in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We point out some useful context and contrast for the reader of this article: PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 3 By using results of a previous paper by the first author and Sorensen ([LS22]) we conceptualize the calculations significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The key input being a systematic way of writing down a p-valuation and an ordered basis for the pro-p Iwahori subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This was missing in literature when the second author published [Ray20], which caused him to resort to the long calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This simplification also allows us to deal with the Chevalley relations without having to break them into too many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We replace the lexicographic ordering on the set of positive roots in [Ray20] by any additive ordering as constructed by Papi in [Pap94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In the process perhaps we shade more light on the seemingly adhoc choice of order in the ordered basis for I in [Ray20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We also provide a more conceptual proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3, which is in fact the technical heart of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In [WE21] the author also gives a ring theoretic presentation of the mod-p Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But their methods are very different from ours, and we do not know the connection between these two approaches yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Applications in Iwasawa theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let E be an elliptic curve without complex multiplication, over a number field F with good ordinary reduction at places of F above p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Consider the trivializing extension F∞ := F[Ep∞] = ∪n≥1F[Epn], where Epn is the pn-torsion points of E(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This trivializing extension has Galois group G which is a compact open subgroup of GL(2, Zp) for all primes p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' and G = GL(2, Zp) for all but finitely many primes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The Pontryagin dual X∞ of the Selmer group � Sel(E/F∞) over this trivializing extension carries several arithmetic information of E and its structure as an Iwasawa module has deep number theoretic consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' It is conjectured (see [Coa99, Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='7]) that X∞ is a finitely generated torsion Λ(G)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' See Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='10 and the example following it of (loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='cit) for cases when the conjecture holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' A partial result giving the explicit structure of this dual Selmer group as an Iwasawa module is also known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' More precisely, X∞ is pseudo-isomorphic to a direct sum of cyclic Iwasawa modules: X∞ ∼ ⊕m i=1Λ(G)/Li, where Li are left reflexive ideals in the Iwasawa algebra Λ(G) (see the structure theorem in [CSS03, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The explicit description of these left reflexive ideals is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So a natural question is whether one can use the explicit presentation of the Iwasawa algebra Λ(G) to explicitly describe these reflexive ideals that are of arithmetic interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Several progress has been made in the literature regarding the nonexistence of the two- sided reflexive ideals and over the mod-p Iwasawa algebra Ω(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' An element x ∈ Ω(G) is called normal if xΩ(G) = Ω(G)x and the ideal generated by a normal element is a two-sided reflexive ideal of Ω(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' When G is the first congruence kernel of a split, semi-simple, simply connected Chevalley group over Zp, under certain hypothesis, the explicit presentation of its Iwasawa algebra given by the second author in [Ray18b] can be used to show that every nonzero normal element of Ω(G) must be a unit (see [HRW21, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence there cannot be any two-sided non-trivial reflexive ideal generated by a normal element 4 ARANYA LAHIRI AND JISHNU RAY in the mod-p Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Note that this technique of using the explicit presentation to prove such a result on the normal element (loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='cit) also works for the prime p = 3 and hence generalizes an earlier result by Ardakov, Wei and Jhang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' their techniques were also entirely different (see [AWZ08, Theorem A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Such an explicit presentation of Iwasawa algebra could also be used to reprove known results regarding the center of the mod-p Iwasawa algebra (see [HRW21, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We must mention that the techniques in [HRW21] concerns only the first congruence kernel and hence the underlying group is uniform pro-p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' It is a natural question, which we believe should have an affirmative answer, to generalize the results in (loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='cit) and investigate whether the explicit presentation of the pro-p Iwahori subgroup presented in this paper could be used to explore one-sided or two-sided reflexive ideals of the correponding Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Acknowledgement The authors thank Peter Schneider and Claus Sorensen for several discussions throughout this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The second author is supported by the Inspire research grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Iwahori factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let p be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For the sake of clarity we will discuss everything with the assumption that our ground field is Qp and just note at the beginning that most of the results in this preliminary section are usually stated in their corresponding sources in an appropriately generalized fashion for a finite extension L/Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We fix a connected reductive group G which is split over Qp, and choose an Qp-split maximal torus T ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let (X∗(T), Φ, X∗(T), Φ∨) be the associated root datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We fix a system of positive roots Φ+ with simple roots ∆ ⊂ Φ+ and let Φ− = −Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We let ht(·) be the height function on Φ, and we recall that the Coxeter number of G is h = 1 + ht(θ) where θ is the highest root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let G = G(Qp) with the usual locally profinite topology, and similarly T = T(Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Further we will denote the maximal compact subgroup by T 0 ⊂ T and its Sylow pro-p-subgroup by T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Recall that to each α ∈ Φ one can attach a unipotent root subgroup Uα normalized by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We let Uα = Uα(Qp) and note that it comes with a natural isomorphism uα : Qp ∼ −→ Uα such that tuα(x)t−1 = uα(α(t)x) for all t ∈ T and x ∈ Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This gives a filtration of Uα by the subgroups Uα,r = uα((pr)) for varying r ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We consider the full Iwahori subgroup J corresponding to Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' If G is assumed to be semisimple and simply connected then J has the geometric interpretation of being the stabilizer of the maximal facet (or alcove) corresponding to the root datum of the Bruhat- Tits building B(G) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We don’t pursue the geometric aspects of J and instead just note that the so-called Iwahori factorization says that multiplication defines a homeomorphism, � α∈Φ− Uα,1 × T 0 × � α∈Φ+ Uα,0 ∼ −→ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 5 We concentrate our attention on the Sylow pro-p subgroup of J, namely the pro-p Iwahori subgroup I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Again multiplication gives a homeomorphism, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0) � α∈Φ− Uα,1 × T 1 × � α∈Φ+ Uα,0 ∼ −→ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Here the products over Φ− and Φ+ are ordered in an arbitrarily chosen way, see the proof of [OS19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3] and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1, part i, in [OS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' From Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4, we will fix an ordering of this product to discuss ordered basis and subsequent developments that depend on the ordered basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' p-saturated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' To make the article somewhat self-contained we collect here a few definitions related to p-valuation on a group G which can be any pro-p, torsion free, compact p-adic Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us recall that a p-valuation [Sch11, Chapter 23, pg 169] ω on the group G is a real valued function, ω : G \\ {1} → ( 1 p−1, ∞) with the convention ω(1) = ∞, satisfying (1) ω(g−1h) ≥ min(ω(g), ω(h)), (2) ω([g, h]) ≥ ω(g) + ω(h), (3) ω(gp) = ω(g) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' A sequence of elements (g1, · · · , gr) in a group G equipped with a p-valuation ω is called an ordered basis [Sch11, Def, Chapter 26, pg 182] of (G, ω) if it satisfies (1) Zd p → G, (x1, · · · , xd) → gx1 1 · · · gxd d is a homeomorphism, (2) ω(gx1 1 · · · gxd d ) = min1≤i≤d(ω(gi) + vp(xi)) for any x1, · · · , xd ∈ Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Finally a p-valuable group (G, ω) equipped with the valuation ω is saturated [Sch11, Def, chapter 26, pg 187] if any g with ω(g) > p p−1 is a p-th power of some element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Additive ordering on the set of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We recollect results from [Pap94] in a form we will need to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let W be the associated Weyl group to the root system Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let ∆ = {δ1, · · · , δn} be a set of simple roots and Φ+ cooresponding set of positive roots and Φ− = −Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' And let s1, · · · , sn be the reflections corresponding to simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' If S := {σ1, · · · , σr} ⊂ Φ+ is a subset of positive roots of Φ we say that S is associated to w ∈ W if there is a reduced expression w = si1 · · · sir of w such that σ1 = δi1, σj = si1 · · · sij−1(δij), δij ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In particular note that, if w0 is the longest element of the Weyl group then Φ+ is associated to w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' An ordered subset (S, <) of Φ+ is said to be compatibly ordered if the following two conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (1) If λ, µ ∈ S with λ < µ, and λ + µ ∈ Φ then λ + µ ∈ S and λ < λ + µ < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (2) If λ + µ ∈ S, λ, µ ∈ Φ+ then λ or µ (or both) belong to S and one of them precedes λ + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 6 ARANYA LAHIRI AND JISHNU RAY The main result of [Pap94] is that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3 (Theorem in pg 662 of [Pap94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' A subset S ⊂ Φ+ can be given a com- patible ordering if and only if it is associated to some w of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Since we already observed that Φ+ is associated to w0, as a corollary of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3 we get Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We can impose a compatible ordering on Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Ordered basis of pro-p Iwahori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We recall what we need from [LS22, Section 3] with slight modification and without any proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In [LS22] it was shown that the pro-p Iwahori subgroup I can be equipped with a p- valuation ω with respect to which I is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For us it will be important to work with a choice of an ordered basis for (I, ω) and to be able to explicitly write down the valuations of these basis elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us fix any ordering on the elements of Φ− with increasing height function on the roots, the elements of ∆ in an arbitrary way and the elements of Φ+ with respect to an arbitrary but fixed compatible ordering in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' And finally we combine this ordering to a totally ordered Φ in a way such that for any β ∈ Φ−, δ ∈ ∆, α ∈ Φ+ we have β < δ < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [LS22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='6] With p − 1 > h, an ordered basis with their respective valuations for (I, ω) is given by uβ(p)β∈Φ− , ω(uβ(p)) = 1 + ht(β) h , δ∨(1 + p)δ∈∆ , ω(δ∨(1 + p)) = 1, uα(1)α∈Φ+, , ω(uα(1)) = ht(α) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' where we order the elements of the ordered basis corresponding to a total ordering of Φ chosen as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Note that the description given in [LS22] chooses ep as an ordered basis of 1 + (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Here to make the calculations more straight forward the natural choice seems to be 1 + p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' From this point onward we change our notation to write hδ instead of δ∨ to match with the notation used in [Ray20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For future reference we note that for any element g of the ordered basis just mentioned above we have 1 h ≤ ω(g) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Iwasawa Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let G be any profinite group and let N (G) the set of open normal subgroups in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then we know that G = lim ←− N∈N (G) G/N is the projective limit, as a topological group, of the finite groups G/N equipped witb the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' By functoriality of associated group rings, the algebraic group rings O[G/N] form, for varying N in N (G), a projective system of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The projective limit Λ(G) := O[[G]] := lim ←− N∈N (G) O[G/N] PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 7 is called the completed group ring or the Iwasawa algebra of G over O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Equipping Λ(G) with the projective limit topology of the natural topologies of O[G/N] makes it a complete pseudo-compact topological ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The embedding of G → Λ(G) given by g → δg promotes to an injective embedding of the group ring O[G] into Λ(G) with dense image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Now let (G, ω) be a p-saturated group over Zp with a p-valuation ω and a choice of ordered basis (h1, · · · , hd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In this situation we get a homeomorphism of analytic manifolds c : Zd p → G (x1, · · · , xd) �→ hx1 1 · · · hxd d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We use the standard notation bj = hj − 1 ∈ Λ(G), and for α = (α1, · · · , αd) ∈ Nd we let bα = bα1 1 · · · bαd d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then any element λ ∈ Λ(G) can be written as λ := � α cαbα with cα ∈ O and we can naturally equip it with the valuation, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0) ˜ωΛ( � α cαbα) = inf α (v(cα) + d � i=1 αiω(hi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' After pulling back and dualizing we get an isomorphism of topological Zp-modules c⋆ : Λ(Zd p) ∼= Λ(G) δei → δhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' By [Sch11, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1] we know that Λ(Zd p) ∼= Zp[[X1, · · · , Xd]] the power series ring in d variables, as a topological ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Using 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 we thus rewrite for I, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 (Lazard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The following are isomorphic as Zp-modules with an isomorphism given by, ˜c : Zp[[Uβ, Vα, Wδ : β ∈ Φ−, α ∈ Φ+, δ ∈ ∆]] ∼= Λ(I) 1 + Vα → uα(1), 1 + Wδ → hδ(1 + p), 1 + Uβ → uβ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The graded structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For any p-valued group G, the valuation ˜ω induces a natural filtration of Λ(G) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Define for v ≥ 0, Λ(G)v := � λ ∈ Λ(G) ��˜ω(λ) ≥ v � , Λ(G)v+ := � λ ∈ Λ(G) ��˜ω(λ) > v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This gives a filtration and the associated graded algebra grvΛ(G) := Λ(G)v/Λ(G)v+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' grΛ(G) = � v≥0 grvΛ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Under this filtration Λ(I) is filtered by 1 hN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 8 ARANYA LAHIRI AND JISHNU RAY This follows from the fact that, grvΛ(I) is generated by the independent elements ps � β∈Φ− U nβ β � δ∈∆ W nδ δ � α∈Φ+ V nα α such that s + � β∈Φ− h + ht(β) h nβ + � δ∈∆ 1 hnδ + � α∈Φ+ ht(α) h nα = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The filtration on Λ(I) induces a natural filtration and thus a grading on Ω(I) = Λ(I)⊗ZpFp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We will use this to compute the dimension of grvΩ(I) as an Fp-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We know that grvΩ(I) is generated by � β∈Φ− U nβ β � δ∈∆ W nδ δ � α∈Φ+ V nα α with � β∈Φ− h + ht(β) h nβ + � δ∈∆ 1 hnδ + � α∈Φ+ ht(α) h nα = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We don’t change the filtered pieces if we replace v ∈ 1 hN by m = vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This change assigns the following valuations (which we still denote by ˜ω by abuse of notation) to the generators of the Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' ˜ω(uβ(p)) = h + ht(β), ˜ω((hδ(1 + p)) = h, ˜ω(uα(1)) = ht(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The remarks above recover the following fact which is really due to Lazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The dimension of grmΩ(I) as an Fp-vector space is equal to the di- mension of space of homogeneous symmetric polynomials of degree m in the variables {Uβ, Vα, Wδ : β ∈ Φ−, α ∈ Φ+, δ ∈ ∆} where the variables are assigned the degrees equal to the corresponding shifted valuations of the ordered basis, deg(Uβ) = h+ht(β), deg(Wδ) = h and deg(Vα) = ht(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Relations in the Iwasawa Algebra 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The Chevalley relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The Chevalley relations on the Chevalley basis of I trans- late to relations between the non-commutative variables defining the Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' As remarked in the introduction the main result of this article is that these are a defining set of relations for the Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The Chevalley relations are the following (see [Ste16, pg 23]): (Diag) hα(u)hβ(t) = hβ(t)hα(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (Diag-uni) hδ(t)xα(u)hδ(t)−1 = xα(t⟨α,δ⟩u) where ⟨α, δ⟩ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (Uni-1) If α + β ̸= 0 and α + β /∈ Φ, then we have xα1(t)xα2(u) = xα2(u)xα1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (Uni-2) If α1+α2 ̸= 0 and α1+α2 ∈ Φ, then we have xα(t)xβ(u) = � i,j>0 xiα+jβ(cijtiuj)xβ(u)xα(t), where cij’s are unique integers depending on α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (Uni-3) Let α ∈ Φ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Writing α = � δi∈∆ niδi, we have hα(t) = � δi∈∆ hδi(t)ni for some PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 9 non-negative integers ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We obtain xα(1)x−α(p) = x−α(p)Q � δi∈∆ hδi(1 + p)nixα(1)Q, where Q = (1 + p)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We include a proof of (Uni-3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' the other relations are exactly as in [Ste16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For any root α ∈ Φ+, we have a homomorphism φα : SL2(Qp) → ⟨xα(t), x−α(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' t ∈ Qp⟩ such that φα(1 + tE12) = xα(t), φα(1 + tE21) = x−α(t), φα(tE11 + t−1E22) = hα(t), where Eij is the 2 × 2 matrix with 1 in the (i, j)-th entry and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In SL2(Zp) we have the following matrix relation (1 + E12)(1 + pE21) = � 1 + p(1 + p)−1E21 �� (1 + p)E11 + (1 + p)−1E22 �� 1 + (1 + p)−1E12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Since φα is a homomorphism we obtain xα(1)x−α(p) = x−α(p(1 + p)−1)hα(1 + p)xα((1 + p)−1), which implies xα(1)x−α(p) = x−α(p)(1+p)−1 � δi∈∆ hδi(1 + p)nixα(1)(1+p)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Relations in the Iwasawa algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The corresponding relations in the Iwasawa algebra are (R-1) (1 + Wδ1)(1 + Wδ2) = (1 + Wδ2)(1 + Wδ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-2) (1 + Wδ)(1 + Vα) = (1 + Vα)M(1 + Wδ), α ∈ Φ+, M = (1 + p)⟨α,δ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-3) (1 + Wδ)(1 + Uβ) = (1 + Uβ)M(1 + Wδ), β ∈ Φ−, M = (1 + p)⟨β,δ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-4) VαUβ = UβVα, α ∈ Φ+, β ∈ Φ−, α + β ̸= 0, α + β /∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-5) For α1, α2 ∈ Φ+, (1 + Vα1)(1 + Vα2) = � i,j>0 iα1+jα2∈Φ (1 + Viα1+jα2)cij(1 + Vα2)(1 + Vα1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-6) For β1, β2 ∈ Φ−, (1+Uβ1)(1+Uβ2) = � i,j>0 iβ1+jβ2∈Φ (1+Uiβ1+jβ2)cijpi+j−1(1+Uβ2)(1+Uβ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-7) For α ∈ Φ+, β ∈ Φ−, (1 + Vα)(1 + Uβ) = � i,j>0 iα+jβ∈Φ− (1 + Uiα+jβ)cijpj−1 � i,j>0 iα+jβ∈Φ+ (1 + Viα+jβ)cijpj(1 + Uβ)(1 + Vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-8) With α = � i niδi, (1 + Vα)(1 + U−α) = (1 + U−α)Q � δi∈∆ (1 + Wδi)ni(1 + Vα)Q, where Q = (1 + p)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 10 ARANYA LAHIRI AND JISHNU RAY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Explicit presentation of the Iwasawa algebra Let |Φ| + |∆| = d and let us totally order S := {Vα, Wδ, Uβ, α ∈ Φ+, δ ∈ ∆, β ∈ Φ−} with an order as prescribed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let X := {X1, · · · , Xd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' It is clear that there is a natural bijection between ψ : X → S sending Xj to the j-th element of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let A be the universal p-adic algebra of non-commutative power series in the variables X1, · · · , Xd with coefficients in Zp with the ordering as in set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us note that any monomial of degree n in A can be expressed as aiXi(1) · · · Xi(n) where i is a map i : {1, · · · , n} → {1, · · · , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus, writing Xi := Xi(1) · · · Xi(n) we can write A := � f = � n � i aiXi �� ai ∈ Zp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The topology of A is given by the valuation vA(f = � n � i aiXi) = inf i (vp(ai) + |i|), where |i| = i(1) + · · ·+ i(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The powers of the maximal ideal mA := (p, X1, · · · , Xd) gives a fundamental system of neighborhoods of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let R ⊂ A be the closed two-sided ideal generated in A by the relations (R-1)-(R-8) (after we have translated them as a relation between the variables Xi’s using ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let A be the reduction modulo p of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The same proof as [Clo11, lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3] gives us Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let R be the image of R in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then R is the closed two-sided ideal in A generated by the relations (R-1)-(R-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let A := Zp{X1, · · · , Xd} be the non-commutative polynomial ring in the variables X1, · · · , Xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The natural inclusion map A → A has dense image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us (by abuse of notation) define a natural map ψ : A → Λ(I) mapping Xi ∈ A to the corresponding i-th element among S := {Vα, Wδ, Uβ, α ∈ Φ+, δ ∈ ∆, β ∈ Φ−} in the Iwasawa algebra Λ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then ψ extends continuously to a surjective homomorphism A → Λ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 and identifying via ψ we can write an element µ ∈ Λ(I) as µ = � i aiψ(Xi) and we have ˜ωΛ(µ) = infi(vp(ai) + �d j=1 i(j)ω(hj)) (from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0)), ≥ infi( vp(ai) h + �d j=1 i(j) 1 h) (as ω(hd) ≥ 1 h, remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2), ≥ 1 hvA(� i ai(Xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence the map is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Surjectivity is almost automatic from the definition of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' □ Consider the the natural filtration of A by powers of the maximal ideal mA, which we denote by F nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We have F nA/F n+1A = grnA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The filtration F n induces a filtration on B = A/R F nB = F nA + R, and hence a gradation grnB = F nB/F n+1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 11 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' With the notation as above, for n ≥ 0, we have dimFp grnB ≤ dimFp grnΩ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The proof of this theorem is the main heart of the paper and occupies the whole of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Bounding the dimension 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Plan for the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The theorem is obviously true for n = 0 as gr0 is Fp and both sides equal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For n = 1, gr1B is a quotient of the space F 1A/F 2A with basis {Vα, Uβ} where α varies over all the simple roots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' ht(α) = 1) and β is the negative of the highest root (hence h + ht(β) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence dimFp gr1B ≤ |∆| + 1 = dimFp gr1Ω(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For the general case, we first reduce all the relations (R-1) − (R-8) modulo appropriate filtrations (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 tells us how the product of two variables in the wrong order is related to the product of the variables in the right order (the right order is the one prescribed by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Next we show that we can arrange the variables in the wrong order in grnB, by a sequence of transpositions and put them in the right order, and at each step we reduce the number of inversions (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This shows that dimFp grnB ≤ dimFp grnΩ(I) since grnΩ(I) consists of homogeneous symmetric polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us start the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3 by analyzing the mod-p reduction of certain bi- nomial coefficients which will be necessary to study the relations between the variables modulo filtrations of an appropriate degree inside R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let 1 ≤ c ≤ p − 1, k ≥ 1, pk ∤ r and 1 ≤ r < pk then we have �cpk r � ≡ 0 (mod p) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For any positive integer n, write n in base p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' n = �d i=0 aipd, where 0 ≤ ai ≤ p−1 and ad ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Define the function sp(n) := �d i=0 ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us recall vp(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=') = n−sp(n) p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus we have vp( �cpk r � ) = cpk−sp(cpk) p−1 − r−sp(r) p−1 − cpk−r−sp(cpk−r) p−1 = sp(cpk−r)+sp(r)−sp(cpk) p−1 From [HLS11, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1] we get sp(cpk − r) = sp(c − 1) + (p − 1)k − sp(r − 1) sp(cpk − r) + sp(r) − sp(cpk) = sp(c − 1) + (p − 1)k − sp(r − 1) + sp(r) − c = c − 1 + (p − 1)k − sp(r − 1) + sp(r) − c = (p − 1)k + sp(r) − sp(r − 1) − 1 Since we assumed r < pk we have sp(r − 1) < (p − 1)k and of course sp(r) − 1 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus we have 12 ARANYA LAHIRI AND JISHNU RAY vp( �cpk r � ) = (p − 1)k + sp(r) − sp(r − 1) − 1 > (p − 1)k − (p − 1)k = 0 And since it is an integer we get vp( �cpk r � ) ≥ 1, which is what we needed to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Relations modulo appropriate filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Recall that deg(Wδ) = h, deg(Vα) = ht(α), deg(Uβ) = h + ht(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The variables in relations (R-1) and (R-4) commute modulo any filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In this fol- lowing proposition we compute the other relations modulo appropriate filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We will see that we are reduced to only three relations (R-5’), (R-7’) and (R-8’) given below where the variables don’t commute modulo filtration of an appropriate order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We have (R-2’) WδVα = VαWδ (mod Filht(α)+h+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-3’) WδUβ = UβWδ (mod Fil2h+ht(β)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-5’) Vα1Vα2 = � i,j>0 iα1+jα2∈Φ cijViα1+jα2 + Vα2Vα1 (mod Filht(α1)+ht(α2)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-6’) Uβ1Uβ2 = Uβ2Uβ1 (mod Fil2h+ht(β1)+ht(β2)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-7’) VαUβ = c11Uα+β + UβVα (mod Filh+ht(α)+ht(β)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (R-8’) VαU−α = � δi∈∆ niWδi + U−αVα (mod Filh+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Consider relation (R-2) (1 + Wδ)(1 + Vα) = (1 + Vα)M(1 + Wδ), α ∈ Φ+, M = (1 + p)⟨α,δ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Note that deg(WδVα) = h + ht(α) and so we are interested in studying relation (R-1) modulo Filht(α)+h+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We want to show that relation (R-2) reduces to (1 + Wδ)(1 + Vα) = (1 + Vα)(1 + Wδ) (mod Filht(α)+h+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We will show that by proving (1 + Vα)M ≡ (1 + Vα) (mod Filht(α)+h+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We note that M ≡ 1 (mod p) and (1 + Vα)M = 1 + MVα + �M 2 � V 2 α + · · · Since degree of V m α = ht(α)m for any m ∈ N we see that V m α ≡ 0 (mod Filht(α)+h+1) as long as ht(α)m > ht(α)+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So we will be done if we can show �M m � ≡ 0 (mod p) for m ≥ 2 and ht(α)m ≤ ht(α) + h or rewriting, ht(α)(m − 1) ≤ h then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But note that, if ht(α)(m − 1) ≤ h then certainly m − 1 ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus if we can prove �M m � ≡ 0 (mod p) for PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 13 m − 1 ≤ h we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' By assumption, h < p − 1, thus if m − 1 ≤ h < p − 1 then m < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So it’s clear in that case �M m � ≡ 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This proves (R-2’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Next we want to show that relation (R-3) (1 + Wδ)(1 + Uβ) = (1 + Uβ)M(1 + Wδ), β ∈ Φ−, M = (1 + p)⟨β,δ⟩, reduces to (1 + Wδ)(1 + Uβ) = (1 + Uβ)(1 + Wδ) (mod Fil2h+ht(β)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Just like the last relation we want to show (1 + Uβ)M ≡ 1 + Uβ (mod Fil2h+ht(β)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' When (ht(β) + h)m > 2h + ht(β) then Um β ≡ 0 (mod Fil2h+ht(β)+1) thus we need to worry about m ≥ 2 and (ht(β) + h)m ≤ 2h + ht(β) or rewriting (ht(β) + h)(m − 1) ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But pretty much by the same argument as above we are reduced to the case m − 1 ≤ h and the above calculation goes through which proves (R-3’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Next consider relation (R-6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We want to show the relations, for β1, β2 ∈ Φ−, (1 + Uβ1)(1 + Uβ2) = � i,j>0 iβ1+jβ2∈Φ (1 + Uiβ1+jβ2)cijpi+j−1(1 + Uβ2)(1 + Uβ1), reduces to (1 + Uβ1)(1 + Uβ2) = (1 + Uβ2)(1 + Uβ1) (mod Fil2h+ht(β1)+ht(β2)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We need to analyze (1 + Uiβ1+jβ2)cijpi+j−1 carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' As usual deg(Um iβ1+jβ2) = m(h + ht(iβ1 + jβ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus for any m such that m(h + ht(iβ1 + jβ2)) > 2h + ht(β1) + ht(β2), we have deg(Um iβ1+jβ2) ≡ 0 (mod Fil2h+ht(β1)+ht(β2) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' From now on we safely make the assumption, m(h + ht(iβ1 + jβ2)) ≤ 2h + ht(β1) + ht(β2) gives m ≤ 2h+ht(β1)+ht(β2) h+ht(iβ1+jβ2) ≤ 2h − 2 (as h + ht(iβ1 + jβ2) ≥ 1) ≤ 2(p − 1) − 2 = 2p − 4 (as p > 1 + h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For any prime p ≥ 3 and any k ≥ 2 we have pk > 2p − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus we have �cijpi+j−1 m � ≡ 0 (mod p), m ≤ 2p − 4 and i + j − 1 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 14 ARANYA LAHIRI AND JISHNU RAY We are left to deal with the case i + j = 2 or i = 1, j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Note that by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1, �cijp m � ≡ 0 (mod p) for m < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus we can make the assumption m ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But in this case we have m(h + ht(β1 + β2)) ≥ p(h + ht(β1 + β2)) > (1 + h)(h + ht(β1 + β2)) = h + ht(β1 + β2) + h(h + ht(β1 + β2)) ≥ h + ht(β1 + β2) + h (as h + ht(β1 + β2) ≥ 1) = 2h + ht(β1 + β2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence this proves (R-6’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Next consider relation (R-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We want to show the relation, α1, α2 ∈ Φ+, (1 + Vα1)(1 + Vα2) = � i,j>0 iα1+jα2∈Φ (1 + Viα1+jα2)cij(1 + Vα2)(1 + Vα1), reduces to Vα1Vα2 = � i,j>0 iα1+jα2∈Φ cijViα1+jα2 + Vα2Vα1 (mod Filht(α1)+ht(α2)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This follows from the observations that (1 + Viα1+jα2)cij ≡ 1 + cijViα1+jα2 (mod Filht(α1)+ht(α2)+1) by degree calculation and that deg(Viα1+jα2) + deg(Vαi) > ht(α1) + ht(α2) for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This shows (R-5’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Next consider the relation (R-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For α ∈ Φ+, β ∈ Φ−, (1 + Vα)(1 + Uβ) = � i,j>0 iα+jβ∈Φ− (1 + Uiα+jβ)cijpj−1 � i,j>0 iα+jβ∈Φ+ (1 + Viα+jβ)cijpj(1 + Uβ)(1 + Vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let us first deal with the case iα + jβ ∈ Φ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For deg Um iα+jβ = m(h + ht(iα + jβ)) > h + ht(β) + ht(α), we have Um iα+jβ = 0 (mod Filh+ht(β)+ht(α)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence we can restrict to the case m(h + ht(iα + jβ)) ≤ h + ht(β) + ht(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' m(h + ht(iα + jβ)) ≤ h + ht(β) + ht(α) gives m ≤ h+ht(β)+ht(α) h+ht(iα+jβ) ≤ 2h − 2 (as h + ht(iα + jβ) ≥ 1, ht(β) ≤ −1, ht(α) ≤ h − 1) ≤ 2(p − 1) − 2 = 2p − 4 (as p > 1 + h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1, similar to the argument we gave while deducing (R-6’), we see that if j − 1 ≥ 2, then with this restriction on m, none of the corresponding terms will contribute modulo the filtration as the coefficients become trivial modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' When j − 1 = 1 or j = 2 then we potentially will have contribution from m = p term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In that case we have, deg(Uiα+2β)p = p(h + ht(iα + 2β)) > (h + 1)(h + ht(iα + 2β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 15 Note that as h + ht(iα + 2β) > 1, (h + 1)(h + ht(iα + 2β)) ≥ 2(h + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So, of course, (h + 1)(h + ht(iα + 2β)) > h + ht(α) + ht(β) if ht(α) + ht(β) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So we assume ht(α) + ht(β) ≥ 1 ht(α) + ht(β) ≥ 1 hence iht(α) ≥ i(1 − ht(β)) which gives iht(α) + 2ht(β) ≥ i(1 − ht(β)) + 2ht(β) = i + (2 − i)ht(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So if 2 − i ≤ 0 then the right hand side is definitely > 0, but this is not possible as iα + 2β was assumed to be a negative root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus we see that only value i can take is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But in that case as p > 1 + h, p(h + ht(α) + 2ht(β)) > (h + 1)(h + ht(α) + 2ht(β)) = (h + ht(α) + ht(β)) + h � h + ht(α) + 2ht(β) � + ht(β) = (h + ht(α) + ht(β)) + h � h + ht(α + 2β) � + ht(β) > (h + ht(α) + ht(β)) + (h + ht(β)) (as h + ht(α + 2β) > 1) > h + ht(α) + ht(β) (as h + ht(β) > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' When j = 1 then m(h+ ht(iα + β)) ≥ h + ht(α) + ht(β) for all m, with equality only when m = 1, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus the only contribution we can possibly have is (1 + c11Uα+β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Now to look at the contribution coming from positive roots part of the equation (R-7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' from the terms � i,j>0 iα+jβ∈Φ+ (1 + Viα+jβ)cijpj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' By the same logic as above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' bounding m by 2p − 4, then showing that the binomial coefficients become trivial modulo p) this time also j ≥ 2 gets ruled out immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The calculation is similar, hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' When j = 1 we are again concerned with the m = p term only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In this case we have deg(Viα+β)p = p � ht(iα + β) � > (h + 1) � iht(α) + ht(β) � = iht(α) + ht(β) + h � ht(iα + β) � ≥ h + htα + ht(β) (as i > 0, ht(iα + β) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus relation (R-7) reduce to (1+Vα)(1+Uβ) = (1+c11Uα+β)(1+Uβ)(1+Vα) (mod Filh+ht(β)+ht(α)+1), when α+β ∈ Φ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Now, deg(Uα+βVα) = h + 2ht(α) + ht(β) > h + ht(β) + ht(α) and deg(Uα+βUβ) = 2h + ht(α) + 2ht(β) = h + ht(β) + ht(α) + (h + ht(β)) > h + ht(β) + ht(α) (as h + ht(β) > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' hence, after further simplifying we obtain 16 ARANYA LAHIRI AND JISHNU RAY VαUβ = c11Uα+β + UβVα (mod Filh+ht(β)+ht(α)+1) which is exactly relation (R-7’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' □ Now consider relation (R-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' First, we will show that (1+U−α)Q = (1+U−α) (mod Filh+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' If (h + ht(−α))m > h then Um −α = 0 (mod Filh+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So we need to worry about the case m ≥ 2 and (h + ht(−α))m ≤ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' But then again m ≤ h < p − 1 and so �Q m � = 0 (mod p) since Q = 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Next, we will show that (1 + Vα)Q = (1 + Vα) (mod Filh+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' If mht(α) > h, the the degree of V m α is strictly larger than h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' If mht(α) ≤ h, then m ≤ h < p − 1 and so again �Q m � = 0 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence relation (R-8) reduces to (1 + Vα)(1 + U−α) = (1 + U−α) � δi∈∆ (1 + Wδi)ni(1 + Vα) (mod Filh+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Since deg(Wδi) = h and ni ≥ 0, further simplification yields VαU−α = � δi∈∆ niWδi + U−αVα (mod Filh+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Computing inversions modulo filtrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Recall that {X1, · · · , Xd} was an or- dered set of variables generating the noncommutative p-adic power series ring A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We set Xi := Xi1 · · · Xiw to mean the monomial Xi(1) · · ·Xi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then we can change Xi to a well- ordered monomial (b → ib increasing) by a sequence of transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Consider a move (b, b + 1) → (b + 1, b) and assume ib > ib+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We write Xi = XfXbXb+1Xe, and we look at Xi (mod Fildeg(Xi)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Suppose (Xb, Xb+1) is any pair of variables among {Uβ, Wδi, Vα, β ∈ Φ−, δi ∈ ∆, α ∈ Φ+} but in the wrong ordering (for example, let (Xb, Xb+1) = (Wδ, Vα) from relation (R-2’)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' From Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1, we see that variables in the wrong ordering commute module filtration of an appropriate degree except relations (R-5’), (R-7’) and (R-8’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence, except for these three relations, XbXb+1 = Xb+1Xb (mod Fildeg(Xb)+deg(Xb+1)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Therefore, we can reduce the number of inversions and change the wrong ordered variables into right order, modulo suitable filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' For the three relations (R-5’), (R-7’) and (R-8’), we have to compute the number of inversions explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Suppose (Xb, Xb+1) be either of the three pairs (Vα1, Vα2), (Vα, Uβ) or (Vα, U−α) in the wrong ordering coming from left hand side of the relations (R-5’), (R-7’) and (R-8’) satisfying equations of the form XbXb+1 = Xb+1Xb + linear terms in Xi (mod Fildeg(Xb)+deg(Xb+1)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then the number of inversions in XfXb+1XbXe and XfXiXe are both strictly less than the number of inversions of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' That is, the two moves XbXb+1 → Xb+1Xb, and XbXb+1 → Xi reduces the number of inversions modulo Fildeg(Xb)+deg(Xb+1)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' First let us compute the number of inversions of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Let ξ and µ be denote indices ̸= b, b + 1 in the product Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Consider relation (R-5’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The number of inversions in Xi was originally as follows: inv = � ξ<µ iξ>iµ 1 + � µ>b+1 iµ∈[I(Vα2 ),I(Vα1 )) 1 + � 2 � µ>b+1 iµI(Vα1 ) 1 � + 1, where I(∗) := index(∗) is the index of the variable ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' As I(Vα2) < I(Vα1), the transition Vα1Vα2 → Vα2Vα1 clearly reduces the number of inver- sions, and by changing Vα1Vα2 to cijViα1+jα2 we get inv′ = � ξ<µ iξ>iµ 1 + � µ>b+1 iµI(Viα1+jα2 ) 1 Because of the choice of an additive ordering on the set of positive roots in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1) I(Vα2) < I(Viα1+jα2) < I(Vα1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence the second term of inv′ is � µ>b+1 iµb+1 iµb+1 iµ∈[I(Vα2 ),I(Vα1 )) 1 + � µ>b+1 iµb+1 iµ∈[I(Vα2 ),I(Vα1 )) 1 + � 2 � µ>b+1 iµI(Viα1+jα2 ) 1 ≤ � ξI(Vα2 ) 1 = � ξI(Vα1 ) 1 ≤ � ξI(Vα1 ) 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Hence inv′ < inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' for the relation (R-5’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The proofs for the relations (R-7’) and (R-8’) are similar and hence omitted, the key necessary things corresponding to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1) being I(Uβ) < I(Uα+β) < I(Vα) for the relation (R-7’) and I(U−α) < I(Wδi) < I(Vα) for the relation (R-8’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proof of the main theorem In this section we recall and prove the main theorem in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Both the proofs are exactly same proof as that of Clozel (see [Clo11, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 554, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='4]), we recall them here for the purpose of completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The mod-p Iwasawa algebra Ω(I) is naturally isomorphic to A/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The natural map ψ : A → Λ(I) sends Mn A to Mn Λ(I), where MΛ(I) is the maximal ideal of Λ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The reduction mod-p of ψ respects the natural filtration on both sides and thus induce a natural map gr•ψ : gr•A → gr•Λ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The induced map on the associated graded is surjective since ψ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Thus it is an isomorphism by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' This then imply the theorem since the filtration on B is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The Iwasawa algebra Λ(I) is naturally isomorphic to A/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' 18 ARANYA LAHIRI AND JISHNU RAY We replicate the proof which is also the same argument as Clozel’s corresponding statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The reduction of ψ : A/R → Λ(I) is ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Recall that R is the image of R in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Assume f ∈ A satisfies ψ(f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' We then have f ∈ R by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' So f = r1 +pf1, r1 ∈ R, f1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Then by applying ψ to both sides we get ψ(f1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Inductively, we obtain an expression f = rn + pnfn with rn ∈ R and fn ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Since pnfn tends to 0 in A and R is closed, we see that f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' □ References [AB06] K Ardakov and K Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Ring-theoretic properties of Iwasawa algebras: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Documenta Mathematica, Extra Volume: John H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Coates’ Sixtieth Birthday (2006):7–33, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [AWZ08] Konstantin Ardakov, Feng Wei, and James J Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Reflexive ideals in Iwasawa algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Ad- vances in Mathematics, 218(3):865–901, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Clo11] Laurent Clozel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Presentation of an Iwasawa algebra: the case of Γ1SL(2, Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Doc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', 16:545– 559, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Clo17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Clozel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Globally analytic p-adic representations of the pro-p-Iwahori subgroup of GL(2) and base change, I: Iwasawa algebras and a base change map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Iranian Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', 43(4):55–76, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Clo18] Laurent Clozel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Globally analytic p-adic representations of the pro-p Iwahori subgroup of GL(2) and base change, II: A Steinberg tensor product theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In Cohomology of arithmetic groups, volume 245 of Springer Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', pages 1–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Springer, Cham, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Coa99] John Coates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Fragments of the GL2 Iwasawa theory of elliptic curves without complex multipli- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', pages 1–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Springer, Berlin, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [CSS03] John Coates, Peter Schneider, and Ramdorai Sujatha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Modules over Iwasawa algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Jussieu, 2(1):73–108, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [HLS11] Kevin Hare, Shanta Laishram, and Thomas Stoll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Stolarsky’s conjecture and the sum of digits of polynomial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proceedings of the American Mathematical Society, 139(1):39–49, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [HRW21] Dong Han, Jishnu Ray, and Feng Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Normal elements in the Iwasawa algebras of Chevalley groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' manuscripta mathematica, 165(3):415–451, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [LS22] Aranya Lahiri and Claus Sorensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Rigid vectors in p-adic principal series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' to appear in Israel Journal of Mathematics, arXiv: 2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='02952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [OS19] Rachel Ollivier and Peter Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' The modular pro-p Iwahori-Hecke Ext-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' In Repre- sentations of reductive groups, volume 101 of Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', pages 255–308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', Providence, RI, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Pap94] Paolo Papi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' A characterization of a special ordering in a root system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Proceedings of the American Mathematical Society, 120(3):661–665, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Ray18a] Jishnu Ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Explicit ring-theoretic presentation of Iwasawa algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Comptes Rendus Mathe- matique, 356(11):1075–1080, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Ray18b] Jishnu Ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Journal of Algebra, 511:405–419, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Ray20] Jishnu Ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Explicit presentation of an Iwasawa algebra: the case of pro-p Iwahori subgroup of SLn(Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=', 32(2):319–338, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Sch11] Peter Schneider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' p-adic Lie groups, volume 344 of Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Springer, Heidelberg, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' [Ste16] Robert Steinberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Lectures on Chevalley groups, volume 66 of University Lecture Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Ameri- can Mathematical Society, Providence, RI, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Notes prepared by John Faulkner and Robert Wilson, Revised and corrected edition of the 1968 original [ MR0466335], With a foreword by Robert R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Snapp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' PRESENTATION OF AN IWASAWA ALGEBRA: THE PRO-p-IWAHORI OF REDUCTIVE GROUPS 19 [WE21] Carl Wang-Erickson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' Higher Yoneda product structures and Iwasawa algebras modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='06295, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' (Lahiri) University of California San Diego Email address: arlahiri@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='edu (Ray) Harish Chandra Research Institute, Chhatnag Road, Jhunsi, Prayagraj (Allahabad) 211 019 India Email address: jishnuray@hri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content=' jishnuray1992@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9AyT4oBgHgl3EQflvh2/content/2301.00458v1.pdf'} diff --git a/odAyT4oBgHgl3EQflvgJ/vector_store/index.pkl b/odAyT4oBgHgl3EQflvgJ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..897b878a65c44edcd9275f2ed1b696603588603c --- /dev/null +++ b/odAyT4oBgHgl3EQflvgJ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:210e0ff4839aa61c726a3e130960ab701797cce27f67761649a8a2ca5d8128b1 +size 149783 diff --git a/odAzT4oBgHgl3EQfqv0U/vector_store/index.faiss b/odAzT4oBgHgl3EQfqv0U/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..df32484bc20a19718778952a058ec3193d985f2a --- /dev/null +++ b/odAzT4oBgHgl3EQfqv0U/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c2c0e3debea71aa6a2b018e1290d7066d7747809b06d4daa3f91c4ae1177c29f +size 1114157 diff --git a/pNFKT4oBgHgl3EQfyS7t/content/tmp_files/2301.11907v1.pdf.txt b/pNFKT4oBgHgl3EQfyS7t/content/tmp_files/2301.11907v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa0dde17c0f3ed775b9f6af19f8b82c2f7d4e583 --- /dev/null +++ b/pNFKT4oBgHgl3EQfyS7t/content/tmp_files/2301.11907v1.pdf.txt @@ -0,0 +1,1022 @@ +arXiv:2301.11907v1 [math.RA] 27 Jan 2023 +UNIVERSAL ENVELOPING OF A GRADED LIE +ALGEBRA +FELIPE YUKIHIDE YASUMURA +Abstract. In this paper we construct a graded universal enveloping +algebra of a G-graded Lie algebra, where G is not necessarily an abelian +group. If the grading group is abelian, then it coincides with the classical +construction. We prove the existence and uniqueness of the graded en- +veloping algebra. As consequences, we prove a graded variant of Witt’s +Theorem on the universal enveloping algebra of the free Lie algebra, and +the graded version of Ado’s Theorem, which states that every finite- +dimensional Lie algebra admits a faithful finite dimensional represen- +tation. Furthermore we investigate if a Lie grading is equivalent to an +abelian grading. +1. Introduction +The universal enveloping algebra of a Lie algebra is a classical and im- +portant construction, and it associates the representations of a Lie algebra +to representations of an associative algebra, see any standard book on Lie +algebra, for instance, [6]. In this paper, we investigate the construction but +starting with a G-graded Lie algebra where G is a group. It is well-known +that, if G is abelian, then the universal enveloping algebra of a Lie algebra +inherits a natural G-grading. However, it is known that the universal en- +veloping algebra need not be a graded algebra if the grading group G is not +abelian. +Let G be an abelian group and L a G-graded Lie algebra. Then we can +consider an ordered homogeneous basis {ei | i ∈ I} of L. +Let [ei, ej] = +� +ℓ αijℓeℓ. Denote by XG = � +g∈G Xg where Xg = {x(g) +1 , x(g) +2 , . . .}, and by +F⟨XG⟩ the free G-graded associative algebra over the field F. The universal +enveloping algebra of L is U(L) ∼= F⟨XG⟩/J, where J is the (ordinary) ideal +generated by all {xixj − xjxi − � +ℓ∈I αijℓxℓ | i, j ∈ I}. Since the elements +generating J are homogeneous, we get that J is a graded ideal. Thus, U(L) +is a G-graded algebra. +We extend the above construction for a G-graded Lie algebra where G +is a not necessarily abelian group (see Theorem 9 and Theorem 12). Our +construction agrees with the classical one when the group is abelian. We +find a PBW basis (Theorem 16). As a consequence, we prove an adapted +version of Witt’s Theorem in the context of the free G-graded Lie algebra +Supported by S˜ao Paulo Research Foundation (FAPESP), grant 2018/23690-6. +1 + +2 +FELIPE YUKIHIDE YASUMURA +(Corollary 23). We also deduce a graded version of Ado’s Theorem, that +is, we show that every finite-dimensional G-graded Lie algebra is a graded +vector subspace of a finite-dimensional G-graded associative algebra (The- +orem 34). We study the problem if every group grading on a Lie algebra +is equivalent to an abelian group grading. We apply our constructions and +find an equivalent formulation to this problem (Corollary 41). +2. Notations and Preliminaries +Let G be a group and A an algebra (not necessarily associative nor Lie), +over a field F. +A G-grading on A is a vector space decomposition A = +� +g∈G Ag such that AgAh ⊆ Agh, for all g, h ∈ G. A G-graded algebra +is an algebra endowed with a G-grading. The vector subspace Ag is called +the homogeneous component of degree g, and its nonzero elements are called +homogeneous of degree g. When the decomposition is not explicitly given, +we shall write (A)g to denote the homogeneous component of degree g. +Given 0 ̸= x ∈ Ag, we denote degG x = g, or simply deg x = g when there is +no risk of ambiguity. The support of the grading is defined as +SuppΓ = {g ∈ G | Ag ̸= 0}. +A vector subspace S ⊆ A is said to be graded if S = � Ag ∩ S. A graded +subalgebra is a subalgebra which is a graded subspace. Analogously we define +a graded ideal. If S is a graded ideal of A, then the quotient A/S inherits +the structure of G-graded algebra. +Given two G-graded algebras A = � +g∈G Ag and C = � +g∈G Cg, a G- +graded homomorphism is an algebra homomorphism ψ : A → C such that +ψ(Ag) ⊆ Cg, for all g ∈ G. +Let Γ : A = � +g∈G Ag and Γ′ : A = � +h∈H A′ +h be two gradings on the +same algebra A. +We say that Γ and Γ′ are equivalent if there exists an +algebra automorphism ψ of A, such that for each g ∈ G there exists h ∈ H +satisfying ψ(Ag) = A′ +h. We say that a grading Γ is realized by a group G if +there exists a G-grading equivalent to Γ. We say that a grading is abelian +if it is realized by an abelian group. +It is worth mentioning that if two gradings on the same algebra A are +equivalent by an isomorphism ψ and I ⊆ A is a graded ideal, then I and +ψ(I) are equivalent and so are A/I and A/ψ(I). +2.1. Universal group of a grading. We follow [2] in this and next subsec- +tions. Given a G-grading, it is relevant to consider the group with minimal +amount of relations that realizes the grading. Here is the formal definition. +Definition 1 ([2, Definition 1.17]). Let Γ be a G-grading on an algebra +A. The universal grading group of Γ is the group U(Γ) such that, for every +realization of Γ as an H-grading, there exists a unique group homomorphism +U(Γ) → H that is the identity map on SuppΓ. + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +3 +The universal grading group of a group grading Γ always exists. It may be +taken as the group with the set of generators Supp Γ, and relations s1s2 = s3, +for s1, s2, s3 ∈ Supp Γ whenever 0 ̸= As1As2 ⊆ As3 (see [2, Proposition +1.18]). +It is also relevant to consider the universal abelian grading group of a +grading Γ. It is defined by the abelianization of U(Γ), that is, Uab(Γ) = +U(Γ)/U(Γ)′, where G′ is the commutator group of the group G. It is not +always true that the universal abelian group grading realizes the grading +Γ. +Indeed, some homogeneous components may coalesce under this new +group. The grading is realized by Uab(Γ) if and only if the initial grading Γ +is abelian. +2.2. Refinement and coarsening. Let Γ : A = � +g∈G Ag and Γ′ : A = +� +h∈H A′ +h be two gradings on A. We say that Γ′ is a refinement of Γ (or +that Γ is a coarsening of Γ′) if for every h ∈ SuppΓ′, there exists g ∈ Supp Γ +such that A′ +h ⊆ Ag. The following is relevant to us: +Lemma 2 ([2, Part of Proposition 1.25]). Let Γ and Γ′ be gradings on A, and +assume that Γ′ is realized as a H-grading and is a coarsening of Γ. Then, +there exists a group epimorphism p : U(Γ) → H such that p(SuppΓ) = +SuppΓ′. +2.3. Free algebra. For each g ∈ G, we let Xg = {x(g) +1 , x(g) +2 , . . .}, and XG = +� +g∈G Xg. Then, the free associative algebra F⟨XG⟩, freely generated by XG, +has a natural G-grading where the monomials are homogeneous and +degG x(g1) +i1 +· · · x(gm) +im += g1 · · · gm. +It satisfies the following universal property: for any associative G-graded +algebra A, and each map ψ0 : XG → A respecting the degrees (that is, +ψ0(x(g) +i ) ∈ Ag), there exists a unique G-graded algebra homomorphism +ψ: F⟨XG⟩ → A extending ψ0. Hence, we can define a G-graded polyno- +mial identity of a G-graded associative algebra as an element of the set +� +ψ: F⟨XG⟩→A +G-graded homomorphism +Ker ψ. +2.4. Free pair. We recall the notion of a variety of associative-Lie pairs. +An associative Lie-pair is a pair (L, A) where A is an associative algebra +generated by L, and L is a Lie algebra which is a vector subspace of A, and +the product of the L coincides with the restriction of the commutator of A +to L. +Given two pairs (L, A) and (H, C), a homomorphism of pairs is an algebra +homomorphism ψ: A → C such that ψ(L) ⊆ H. Hence, ψ restricts to a Lie +homomorphism L → H. +Let W be a class of associative-Lie pairs. A pair (L, F) is free in the class +W , freely generated by X, if for any pair (H, C) in W , and for any map + +4 +FELIPE YUKIHIDE YASUMURA +ψ0 : X → H, there exists a unique homomorphism of pairs from (L, F) to +(H, B) extending ψ0. It is clear that (L(X), F⟨X⟩), where L(X) is the Lie +subalgebra of the free associative algebra generated by X (that is, the free Lie +algebra) is a free pair, freely generated by X, in the class of all associative- +Lie pairs. Now, due to the existence of a free associative-Lie pair, we can +speak of polynomial identities of pairs. By tradition these are called weak +polynomial identities of the pair (L, A) (or identities of representations of +Lie algebras). +The above discussion can be extended to the context of G-graded associative- +Lie pairs (L, A): a G-graded associative algebra A, and a G-graded Lie al- +gebra L that is a G-graded subspace of A, where the product of L is the +commutator of A restricted to L. We shall prove below that there exists a +G-graded free associative-Lie pair. +3. Graded Universal Enveloping Algebra +We let G be a non-necessarily abelian group. +Notation. By a G-pair (L, A), we understand a G-graded associative-Lie +pair, i.e. A is a G-graded associative algebra, L is a G-graded subspace of A +such that L is also a G-graded Lie algebra with respect to the commutator +of A restricted to L. +Equivalently, a G-pair is a pair (L, ι), where L is +a G-graded Lie algebra, A is a G-graded associative algebra, and ι: L → +A is a graded linear map such that ι: L → A(−) is an (ungraded) Lie +monomorphism. +Note that the G-grading on the vector space A does not necessarily define +a G-graded Lie algebra with respect to the commutator. If this is the case, +then we shall call the G-pair (L, A) a strong G-pair. +Definition 3. Let L be a G-graded Lie algebra. +The G-graded univer- +sal enveloping algebra of L is a G-pair (L, UG(L)) such that: (i) UG(L) is +generated by L, and (ii) for every G-pair (H, A) and every G-graded Lie +homomorphism L → H, there exists an extension to a G-graded algebra +homomorphism UG(L) → A. +As it is customary, we shall call the algebra UG(L) a G-graded universal +enveloping algebra of L. We shall prove the existence and uniqueness of +UG(L). It will be an appropriate quotient of the usual universal enveloping +algebra U(L). +Remark. In what follows, we may weaken the definition of a pair, replacing +the condition of ι : L → A(−) being a monomorphism to ask ι to be only +a homomorphism. This is because we cannot guarantee (until Theorem 16) +that L → UG(L) is an embedding. +Lemma 4. Let (L, A) be a G-pair and let x, y ∈ L be homogeneous elements +such that deg x = g, deg y = h, and gh ̸= hg. Then xy = 0. +Proof. Write L = � +g∈G Lg and A = � +g∈G Ag, then Lg ⊆ Ag, for all g ∈ G. + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +5 +By g = deg x and h = deg y we get yx ∈ Ahg and xy ∈ Agh. On the other +hand, +−[y, x] = −yx + xy ∈ Lhg ⊆ Ahg. +Thus xy = yx + (−yx + xy) ∈ Ahg ∩ Agh = 0. +□ +In the language of polynomial identities, we have: +Corollary 5. Let (L, A) be a G-pair. Then the pair satisfies the weak G- +graded polynomial identities +x(g) +1 x(h) +2 += 0, +[g, h] ̸= 1. +□ +Let L be a G-graded Lie algebra with a homogeneous vector space basis +B = {ei | i ∈ N}, where N is ordered, and denote [ei, ej] = � +ℓ∈N α(k) +ij ek. +Let XB = {xi | i ∈ I}. Let F⟨XB⟩ be the free associative G-graded algebra, +freely generated by XB, where deg xi = degG ei, for each i ∈ N. Define IB +to be the graded ideal generated by +(1) +{xixj − xjxi − +� +k +α(k) +ij xk | i, j ∈ N}, +that is, IB is the least graded ideal containing all the elements above. Denote +further by J the ungraded ideal generated by the same elements, so that +U(L) ∼= F⟨XB⟩/J. The algebra F⟨XB⟩/IB is graded, and (L, F⟨XB⟩/IB) is a +G-pair. +Lemma 6. Let JB be the ideal of U(L) generated by +{eiej | i, j ∈ N, [deg ei, deg ej] ̸= 1}. +Then +F⟨XB⟩/IB ∼= U(L)/JB. +Proof. Since J ⊆ IB, there exists a surjective algebra homomorphism π : +U(L) → F⟨XB⟩/IB. By Lemma 4, since (L, F⟨XB⟩/IB) is a G-pair, then +JG ⊆ Ker π. So, we get that +IB ⊇ J + K, +where K is the (possibly ungraded) ideal ⟨xixj | i, j ∈ N, [deg xi, deg xj] ̸= +1⟩. Note that K is actually graded, since its generators are homogeneous +elements. Now, each of the elements (1) is either homogeneous (this is the +case whenever [deg xi, deg xj] = 1), or it belongs to K (if [deg xi, deg xj] ̸= +1). Hence, J +K is a graded ideal. Thus we obtain that J +K ⊇ IB, proving +the result. +□ +Now we can prove the existence of a G-graded universal enveloping algebra +of L. +Lemma 7. The G-pair (L, F⟨XB⟩/IB) is a G-graded universal enveloping +algebra of L. + +6 +FELIPE YUKIHIDE YASUMURA +Proof. Let (H, A) be a G-pair and ϕ: L → H be a G-graded Lie homo- +morphism. Then ϕ admits an extension (also denoted by ϕ) to an algebra +homomorphism ϕ: U(L) → alg(H) ⊆ A. +Here alg(H) is the associative +subalgebra generated by H. If [deg ei, deg ej] ̸= 1, then Lemma 4 tells us +that +0 = ϕ(ei)ϕ(ej) = ϕ(eiej). +Thus, ϕ factors through JB, that is, there exists a map ¯ϕ: U(L)/JB → A. +By Lemma 6, U(L)/JB ∼= F⟨XB⟩/IB. +The map ¯ϕ is a graded map and +extends ϕ, concluding the proof. +□ +Finally, it is a standard exercise to prove that the G-graded universal +enveloping algebra is unique, up to an isomorphism. +Lemma 8. Let UG and VG be G-graded universal enveloping algebras of L. +Then there exists a G-graded isomorphism ι: UG → VG such that ι(x) = x, +for each x ∈ L. +Proof. Since (L, VG) is a pair, the identity map L → L admits an extension +to a G-graded algebra homomorphism ι: UG → VG. Conversely, the identity +map also admits an extension to a G-graded homomorphism j : VG → UG. +Since the composition jι: UG → UG fixes all the elements of L, it must be +the identity map. Similarly, ιj is the identity map. +□ +We summarize our results. Of course, from a vector space basis of U(L), +we obtain a set of generators of UG(L) as a vector space. +Theorem 9. Let L be a G-graded Lie algebra, where G is a non-necessarily +abelian group. Then it admits a unique, up to an isomorphism, G-graded +universal enveloping algebra UG(L). If {ei | i ∈ I} is an ordered homoge- +neous basis of L, then +ei1 · · · eim, m ≥ 0, i1 ≤ · · · ≤ im, [deg eij, deg eij+1] = 1, j ∈ {1, 2, . . . , m−1}, +is a set of homogeneous elements that spans UG(L). +□ +Example 1. If G is abelian and L is a G-graded algebra, then UG(L) = U(L). +Example 2. Let G = C2 ∗ C2 = ⟨g, h | g2 = h2 = 1⟩, and L = Span{x, y} +be the 2-dimensional abelian Lie algebra. Define a G-grading on L where +deg x = g and deg y = h. Then, it is known that U(L) ∼= F[X, Y ], the poly- +nomial algebra in two commuting variables. Since UG(L) ∼= F[X, Y ]/⟨XY ⟩, +one obtains that UG(L) is the associative unital subalgebra of F[X] ⊕ F[Y ] +generated by X and Y , that is, +UG(L) = {α(1, 1) + (F(X), G(Y )) | α ∈ F, F(X) ∈ F[X], G(Y ) ∈ F[Y ]}. +4. Strong pair +We assume that G is a not necessarily abelian group. We investigate when +UG(L) is a G-graded Lie algebra with respect to the commutator. To this +end, we need to investigate strong pairs, that is, G-pairs (L, A), where A is + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +7 +a G-graded Lie algebra with respect to the commutator. Let VG denote the +variety of G-graded associative algebras satisfying the polynomial identities +� +x(g) +1 x(h) +2 += 0 | g, h ∈ G, [g, h] ̸= 1 +� +. +Whenever we have a strong pair (L, A), from Lemma 4, we have A ∈ VG. +Denote by XG = � +g∈G Xg, where Xg = {x(g) +1 , x(g) +2 , . . .}, for each g ∈ G. +Let FG(XG) be the relatively free algebra in VG, freely generated by XG. +Then +FG(XG) ∼= F⟨XG⟩/⟨x(g) +1 x(h) +2 +| [g, h] ̸= 1⟩TG. +We slightly modify the definition of G-graded universal enveloping algebra +of a Lie algebra. +Definition 10. Let L be a G-graded Lie algebra. +The strong G-graded +universal enveloping of L is a strong pair (L, SUG(L)) such that for every +strong pair (H, A), each G-graded Lie homomorphism L → H admits a +unique extension to a G-graded algebra homomorphism SUG(L) → A. +A standard argument shows that if a strong G-graded universal enveloping +algebra exists, then it is unique up to an isomorphism that is the identity +map on L. Moreover, L generates SUG(L). If it exists, it is a quotient of +the G-graded universal enveloping algebra. +Lemma 11. If a strong G-graded universal enveloping algebra of L exists, +then it is a quotient of UG(L). +Proof. If (L, SUG(L)) is a strong pair, then it is also a G-pair. Hence there +exists an extension of the identity map L → L to an algebra homomorphism +UG(L) → SUG(L). The statement follows since L generates SUG(L). +□ +Example 3. If G is an abelian group, then every G-pair is a strong pair. +Hence, SUG(L) exists and it coincides with UG(L) (which in turn equals +U(L)). +Example 4. Consider once again Example 2: let G = C2 ∗ C2 = ⟨g, h | +g2 = h2 = 1⟩ and L = Span{x, y} the 2-dimensional abelian Lie algebra, +where deg x = g and deg y = h. The G-graded universal enveloping algebra +of L is a Lie algebra with respect to the commutator (indeed, UG(L) is a +commutative algebra). In particular, it satisfies the definition of the strong +G-graded universal enveloping algebra, so SUG(L) = UG(L). +Example 5. More generally, if UG(L) happens to be a G-graded Lie algebra +with respect to the commutator, then SUG(L) exists and it coincides with +UG(L). Conversely, if SUG(L) = UG(L), then UG(L)(−) is a G-graded Lie +algebra. +Now, we shall prove the existence of a strong G-graded universal envelop- +ing algebra. The construction is similar to that of the G-graded universal +enveloping algebra, but we need to work in the variety VG. + +8 +FELIPE YUKIHIDE YASUMURA +Let L be a G-graded Lie algebra, and let B = {ei | i ∈ N} be a homoge- +neous basis of L. Consider the structure constants α(k) +ij , i, j, k ∈ N, that +is, +[ei, ej] = +� +k∈N +α(k) +ij ek. +We let XB = {zi = x(gi) +i +| i ∈ N, gi = deg ei}, and let FG(XB) be the +relatively free algebra in VG, freely generated by XB. Let JB be the ideal of +FG(XB) generated by all +zizj − zjzi − +� +k∈N +α(k) +ij zk, +i, j ∈ N. +The above elements are homogeneous. Indeed, either [deg zi, deg zj] = 1, or +zizj = zjzi = [zi, zj] = 0. Thus JB is a graded ideal. +Theorem 12. SUG(L) = FG(XB)/JB. +Proof. Let (H, A) be a strong pair, and let f0 : L → H be a G-graded Lie +homomorphism. Then f0 restricts to a map XB → H ⊆ A which respects +the degrees. +Since A ∈ VG, the restriction of f0 extends to a G-graded +algebra homomorphism f : FG(XB) → A. Since both FG(XB) and A are G- +graded Lie algebras with respect to the commutators, then f factors through +JB. +Thus f0 admits an extension to a G-graded algebra homomorphism +FG(XB)/JB → A. +□ +As a consequence, every G-graded Lie algebra is a subalgebra of some +A(−), where A is a G-graded associative algebra such that A(−) is a G- +graded Lie algebra. +Notation. Whenever it is convenient, we shall identify the homogeneous +variables zi ∈ XB with the basis elements ei ∈ B. So, SUG(L) is spanned by +all elements ei1 · · · eim, where ei1, . . . , eim ∈ B. +5. A PBW basis for SUG(L) +In this section, we shall find a homogeneous vector space basis for SUG(L). +For, we let XG = � +g∈G Xg be a set of G-homogeneous variables (where each +Xg is finite or not). We start finding a homogeneous vector space basis for +FG(XG). +Notation. If {g1, . . . , gm} is a subset of a group, then the abbreviation +g.a.s. means that {g1, . . . , gm} generates an abelian subgroup. +Lemma 13. The relatively free algebra FG(XG) has, as a G-homogeneous +vector space basis, all monomials of the kind +x(g1) +i1 +· · · x(gm) +im , +{g1, . . . , gm} g.a.s. . +Proof. It is clear that such a set generates FG(XG) as a vector space. So, it +is sufficient to prove that it is linearly independent. Consider a finite subset +S of the above monomials. We may assume that a fixed set of variables, + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +9 +say Y = {x(g1) +i1 +, . . . , x(gm) +im }, appears in all the monomials of S. +We can +assume that the subgroup generated by {g1, . . . , gm} is abelian, otherwise +every monomial in S would be zero. So, let H = ⟨g1, . . . , gm⟩ be the abelian +subgroup of G generated by g1, . . . , gm. +The free associative H-graded +algebra F⟨Y ⟩, freely generated by Y , may be seen as a G-graded algebra, +where the homogeneous components in G \ H are 0. Therefore, there exists +a surjective G-graded algebra homomorphism p : FG(XG) → F⟨Y ⟩ via +p(x(g) +j ) = +� +x(g) +j , +if x(g) +j +∈ Y , +0, +otherwise. +Now, the set S is linearly independent under the image of p. So, S is linearly +independent as well. The proof is complete. +□ +Now, let G be any group and L a G-graded Lie algebra over an arbitrary +field F, and let B = {ei | i ∈ N} be a homogeneous vector space basis of L, +where N is ordered. Let +[ei, ej] = +� +ℓ∈N +α(ℓ) +ij eℓ +be the structure constants of L with respect to B. We shall consider the set +of free variables XB = {ei | i ∈ N}, and use the same letters to denote the +homogeneous variables and the elements of B. Let FG(XB) be the relatively +free G-graded algebra in VG, freely generated by XB, and F⟨XB⟩ be the free +associative G-graded algebra, freely generated by XB. +We let R be the vector subspace of F⟨XB⟩ spanned by all +ei1 · · · eim, +m ≥ 0, i1 ≤ · · · ≤ im. +The following is a classical result. +Lemma 14 ([6, Lemma 5.2]). There exists a linear map σ0 : F⟨XB⟩ → R +such that +σ01 = 1, +σ0(ei1 · · · eim) = ei1 · · · eim, if i1 ≤ · · · ≤ im, +σ0(ej1 · · · ejm − ej1 · · · ejk+1ejk · · · ejm) = σ0(ej1 · · · +�� +ℓ∈N +α(ℓ) +jkjk+1eℓ +� +· · · ejm). +As an immediate consequence, we have the following graded version. Let +B be the vector subspace of FG(XB) spanned by all +ei1 · · · eim, +m ≥ 0, i1 ≤ · · · ≤ im, {degG ei1, . . . , degG eim} g.a.s. +Lemma 15. There exists a linear map σ : FG(XB) → B such that +σ(ei1 · · · eim) = 0, + +10 +FELIPE YUKIHIDE YASUMURA +if {degG ei1, . . . , degG eim} does not generate an abelian subgroup, and oth- +erwise, +σ1 = 1, +σ(ei1 · · · eim) = ei1 · · · eim, if i1 ≤ · · · ≤ im, +σ(ej1 · · · ejm − ej1 · · · ejk+1ejk · · · ejm) = σ(ej1 · · · +�� +ℓ∈N +α(ℓ) +jkjk+1eℓ +� +· · · ejm). +Proof. The projection F⟨XB⟩ → FG(XB) induces a map R → B. Let σ0 : +F⟨XB⟩ → R be the linear map of Lemma 14. Then, we have the diagram +0 +0 +F⟨XB⟩ +R +B +FG(XB) +Since the kernel of F⟨XB⟩ → B lies inside the kernel of F⟨XB⟩ → FG(XB) +(see Lemma 13), we obtain the required map σ : FG(XB) → B. +□ +This gives a PBW basis for SUG(L). More precisely, we have +Theorem 16. Let G be any group and L be a G-graded Lie algebra over a +field F. Let {ei | i ∈ N} be a homogeneous basis of L, where N is ordered. +Then, a homogeneous vector space basis of SUG(L) is given by +ei1 · · · eim, +i1 ≤ · · · ≤ im, {degG ei1, . . . , degG eim} g.a.s. +Proof. It is clear, as in the classical case, that the above monomials span +SUG(L). Now, the map σ : FG(XB) → B from Lemma 15 factors through +SUG(L) → B. So, we obtain a surjective linear map SUG(L) → B, where +the above set of monomials is sent to a linearly independent set. Hence, it +is a basis. +□ +As a consequence, we obtain that (L, UG(L)) and (L, SUG(L)) are indeed +pairs. That is, we have the following. +Corollary 17. Let G be any group and L a G-graded Lie algebra. Then +(i) there is an embedding L ֒→ SUG(L), +(ii) there is an embedding L ֒→ UG(L). +Proof. The assertion (i) follows directly from the previous theorem. To prove +(ii), one should consider the diagram + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +11 +0 +0 +L +UG(L) +SUG(L) +where the surjective map UG(L) → SUG(L) is from Lemma 11. +□ +6. Free graded Lie algebra +There is a direct way to construct the G-graded free Lie algebra. It was +mentioned, for instance, in [3]. +We include its construction here for the +sake of completeness. Let F{XG} be the absolutely free (nonassociative) +G-graded algebra, and let I be the graded T-ideal generated by all elements +of the following types: +x(g1) +1 +x(g2) +2 ++ x(g2) +2 +x(g1) +1 +, +(x(g1) +1 +x(g2) +2 +)x(g3) +3 ++ (x(g2) +2 +x(g3) +3 +)x(g1) +1 ++ (x(g3) +3 +x(g1) +1 +)x(g2) +2 +. +Denote L(XG) = F{XG}/I. By construction, L(XG) is a G-graded algebra, +and it is clearly a Lie algebra. +Proposition 18. L(XG) is free in the class of all G-graded Lie algebras, +and it is freely generated by the set XG. +Proof. Let H be a G-graded Lie algebra, and let ϕ: XG → H be a degree- +preserving map, that is, deg ϕ(x(g) +i ) = g. Then ϕ admits an extension to a +G-graded homomorphism ϕ: F{XG} → H. Then Ker ϕ is a graded ideal; +and, since H is a graded Lie algebra, it contains the generators of I. Thus, +ϕ factors through F{XG}/I, that is, ϕ induces a graded Lie homomorphism +L(XG) → H that extends the map XG → H. +□ +6.1. Free graded algebras and pairs. We fix a group G. As a conse- +quence of Corollary 5, we may consider the variety WG of pairs satisfying +the weak G-graded polynomial identities +x(g) +1 x(h) +2 += 0, +[g, h] ̸= 1. +Let L(XG) be the free G-graded Lie algebra, and let UG(L(XG)) be its +respective G-graded universal enveloping algebra. +Proposition 19. The pair (L(XG), UG(L(XG))) is free in the variety of +pairs WG, and XG is a set of free generators. +Proof. Let (H, A) be a G-pair, and let ψ0 : XG → H be a degree-preserving +map. +Since L(XG) is free, then ψ0 admits a unique extension to a Lie +homomorphism L(XG) → H. So, it admits a unique extension to an algebra +homomorphism UG(L(XG)) → A. Thus, (L(XG), UG(L(XG))) is free in the +variety of pairs WG, freely generated by XG. +□ + +12 +FELIPE YUKIHIDE YASUMURA +Conversely, we may obtain the free G-graded Lie algebra from a free pair +in WG. So, let (L, F) be a free pair in the variety of G-graded pairs in WG, +freely generated by X. We let L(X) be the G-graded Lie algebra generated +by X. Then, L(X) = L. More precisely, we have: +Proposition 20. The algebra L(X) is the free G-graded Lie algebra, freely +generated by X, and the G-pair (L(X), UG(L(X))) is free in the variety of +pairs WG, and X is a set of free generators. +Proof. Let (H, A) be a G-pair, and let ψ0 : X → H be a degree-preserving +map. Since the pair belongs to WG (Corollary 5) and (L, F) is free, there +exists an extension of ψ0 to a G-graded algebra homomorphism ψ: F → A, +which restricts to a Lie homomorphism ¯ψ: L → H. In particular, ψ restricts +to a Lie homomorphism from L(X) → H. Every G-graded Lie algebra may +be thought of as the pair given by itself and its G-graded universal enveloping +algebra, we get that L(X) is a free G-graded Lie algebra. Now ¯ψ extends +uniquely to a homomorphism UG(L(X)) → A. Hence (L(X), UG(L(X))) is +a free pair in WG, and it is freely generated by X. +□ +As a consequence of Proposition 19 and Proposition 20, we obtain an +alternative description of the free G-graded Lie algebra, and a relation with +the free G-pair in the variety WG. This is a graded version of Witt’s Theorem +(see, for instance, [1, Theorem 1.3.5] or [6, Theorem V.7] for the classical +case). +Corollary 21. Let (L, F) be free in the variety of pairs WG, freely generated +by X. Then L coincides with the Lie algebra L(X), generated by X, and it +is the free G-graded Lie algebra, freely generated by X, and F = UG(L). +□ +6.2. Strong Witt’s Theorem. Now, we obtain an outright graded version +of Witt’s Theorem. Let FG(XG) be the relatively free G-graded associative +algebra in the variety VG, and let L(XG) be its Lie subalgebra generated by +XG. Then we have +Theorem 22. The algebra L(XG) is a free G-graded Lie algebra, freely +generated by XG. Moreover, SUG(L(XG)) = FG(XG). +Proof. We let H be a G-graded Lie algebra and f0 : XG → H a map respect- +ing degrees. Now, since H ⊆ SUG(L), the map f0 extends to a G-graded +algebra homomorphism f : FG(XG) → SUG(H). Such maps restricts to a +G-graded Lie homomorphism L(XG) → Lie(H) = H. So, L(XG) is free, +freely generated by XG. +Now, let (H, A) be a strong pair, and f : L(XG) → H a G-graded Lie +homomorphism. Then, f restricts to a graded map XG → H, which admits +an extension to a G-graded algebra homomorphism ¯f : FG(XG) → A. Since +the restriction of ¯f to L(XG) is a G-graded Lie homomorphism, and f is +the unique extension of the map XG → H, we get that the restriction of ¯f +coincides with f. It means that ¯f is an extension of f. So, SUG(L(XG)) = +FG(XG). +□ + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +13 +Combining Corollary 21 and Theorem 22 we obtain a complete version of +the graded version of Witt’s Theorem. We summarize the results, recalling +the main definitions. +Corollary 23. Let G be a group, and consider the set of G-graded polyno- +mials +(2) +x(g) +1 x(h) +2 , +[g, h] ̸= 1, g, h ∈ G. +Let VG be the variety of G-graded associative algebras satisfying the identi- +ties (2), and WG be the variety of G-graded associative-Lie pairs satisfying +the weak-polynomial identities (2). Let XG be a set of G-graded variables, +FG(XG) be the relatively free algebra in VG, freely generated by XG, and +(L, F) be a free pair in WG, freely generated by XG. Then: +(i) The Lie subalgebra LG(XG) of FG(XG) generated by XG, is the +free G-graded Lie algebra, freely generated by XG. +Furthermore, +SUG(LG(XG)) = FG(XG). +(ii) The Lie subalgebra of F generated by XG coincides with L, and it +is the free G-graded Lie algebra, freely generated by XG. Moreover, +UG(L) = F. +□ +7. A graded version of Ado’s Theorem +In this section, we assume that char F = 0. +If L is a G-graded Lie algebra, then it is known that its center is a graded +ideal. +Moreover, the following result due to Gordienko (also proved by +Pagon, Repovˇs and Zaicev in [7]) is useful: +Theorem 24 ([4, Corollary 4.2 and Corollary 4.3]). Let L be a finite- +dimensional G-graded Lie algebra over a field of characteristic zero, and +R its solvable radical. +(1) The nilradical and the solvable radical of L are graded ideals. +(2) (Graded Levi’s decomposition) There exists a graded subalgebra L1 +such that L1 ∩ R = 0 and L = R + L1. +We shall follow the classical ungraded version of the proof of Ado’s Theo- +rem. The main idea is to find a graded ideal I ⊆ UG(L) of finite codimension +such that L ∩ I = 0. +Lemma 25. Let L be a G-graded solvable Lie algebra, R its nilradical, and +I ⊆ UG(L) a graded ideal of finite codimension. Assume that every x ∈ R is +nilpotent modulo I. Then there exists a graded ideal J ⊆ UG(L) such that: +(1) J ⊆ I, +(2) J has finite codimension in UG(L), +(3) every x ∈ R is nilpotent modulo J, +(4) every derivation D of L, that can be extended to UG(L), satisfies +DJ ⊆ J. + +14 +FELIPE YUKIHIDE YASUMURA +Proof. Let I′ be the graded ideal of UG(L) generated by I and R. Then +L/L ∩ I ∼= (L + I)/I is a G-graded Lie algebra, and I′/I is one of its graded +ideals. Since dim L/L ∩ I < ∞ and I′/I contains a basis constituted of +nilpotent elements, then I′/I is nilpotent. Hence there exists m ∈ N such +that +J := (I′)m ⊆ I. +Clearly J satisfies (1)–(3). On the other hand, since char F = 0, it is known +that DL ⊆ R, for any derivation D of L. +Hence DUG(L) ⊆ I′. +Thus +DJ ⊆ J. +□ +The following lemma is easy to deduce, and we include its statement for +the sake of completeness. +Lemma 26. Let L be a finite-dimensional G-graded Lie algebra. +Then, +IDer(L), the set of inner derivations of L, is a G-graded Lie algebra. +□ +Lemma 27. Let L be a finite-dimensional G-graded Lie algebra, and H ⊆ +EndFL be a G-graded Lie algebra contained in Der(L). Then the Lie subal- +gebra generated by H and IDer(L) is a G-graded Lie algebra. +Proof. Since IDer(L) is a Lie ideal of Der(L), the vector space H + IDer(L) +is a Lie algebra. Since both spaces are graded, their sum is a graded sub- +space as well. +As each one of H and IDer(L) is a graded Lie subalge- +bras, it is enough to show the following property. If D ∈ H and x ∈ L +are homogeneous, then [D, ad x] is either 0 or is homogeneous of degree +degG D degG x. This is clear, since [D, ad x] = ad(D(x)), and D(x) is either +0 or degG D(x) = degG D degG x. +□ +Lemma 28. Let L and H be finite-dimensional G-graded Lie algebras. Let +ρ: H → EndFL be a Lie homomorphism and a G-graded linear map. Then +Im ρ is a G-graded Lie algebra, and ρ is a graded Lie homomorphism. +Proof. We know that Im ρ is a graded subspace. So, it is enough to show +that, given homogeneous u, v ∈ Im ρ, either [u, v] = 0 or [u, v] is homo- +geneous of degree h := degG u degG v. +Let u0, v0 ∈ H be homogeneous +elements such that u = ρ(u0) and v = ρ(v0). Thus, [u, v] = ρ[u0, v0], so it +is enough to show that this last one is a graded linear map of degree h. If +[degg u, degg v] ̸= 1, then [u0, v0] = 0, and there is nothing to do. Otherwise, +if x ∈ L is homogeneous then uvx and vux are both homogeneous elements +of degree h degG x. Hence, [u, v] is homogeneous of degree h. +□ +Example 6. Let G = ⟨α1, α2, α3⟩ be the free group, freely generated by +{α1, α2, α3}. Let L be the 3-dimensional abelian Lie algebra, and assume +that {x1, x2, x3} is a basis of L. Then, imposing degG xi = αi, for i = 1, +2, 3, we obtain a (Lie algebra) G-grading on L. Note that IDer L = 0 and +Der L = EndFL. However, Der L is not a G-graded Lie algebra. Indeed, let +eij ∈ EndFL be the map defined by +eijxℓ = δjℓxi. + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +15 +Then eij is a homogeneous map of degree αiα−1 +j . On one hand, e12e23 = +e13 ̸= 0. On the other hand, [degG e12, degG e23] ̸= 1. Thus, by Lemma 4, it +is not possible to have a structure of a G-graded Lie algebra on Der L, given +by the commutator. +Note that Der L contains several G-graded Lie algebras, with respect to +the commutator. For instance, if 1 ≤ i < j ≤ 3, then Span{eii, eij, eji, ejj} +is a G-graded Lie algebra. +The proposition below is the main step in the proof of our main result of +this section. +Proposition 29. Let L = R + L1 be a finite-dimensional G-graded Lie +algebra, where R is a graded solvable ideal, and L1 is a graded subalgebra. +Assume that J ⊆ UG(R) is a graded ideal satisfying (ii)–(iv) of Lemma 25. +Then there exists a graded ideal J′ ⊆ UG(L) such that +(i) R ∩ J′ ⊆ R ∩ J, +(ii) if x ∈ n(R) and y ∈ L1 is such that adR y is nilpotent, then x + y is +nilpotent modulo J′. +Proof. Consider the map +IDer L → Der(R/R ∩ J), +given by the restriction of the inner derivation to R, followed by the induced +map on the quotient. Since R ∩ J is an ideal of R, the map is well-defined. +Since R is a graded ideal of L, it follows that the map is graded. Thus, by +Lemma 28, its image is a G-graded Lie algebra, say H. Now, consider the +composition +ρ: L → IDer L → H. +Therefore ρ is a G-graded homomorphism of Lie algebras, and it admits +an extension ¯ρ: UG(L) → UG(H). +Let J′ = Ker ¯ρ ∩ ⟨J⟩, where ⟨J⟩ = +UG(L)JUG(L). Note that, by construction, R ∩ J′ ⊆ R ∩ J. Moreover, if +y ∈ L1 is such that adR y is nilpotent, then by construction ¯ρ(y) is nilpotent +too. Thus, y is nilpotent modulo J′. +□ +Now we proceed with the proof of our main result. We divide the proof +in several steps. +Lemma 30. Let z be a finite-dimensional G-graded abelian Lie algebra. +Then there exists a graded ideal of finite codimension I ⊆ UG(z) such that +(i) I ∩ z = 0, and +(ii) every x ∈ z is nilpotent modulo I. +Proof. Since z has zero product, it is also an associative algebra with trivial +product. +So (z, z) is a pair. +Hence, the identity map z → z admits an +extension ρ: UG(z) → z. Let I = Ker ρ. Since dim z < ∞, then I has finite +codimension. Since ρ is a graded homomorphism, I is also a graded ideal. +As ρ is 1–1 in z, we see that I ∩z = 0. Finally, once z has the trivial product, +ρ(UG(z)2) ⊆ z2 = 0. + +16 +FELIPE YUKIHIDE YASUMURA +In particular, z2 ⊆ I. +□ +Lemma 31. Let n be a finite-dimensional G-graded nilpotent Lie algebra, +and let z be its center. Then there exists a graded ideal of finite codimension +I ⊆ UG(z) such that +(i) I ∩ z = 0, and +(ii) every x ∈ n is nilpotent modulo I. +Proof. Consider a chain +z = n0 ⊆ n1 ⊆ · · · ⊆ nt = n, +where each ni is a graded ideal and dim ni = i + dim z. We shall prove the +result by induction on i. If i = 0, then we may apply Lemma 30. So, for +some 0 ≤ i < i, assume that Ii ⊆ UG(ni) is a graded ideal satisfying (i), and +such that each x ∈ ni is nilpotent modulo Ii. Write ni+1 = ni + Li, where +Li is a graded Lie algebra. By Proposition 29, there exists a graded ideal +Ii+1 ⊆ UG(ni+1) satisfying (ii) and such that Ii+1 ∩ zi ⊆ Ii ∩ zi. Thus, +Ii+1 ∩ z = Ii+1 ∩ z ∩ ni ⊆ Ii ∩ ni ∩ z = 0. +Hence, Ii+1 satisfies (i) as well. +□ +Lemma 32. Let R be a finite-dimensional G-graded solvable Lie algebra, +n its nilradical, and z its center. Then there exists a graded ideal of finite +codimension I ⊆ UG(R) such that +(i) I ∩ z = 0, and +(ii) every x ∈ n is nilpotent modulo I. +Proof. Consider a chain of graded subalgebras +n = R0 ⊆ R1 ⊆ · · · ⊆ Rs = R, +where Ri is an ideal of Ri+1, and dim Ri = i + dim n. Then, it is known +that the nilradical of each Ri is n. We may proceed by induction on i, where +the validity for i = 0 holds thanks to Lemma 31. Then, we may repeat the +argument given in the proof of the previous lemma. +□ +Corollary 33. Let L be a finite-dimensional G-graded Lie algebra and z its +center. Then there exists a graded ideal of finite codimension I ⊆ UG(L) +such that I ∩ z = 0. +Proof. It is enough to combine the graded Levi’s decomposition (Theorem +24), Lemma 32 and Proposition 29. +□ +Now, we are in a position to prove the main result of this section. +Theorem 34. Let L be a finite-dimensional G-graded Lie algebra over a +field of characteristic zero, where G is a non-necessarily abelian group. Then +there exists a finite-dimensional G-graded associative algebra A such that +(L, A) is a G-pair. In other words L ⊆ A is a graded vector subspace, and +L is a G-graded Lie algebra with respect to the commutator of A. + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +17 +Proof. It is enough to find a graded ideal I ⊆ UG(L) of finite codimension +such that I ∩ L = 0. If I is such an ideal then A = UG(L)/I is a finite- +dimensional G-graded associative algebra, and the natural graded linear map +L → A has L ∩ I = 0 as its kernel. Thu, (L, A) is a pair. +Consider the adjoint map L → IDer L, and its extension UG(L) → +EndF(L). Let I1 be the kernel of the latter map. Then it is known that +I1 ∩ L = z, the center of L. Since dim EndF L < ∞, we see that I1 has finite +codimension. Let I2 be the ideal of Corollary 33, and let I = I1 ∩ I2. Then, +by construction, I is a graded ideal of finite codimension. Moreover, +I ∩ L = (I1 ∩ L) ∩ I2 = z ∩ I2 = 0. +The proof is complete. +□ +Remark. There is no guarantee that we have a strong pair (L, A) where A is +finite-dimensional. This is because we cannot guarantee that (IDer L, EndF(L)) +is a strong pair. +8. Group gradings on Lie algebras +The following question was posed by I. Shestakov: is any G-grading on +a Lie algebra equivalent to an abelian group grading? +We shall use our +constructions to provide a partial answer to this question. To this end we +need several technical lemmas. +First we need a map that behaves like a homomorphism from a free abelian +group to an arbitrary group. +This construction is useful for finding an +equivalence between an abelian grading and a G-grading, where G is not +necessarily abelian. +Lemma 35. Let G be a group, g1, . . . , gm ∈ G, and Z = ⟨a1, . . . , am | +[ai, aj] = 1, ∀i, j⟩ be the free abelian group of rank m. Let +S = {ai1 · · · air ∈ Z | {gi1, . . . , gir} g.a.s. }. +Then there exists a well-defined map α : S → G such that +α(ai1 · · · air) = gi1 · · · gir. +Proof. Let +F = {{i1, . . . , ir} ⊆ {1, . . . , m} | {gi1, . . . , gir} g.a.s. } . +If S1, S2 ∈ F, then S1 ∩ S2 ∈ F. For each S0 = {i1, . . . , ir} ∈ F, let +GS0 = ⟨gi1, . . . gir⟩ ⊆ G and ZS0 = ⟨ai1, . . . , air⟩. Then ZS0 is a free abelian +group of rank r, and GS0 is an abelian subgroup of G. Thus there is a well- +defined group homomorphism αS0 : ZS0 → GS0 such that aij �→ gij, for each +ij ∈ S0. Note that S ⊆ � +S0∈F ZS0. Now for each s ∈ S, let s ∈ S0 ∈ F, +and set +α(s) = αS0(s). + +18 +FELIPE YUKIHIDE YASUMURA +If s ∈ S1 ∩ S2, then since the maps are uniquely defined by their values on +the generators, we have +αS1(s) = αS1∩S2(s) = αS2(s). +Therefore α is well-defined. +□ +Let us consider once again the variety VG, defined in Section 4, that is, +the variety of G-graded associative algebras satisfying the identities +x(g) +1 x(h) +2 += 0, +[g, h] ̸= 1, g, h ∈ G. +Let L be a G-graded Lie algebra and B = {ei | i ∈ N} a homogeneous +vector space basis. We let XB = {x(gi) +i +| i ∈ N, gi = degG ei}. Assume +that {gi | i ∈ N} has m elements, and denote these by {g1, . . . , gm}. Let +Z = ⟨a1, . . . , am | [ai, aj] = 1, ∀i, j⟩ be the free abelian group of rank m. We +consider the variables XZ = {x(zj) +i +| deg ei = gj, i ∈ N}, the “lifting” of +the variables XB to Z-graded variables. Let F⟨XZ⟩ be the free associative +Z-graded algebra, freely generated by XZ, and FG(XB) the relatively free +algebra in VG, freely generated by XB. We have an algebra epimorphism +ψ: F⟨XZ⟩ → FG(XB), where ψ(x(zj) +i +) = x(gj) +i +, for each i ∈ N. Following +Lemma 35, let +S = {zi1 · · · zir ∈ Z | {gi1, . . . , gir} g.a.s. }, +and let α: S → G be defined by +α(zi1 · · · zir) = gi1 · · · gir. +Proposition 36. The map ψ is α-graded, that is, +ψ (F⟨XZ⟩)z ⊆ (FG(XL))α(z) , +∀z ∈ S, +and ψ (F⟨XZ⟩)z = 0, for z ∈ Z \ S. +Proof. Let m = x +(zi1) +i1 +· · · x(zir ) +ir +∈ F⟨XZ⟩. It is enough to show that either +ψ(m) = 0 or ψ(m) is homogeneous of degree α(zi1) · · · α(zir) in the G- +grading. The proof is by induction on r, with obvious basis the case r = 1. +Let m0 = x +(zi1) +i1 +· · · x +(zir−1) +ir−1 +. There is nothing to do if ψ(m0) = 0. Otherwise, +the subset {α(zi1), . . . , α(zir−1)} generates an abelian group. If, for some j, +[α(zij)α(zij+1) · · · α(zir−1), α(zir)] ̸= 1, +then ψ(m) = 0. Otherwise, {α(zi1), . . . , α(zir−1), α(zir)} generates an abelian +subgroup as well. Hence, either ψ(m) = 0 or degG ψ(m) = α(degZ m). +□ +Corollary 37. The grading of FG(XB) is a coarsening of a grading realized +by the respective universal abelian grading group. +Proof. It follows from Proposition 36 that there is a fine abelian grading on +FF(XZ) (obtained as a quotient of F⟨XZ⟩) that is a refinement of FG(XB). +□ + +UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA +19 +We do not know yet a proof for the following. +Conjecture 38. Let Γ be a group grading on a Lie algebra given by a +coarsening of an abelian grading. Then Γ is abelian. +Example 7. Conjecture 38 is false in a context other than of a Lie algebra. +Indeed, we know that on any matrix algebra Mn(F) there is a unique fine ele- +mentary grading, up to an equivalence, and it may be realized by an abelian +group (see, for instance, [2, Proposition 2.31]). Thus, any elementary grad- +ing on a matrix algebra is a coarsening of an abelian grading. On the other +hand, it is not hard to see that an abelian grading on a finite-dimensional +algebra is equivalent to a finite group grading (see also [5, Proposition 4.11]). +In [5, Corollary 5.6], it is proved that there exists an elementary grading Γ +on a matrix algebra Mn(F) that cannot be realized by a finite group. As a +consequence, Γ cannot be realized by an abelian group, and it is a coarsening +of an abelian grading. +Putting the pieces together (Corollary 37 and Conjecture 38), we obtain +the following lemma. +Lemma 39. Let L be a G-graded Lie algebra. Then, if Conjecture 38 is +true, the grading on SUG(L) is equivalent to an abelian grading. +Proof. From Corollary 37 and Conjecture 38, we obtain that the grading on +FG(XB) is equivalent to an abelian grading. Now the equivalence holds for +the ideal JB and its quotient FG(XB)/JB ∼= SUG(L). +□ +As a consequence, we obtain the main result of this section. +Theorem 40. If Conjecture 38 is true, then every G-grading on a Lie al- +gebra is equivalent to an abelian grading. +Proof. Let L be a G-graded algebra. By construction (Theorem 12), L is a +graded subalgebra of SUG(L). From Lemma 39, the grading on SUG(L) is +equivalent to an abelian grading. Hence the grading on L is equivalent to +an abelian grading as well. +□ +We summarize the results of this section. We obtained the equivalence of +the following two problems. +Corollary 41. The following assertions are equivalent. +(1) Every group grading on a Lie algebra is equivalent to an abelian +group grading. +(2) Every coarsening of an abelian group grading on a Lie algebra is +equivalent to an abelian group grading. +Proof. Clearly (1) implies (2). Now, assuming that (2) holds valid, Theo- +rem 40 proves (1). +□ + +20 +FELIPE YUKIHIDE YASUMURA +Acknowledgements +We are thankful to Professor Ivan Shestakov, Plamen Koshlukov, Lucia +Ikemoto and Mikhail Kochetov for the useful discussion and encouragement. +References +[1] V. Drensky, Free algebras and PI-algebras. Graduate course in algebra, Springer-Verlag +Singapore, Singapore, 2000. +[2] A. Elduque, M. Kochetov, Gradings on simple Lie algebras, Mathematical Surveys and +Monographs, 189. American Mathematical Society (2013). +[3] A. Gordienko, Amitsur’s conjecture for polynomial H-identities of H-module Lie alge- +bras, Transactions of the American Mathematical Society 367 (2015), 313–354. +[4] A. Gordienko, Co-stability of radicals and its applications to PI-theory, Algebra Col- +loquium 23 (2016), 481–492. +[5] A. Gordienko, O. Schnabel, On weak equivalences of gradings, Journal of Algebra 501 +(2018), 435–457. +[6] N. Jacobson, Lie algebras. Republication of the 1962 original. Dover Publications, Inc., +New York, 1979. +[7] D. Pagon, D. Repovˇs, M. Zaicev, Group gradings on finite dimensional Lie algebras, +Algebra Colloquium 20 (2013), 573–578. +Department of Mathematics, Instituto de Matem´atica e Estat´ıstica, Uni- +versidade de S˜ao Paulo, SP, Brazil +Email address: fyyasumura@ime.usp.br + diff --git a/pNFKT4oBgHgl3EQfyS7t/content/tmp_files/load_file.txt b/pNFKT4oBgHgl3EQfyS7t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72685c14ae330531a9007a02b3bfa1961199acc8 --- /dev/null +++ b/pNFKT4oBgHgl3EQfyS7t/content/tmp_files/load_file.txt @@ -0,0 +1,714 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf,len=713 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='11907v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='RA] 27 Jan 2023 UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA FELIPE YUKIHIDE YASUMURA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In this paper we construct a graded universal enveloping algebra of a G-graded Lie algebra, where G is not necessarily an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If the grading group is abelian, then it coincides with the classical construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We prove the existence and uniqueness of the graded en- veloping algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As consequences, we prove a graded variant of Witt’s Theorem on the universal enveloping algebra of the free Lie algebra, and the graded version of Ado’s Theorem, which states that every finite- dimensional Lie algebra admits a faithful finite dimensional represen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Furthermore we investigate if a Lie grading is equivalent to an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Introduction The universal enveloping algebra of a Lie algebra is a classical and im- portant construction, and it associates the representations of a Lie algebra to representations of an associative algebra, see any standard book on Lie algebra, for instance, [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In this paper, we investigate the construction but starting with a G-graded Lie algebra where G is a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is well-known that, if G is abelian, then the universal enveloping algebra of a Lie algebra inherits a natural G-grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' However, it is known that the universal en- veloping algebra need not be a graded algebra if the grading group G is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G be an abelian group and L a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then we can consider an ordered homogeneous basis {ei | i ∈ I} of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let [ei, ej] = � ℓ αijℓeℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Denote by XG = � g∈G Xg where Xg = {x(g) 1 , x(g) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' }, and by F⟨XG⟩ the free G-graded associative algebra over the field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The universal enveloping algebra of L is U(L) ∼= F⟨XG⟩/J, where J is the (ordinary) ideal generated by all {xixj − xjxi − � ℓ∈I αijℓxℓ | i, j ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since the elements generating J are homogeneous, we get that J is a graded ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, U(L) is a G-graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We extend the above construction for a G-graded Lie algebra where G is a not necessarily abelian group (see Theorem 9 and Theorem 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Our construction agrees with the classical one when the group is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We find a PBW basis (Theorem 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As a consequence, we prove an adapted version of Witt’s Theorem in the context of the free G-graded Lie algebra Supported by S˜ao Paulo Research Foundation (FAPESP), grant 2018/23690-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 1 2 FELIPE YUKIHIDE YASUMURA (Corollary 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We also deduce a graded version of Ado’s Theorem, that is, we show that every finite-dimensional G-graded Lie algebra is a graded vector subspace of a finite-dimensional G-graded associative algebra (The- orem 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We study the problem if every group grading on a Lie algebra is equivalent to an abelian group grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We apply our constructions and find an equivalent formulation to this problem (Corollary 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Notations and Preliminaries Let G be a group and A an algebra (not necessarily associative nor Lie), over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A G-grading on A is a vector space decomposition A = � g∈G Ag such that AgAh ⊆ Agh, for all g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A G-graded algebra is an algebra endowed with a G-grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The vector subspace Ag is called the homogeneous component of degree g, and its nonzero elements are called homogeneous of degree g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' When the decomposition is not explicitly given, we shall write (A)g to denote the homogeneous component of degree g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Given 0 ̸= x ∈ Ag, we denote degG x = g, or simply deg x = g when there is no risk of ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The support of the grading is defined as SuppΓ = {g ∈ G | Ag ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A vector subspace S ⊆ A is said to be graded if S = � Ag ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A graded subalgebra is a subalgebra which is a graded subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Analogously we define a graded ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If S is a graded ideal of A, then the quotient A/S inherits the structure of G-graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Given two G-graded algebras A = � g∈G Ag and C = � g∈G Cg, a G- graded homomorphism is an algebra homomorphism ψ : A → C such that ψ(Ag) ⊆ Cg, for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let Γ : A = � g∈G Ag and Γ′ : A = � h∈H A′ h be two gradings on the same algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We say that Γ and Γ′ are equivalent if there exists an algebra automorphism ψ of A, such that for each g ∈ G there exists h ∈ H satisfying ψ(Ag) = A′ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We say that a grading Γ is realized by a group G if there exists a G-grading equivalent to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We say that a grading is abelian if it is realized by an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is worth mentioning that if two gradings on the same algebra A are equivalent by an isomorphism ψ and I ⊆ A is a graded ideal, then I and ψ(I) are equivalent and so are A/I and A/ψ(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Universal group of a grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We follow [2] in this and next subsec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Given a G-grading, it is relevant to consider the group with minimal amount of relations that realizes the grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Here is the formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Definition 1 ([2, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let Γ be a G-grading on an algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The universal grading group of Γ is the group U(Γ) such that, for every realization of Γ as an H-grading, there exists a unique group homomorphism U(Γ) → H that is the identity map on SuppΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 3 The universal grading group of a group grading Γ always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It may be taken as the group with the set of generators Supp Γ, and relations s1s2 = s3, for s1, s2, s3 ∈ Supp Γ whenever 0 ̸= As1As2 ⊆ As3 (see [2, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is also relevant to consider the universal abelian grading group of a grading Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is defined by the abelianization of U(Γ), that is, Uab(Γ) = U(Γ)/U(Γ)′, where G′ is the commutator group of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is not always true that the universal abelian group grading realizes the grading Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Indeed, some homogeneous components may coalesce under this new group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The grading is realized by Uab(Γ) if and only if the initial grading Γ is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Refinement and coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let Γ : A = � g∈G Ag and Γ′ : A = � h∈H A′ h be two gradings on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We say that Γ′ is a refinement of Γ (or that Γ is a coarsening of Γ′) if for every h ∈ SuppΓ′, there exists g ∈ Supp Γ such that A′ h ⊆ Ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The following is relevant to us: Lemma 2 ([2, Part of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let Γ and Γ′ be gradings on A, and assume that Γ′ is realized as a H-grading and is a coarsening of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, there exists a group epimorphism p : U(Γ) → H such that p(SuppΓ) = SuppΓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Free algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' For each g ∈ G, we let Xg = {x(g) 1 , x(g) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' }, and XG = � g∈G Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, the free associative algebra F⟨XG⟩, freely generated by XG, has a natural G-grading where the monomials are homogeneous and degG x(g1) i1 · · x(gm) im = g1 · · · gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It satisfies the following universal property: for any associative G-graded algebra A, and each map ψ0 : XG → A respecting the degrees (that is, ψ0(x(g) i ) ∈ Ag), there exists a unique G-graded algebra homomorphism ψ: F⟨XG⟩ → A extending ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, we can define a G-graded polyno- mial identity of a G-graded associative algebra as an element of the set � ψ: F⟨XG⟩→A G-graded homomorphism Ker ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Free pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We recall the notion of a variety of associative-Lie pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' An associative Lie-pair is a pair (L, A) where A is an associative algebra generated by L, and L is a Lie algebra which is a vector subspace of A, and the product of the L coincides with the restriction of the commutator of A to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Given two pairs (L, A) and (H, C), a homomorphism of pairs is an algebra homomorphism ψ: A → C such that ψ(L) ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, ψ restricts to a Lie homomorphism L → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let W be a class of associative-Lie pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A pair (L, F) is free in the class W , freely generated by X, if for any pair (H, C) in W , and for any map 4 FELIPE YUKIHIDE YASUMURA ψ0 : X → H, there exists a unique homomorphism of pairs from (L, F) to (H, B) extending ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is clear that (L(X), F⟨X⟩), where L(X) is the Lie subalgebra of the free associative algebra generated by X (that is, the free Lie algebra) is a free pair, freely generated by X, in the class of all associative- Lie pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, due to the existence of a free associative-Lie pair, we can speak of polynomial identities of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By tradition these are called weak polynomial identities of the pair (L, A) (or identities of representations of Lie algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The above discussion can be extended to the context of G-graded associative- Lie pairs (L, A): a G-graded associative algebra A, and a G-graded Lie al- gebra L that is a G-graded subspace of A, where the product of L is the commutator of A restricted to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We shall prove below that there exists a G-graded free associative-Lie pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Graded Universal Enveloping Algebra We let G be a non-necessarily abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By a G-pair (L, A), we understand a G-graded associative-Lie pair, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A is a G-graded associative algebra, L is a G-graded subspace of A such that L is also a G-graded Lie algebra with respect to the commutator of A restricted to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Equivalently, a G-pair is a pair (L, ι), where L is a G-graded Lie algebra, A is a G-graded associative algebra, and ι: L → A is a graded linear map such that ι: L → A(−) is an (ungraded) Lie monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Note that the G-grading on the vector space A does not necessarily define a G-graded Lie algebra with respect to the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If this is the case, then we shall call the G-pair (L, A) a strong G-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The G-graded univer- sal enveloping algebra of L is a G-pair (L, UG(L)) such that: (i) UG(L) is generated by L, and (ii) for every G-pair (H, A) and every G-graded Lie homomorphism L → H, there exists an extension to a G-graded algebra homomorphism UG(L) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As it is customary, we shall call the algebra UG(L) a G-graded universal enveloping algebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We shall prove the existence and uniqueness of UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It will be an appropriate quotient of the usual universal enveloping algebra U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In what follows, we may weaken the definition of a pair, replacing the condition of ι : L → A(−) being a monomorphism to ask ι to be only a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' This is because we cannot guarantee (until Theorem 16) that L → UG(L) is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (L, A) be a G-pair and let x, y ∈ L be homogeneous elements such that deg x = g, deg y = h, and gh ̸= hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then xy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Write L = � g∈G Lg and A = � g∈G Ag, then Lg ⊆ Ag, for all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 5 By g = deg x and h = deg y we get yx ∈ Ahg and xy ∈ Agh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' On the other hand, −[y, x] = −yx + xy ∈ Lhg ⊆ Ahg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus xy = yx + (−yx + xy) ∈ Ahg ∩ Agh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ In the language of polynomial identities, we have: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (L, A) be a G-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then the pair satisfies the weak G- graded polynomial identities x(g) 1 x(h) 2 = 0, [g, h] ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Let L be a G-graded Lie algebra with a homogeneous vector space basis B = {ei | i ∈ N}, where N is ordered, and denote [ei, ej] = � ℓ∈N α(k) ij ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let XB = {xi | i ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let F⟨XB⟩ be the free associative G-graded algebra, freely generated by XB, where deg xi = degG ei, for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Define IB to be the graded ideal generated by (1) {xixj − xjxi − � k α(k) ij xk | i, j ∈ N}, that is, IB is the least graded ideal containing all the elements above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Denote further by J the ungraded ideal generated by the same elements, so that U(L) ∼= F⟨XB⟩/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The algebra F⟨XB⟩/IB is graded, and (L, F⟨XB⟩/IB) is a G-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let JB be the ideal of U(L) generated by {eiej | i, j ∈ N, [deg ei, deg ej] ̸= 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then F⟨XB⟩/IB ∼= U(L)/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since J ⊆ IB, there exists a surjective algebra homomorphism π : U(L) → F⟨XB⟩/IB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By Lemma 4, since (L, F⟨XB⟩/IB) is a G-pair, then JG ⊆ Ker π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, we get that IB ⊇ J + K, where K is the (possibly ungraded) ideal ⟨xixj | i, j ∈ N, [deg xi, deg xj] ̸= 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Note that K is actually graded, since its generators are homogeneous elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, each of the elements (1) is either homogeneous (this is the case whenever [deg xi, deg xj] = 1), or it belongs to K (if [deg xi, deg xj] ̸= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, J +K is a graded ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus we obtain that J +K ⊇ IB, proving the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Now we can prove the existence of a G-graded universal enveloping algebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The G-pair (L, F⟨XB⟩/IB) is a G-graded universal enveloping algebra of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 6 FELIPE YUKIHIDE YASUMURA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (H, A) be a G-pair and ϕ: L → H be a G-graded Lie homo- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then ϕ admits an extension (also denoted by ϕ) to an algebra homomorphism ϕ: U(L) → alg(H) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Here alg(H) is the associative subalgebra generated by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If [deg ei, deg ej] ̸= 1, then Lemma 4 tells us that 0 = ϕ(ei)ϕ(ej) = ϕ(eiej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, ϕ factors through JB, that is, there exists a map ¯ϕ: U(L)/JB → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By Lemma 6, U(L)/JB ∼= F⟨XB⟩/IB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The map ¯ϕ is a graded map and extends ϕ, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Finally, it is a standard exercise to prove that the G-graded universal enveloping algebra is unique, up to an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let UG and VG be G-graded universal enveloping algebras of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a G-graded isomorphism ι: UG → VG such that ι(x) = x, for each x ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since (L, VG) is a pair, the identity map L → L admits an extension to a G-graded algebra homomorphism ι: UG → VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Conversely, the identity map also admits an extension to a G-graded homomorphism j : VG → UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since the composition jι: UG → UG fixes all the elements of L, it must be the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Similarly, ιj is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ We summarize our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Of course, from a vector space basis of U(L), we obtain a set of generators of UG(L) as a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded Lie algebra, where G is a non-necessarily abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then it admits a unique, up to an isomorphism, G-graded universal enveloping algebra UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If {ei | i ∈ I} is an ordered homoge- neous basis of L, then ei1 · · · eim, m ≥ 0, i1 ≤ · · · ≤ im, [deg eij, deg eij+1] = 1, j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , m−1}, is a set of homogeneous elements that spans UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If G is abelian and L is a G-graded algebra, then UG(L) = U(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G = C2 ∗ C2 = ⟨g, h | g2 = h2 = 1⟩, and L = Span{x, y} be the 2-dimensional abelian Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Define a G-grading on L where deg x = g and deg y = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, it is known that U(L) ∼= F[X, Y ], the poly- nomial algebra in two commuting variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since UG(L) ∼= F[X, Y ]/⟨XY ⟩, one obtains that UG(L) is the associative unital subalgebra of F[X] ⊕ F[Y ] generated by X and Y , that is, UG(L) = {α(1, 1) + (F(X), G(Y )) | α ∈ F, F(X) ∈ F[X], G(Y ) ∈ F[Y ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Strong pair We assume that G is a not necessarily abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We investigate when UG(L) is a G-graded Lie algebra with respect to the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' To this end, we need to investigate strong pairs, that is, G-pairs (L, A), where A is UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 7 a G-graded Lie algebra with respect to the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let VG denote the variety of G-graded associative algebras satisfying the polynomial identities � x(g) 1 x(h) 2 = 0 | g, h ∈ G, [g, h] ̸= 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Whenever we have a strong pair (L, A), from Lemma 4, we have A ∈ VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Denote by XG = � g∈G Xg, where Xg = {x(g) 1 , x(g) 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' }, for each g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let FG(XG) be the relatively free algebra in VG, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then FG(XG) ∼= F⟨XG⟩/⟨x(g) 1 x(h) 2 | [g, h] ̸= 1⟩TG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We slightly modify the definition of G-graded universal enveloping algebra of a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The strong G-graded universal enveloping of L is a strong pair (L, SUG(L)) such that for every strong pair (H, A), each G-graded Lie homomorphism L → H admits a unique extension to a G-graded algebra homomorphism SUG(L) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A standard argument shows that if a strong G-graded universal enveloping algebra exists, then it is unique up to an isomorphism that is the identity map on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Moreover, L generates SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If it exists, it is a quotient of the G-graded universal enveloping algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If a strong G-graded universal enveloping algebra of L exists, then it is a quotient of UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If (L, SUG(L)) is a strong pair, then it is also a G-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence there exists an extension of the identity map L → L to an algebra homomorphism UG(L) → SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The statement follows since L generates SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If G is an abelian group, then every G-pair is a strong pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, SUG(L) exists and it coincides with UG(L) (which in turn equals U(L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider once again Example 2: let G = C2 ∗ C2 = ⟨g, h | g2 = h2 = 1⟩ and L = Span{x, y} the 2-dimensional abelian Lie algebra, where deg x = g and deg y = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The G-graded universal enveloping algebra of L is a Lie algebra with respect to the commutator (indeed, UG(L) is a commutative algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In particular, it satisfies the definition of the strong G-graded universal enveloping algebra, so SUG(L) = UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' More generally, if UG(L) happens to be a G-graded Lie algebra with respect to the commutator, then SUG(L) exists and it coincides with UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Conversely, if SUG(L) = UG(L), then UG(L)(−) is a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, we shall prove the existence of a strong G-graded universal envelop- ing algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The construction is similar to that of the G-graded universal enveloping algebra, but we need to work in the variety VG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 8 FELIPE YUKIHIDE YASUMURA Let L be a G-graded Lie algebra, and let B = {ei | i ∈ N} be a homoge- neous basis of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider the structure constants α(k) ij , i, j, k ∈ N, that is, [ei, ej] = � k∈N α(k) ij ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We let XB = {zi = x(gi) i | i ∈ N, gi = deg ei}, and let FG(XB) be the relatively free algebra in VG, freely generated by XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let JB be the ideal of FG(XB) generated by all zizj − zjzi − � k∈N α(k) ij zk, i, j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The above elements are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Indeed, either [deg zi, deg zj] = 1, or zizj = zjzi = [zi, zj] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus JB is a graded ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' SUG(L) = FG(XB)/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (H, A) be a strong pair, and let f0 : L → H be a G-graded Lie homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then f0 restricts to a map XB → H ⊆ A which respects the degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since A ∈ VG, the restriction of f0 extends to a G-graded algebra homomorphism f : FG(XB) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since both FG(XB) and A are G- graded Lie algebras with respect to the commutators, then f factors through JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus f0 admits an extension to a G-graded algebra homomorphism FG(XB)/JB → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ As a consequence, every G-graded Lie algebra is a subalgebra of some A(−), where A is a G-graded associative algebra such that A(−) is a G- graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Whenever it is convenient, we shall identify the homogeneous variables zi ∈ XB with the basis elements ei ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, SUG(L) is spanned by all elements ei1 · · · eim, where ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , eim ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A PBW basis for SUG(L) In this section, we shall find a homogeneous vector space basis for SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' For, we let XG = � g∈G Xg be a set of G-homogeneous variables (where each Xg is finite or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We start finding a homogeneous vector space basis for FG(XG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm} is a subset of a group, then the abbreviation g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' means that {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm} generates an abelian subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The relatively free algebra FG(XG) has, as a G-homogeneous vector space basis, all monomials of the kind x(g1) i1 · · x(gm) im , {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm} g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is clear that such a set generates FG(XG) as a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, it is sufficient to prove that it is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider a finite subset S of the above monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We may assume that a fixed set of variables, UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 9 say Y = {x(g1) i1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , x(gm) im }, appears in all the monomials of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We can assume that the subgroup generated by {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm} is abelian, otherwise every monomial in S would be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, let H = ⟨g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm⟩ be the abelian subgroup of G generated by g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The free associative H-graded algebra F⟨Y ⟩, freely generated by Y , may be seen as a G-graded algebra, where the homogeneous components in G \\ H are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Therefore, there exists a surjective G-graded algebra homomorphism p : FG(XG) → F⟨Y ⟩ via p(x(g) j ) = � x(g) j , if x(g) j ∈ Y , 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, the set S is linearly independent under the image of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, S is linearly independent as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Now, let G be any group and L a G-graded Lie algebra over an arbitrary field F, and let B = {ei | i ∈ N} be a homogeneous vector space basis of L, where N is ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let [ei, ej] = � ℓ∈N α(ℓ) ij eℓ be the structure constants of L with respect to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We shall consider the set of free variables XB = {ei | i ∈ N}, and use the same letters to denote the homogeneous variables and the elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let FG(XB) be the relatively free G-graded algebra in VG, freely generated by XB, and F⟨XB⟩ be the free associative G-graded algebra, freely generated by XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We let R be the vector subspace of F⟨XB⟩ spanned by all ei1 · · · eim, m ≥ 0, i1 ≤ · · · ≤ im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The following is a classical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 14 ([6, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' There exists a linear map σ0 : F⟨XB⟩ → R such that σ01 = 1, σ0(ei1 · · · eim) = ei1 · · · eim, if i1 ≤ · · · ≤ im, σ0(ej1 · · · ejm − ej1 · · · ejk+1ejk · · · ejm) = σ0(ej1 · · · �� ℓ∈N α(ℓ) jkjk+1eℓ � · · ejm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As an immediate consequence, we have the following graded version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let B be the vector subspace of FG(XB) spanned by all ei1 · · · eim, m ≥ 0, i1 ≤ · · · ≤ im, {degG ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , degG eim} g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' There exists a linear map σ : FG(XB) → B such that σ(ei1 · · · eim) = 0, 10 FELIPE YUKIHIDE YASUMURA if {degG ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , degG eim} does not generate an abelian subgroup, and oth- erwise, σ1 = 1, σ(ei1 · · · eim) = ei1 · · · eim, if i1 ≤ · · · ≤ im, σ(ej1 · · · ejm − ej1 · · · ejk+1ejk · · · ejm) = σ(ej1 · · · �� ℓ∈N α(ℓ) jkjk+1eℓ � · · ejm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The projection F⟨XB⟩ → FG(XB) induces a map R → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let σ0 : F⟨XB⟩ → R be the linear map of Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, we have the diagram 0 0 F⟨XB⟩ R B FG(XB) Since the kernel of F⟨XB⟩ → B lies inside the kernel of F⟨XB⟩ → FG(XB) (see Lemma 13), we obtain the required map σ : FG(XB) → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ This gives a PBW basis for SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' More precisely, we have Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G be any group and L be a G-graded Lie algebra over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let {ei | i ∈ N} be a homogeneous basis of L, where N is ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, a homogeneous vector space basis of SUG(L) is given by ei1 · · · eim, i1 ≤ · · · ≤ im, {degG ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , degG eim} g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is clear, as in the classical case, that the above monomials span SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, the map σ : FG(XB) → B from Lemma 15 factors through SUG(L) → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, we obtain a surjective linear map SUG(L) → B, where the above set of monomials is sent to a linearly independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, it is a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ As a consequence, we obtain that (L, UG(L)) and (L, SUG(L)) are indeed pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' That is, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Corollary 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G be any group and L a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then (i) there is an embedding L ֒→ SUG(L), (ii) there is an embedding L ֒→ UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The assertion (i) follows directly from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' To prove (ii), one should consider the diagram UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 11 0 0 L UG(L) SUG(L) where the surjective map UG(L) → SUG(L) is from Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Free graded Lie algebra There is a direct way to construct the G-graded free Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It was mentioned, for instance, in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We include its construction here for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let F{XG} be the absolutely free (nonassociative) G-graded algebra, and let I be the graded T-ideal generated by all elements of the following types: x(g1) 1 x(g2) 2 + x(g2) 2 x(g1) 1 , (x(g1) 1 x(g2) 2 )x(g3) 3 + (x(g2) 2 x(g3) 3 )x(g1) 1 + (x(g3) 3 x(g1) 1 )x(g2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Denote L(XG) = F{XG}/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By construction, L(XG) is a G-graded algebra, and it is clearly a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proposition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' L(XG) is free in the class of all G-graded Lie algebras, and it is freely generated by the set XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let H be a G-graded Lie algebra, and let ϕ: XG → H be a degree- preserving map, that is, deg ϕ(x(g) i ) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then ϕ admits an extension to a G-graded homomorphism ϕ: F{XG} → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then Ker ϕ is a graded ideal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' and, since H is a graded Lie algebra, it contains the generators of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, ϕ factors through F{XG}/I, that is, ϕ induces a graded Lie homomorphism L(XG) → H that extends the map XG → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Free graded algebras and pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We fix a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As a conse- quence of Corollary 5, we may consider the variety WG of pairs satisfying the weak G-graded polynomial identities x(g) 1 x(h) 2 = 0, [g, h] ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L(XG) be the free G-graded Lie algebra, and let UG(L(XG)) be its respective G-graded universal enveloping algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The pair (L(XG), UG(L(XG))) is free in the variety of pairs WG, and XG is a set of free generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (H, A) be a G-pair, and let ψ0 : XG → H be a degree-preserving map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since L(XG) is free, then ψ0 admits a unique extension to a Lie homomorphism L(XG) → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, it admits a unique extension to an algebra homomorphism UG(L(XG)) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, (L(XG), UG(L(XG))) is free in the variety of pairs WG, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ 12 FELIPE YUKIHIDE YASUMURA Conversely, we may obtain the free G-graded Lie algebra from a free pair in WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, let (L, F) be a free pair in the variety of G-graded pairs in WG, freely generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We let L(X) be the G-graded Lie algebra generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, L(X) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' More precisely, we have: Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The algebra L(X) is the free G-graded Lie algebra, freely generated by X, and the G-pair (L(X), UG(L(X))) is free in the variety of pairs WG, and X is a set of free generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (H, A) be a G-pair, and let ψ0 : X → H be a degree-preserving map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since the pair belongs to WG (Corollary 5) and (L, F) is free, there exists an extension of ψ0 to a G-graded algebra homomorphism ψ: F → A, which restricts to a Lie homomorphism ¯ψ: L → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In particular, ψ restricts to a Lie homomorphism from L(X) → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Every G-graded Lie algebra may be thought of as the pair given by itself and its G-graded universal enveloping algebra, we get that L(X) is a free G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now ¯ψ extends uniquely to a homomorphism UG(L(X)) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence (L(X), UG(L(X))) is a free pair in WG, and it is freely generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ As a consequence of Proposition 19 and Proposition 20, we obtain an alternative description of the free G-graded Lie algebra, and a relation with the free G-pair in the variety WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' This is a graded version of Witt’s Theorem (see, for instance, [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='5] or [6, Theorem V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='7] for the classical case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Corollary 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let (L, F) be free in the variety of pairs WG, freely generated by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then L coincides with the Lie algebra L(X), generated by X, and it is the free G-graded Lie algebra, freely generated by X, and F = UG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Strong Witt’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, we obtain an outright graded version of Witt’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let FG(XG) be the relatively free G-graded associative algebra in the variety VG, and let L(XG) be its Lie subalgebra generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then we have Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The algebra L(XG) is a free G-graded Lie algebra, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Moreover, SUG(L(XG)) = FG(XG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We let H be a G-graded Lie algebra and f0 : XG → H a map respect- ing degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, since H ⊆ SUG(L), the map f0 extends to a G-graded algebra homomorphism f : FG(XG) → SUG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Such maps restricts to a G-graded Lie homomorphism L(XG) → Lie(H) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, L(XG) is free, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, let (H, A) be a strong pair, and f : L(XG) → H a G-graded Lie homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, f restricts to a graded map XG → H, which admits an extension to a G-graded algebra homomorphism ¯f : FG(XG) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since the restriction of ¯f to L(XG) is a G-graded Lie homomorphism, and f is the unique extension of the map XG → H, we get that the restriction of ¯f coincides with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It means that ¯f is an extension of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, SUG(L(XG)) = FG(XG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 13 Combining Corollary 21 and Theorem 22 we obtain a complete version of the graded version of Witt’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We summarize the results, recalling the main definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Corollary 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G be a group, and consider the set of G-graded polyno- mials (2) x(g) 1 x(h) 2 , [g, h] ̸= 1, g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let VG be the variety of G-graded associative algebras satisfying the identi- ties (2), and WG be the variety of G-graded associative-Lie pairs satisfying the weak-polynomial identities (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let XG be a set of G-graded variables, FG(XG) be the relatively free algebra in VG, freely generated by XG, and (L, F) be a free pair in WG, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then: (i) The Lie subalgebra LG(XG) of FG(XG) generated by XG, is the free G-graded Lie algebra, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Furthermore, SUG(LG(XG)) = FG(XG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' (ii) The Lie subalgebra of F generated by XG coincides with L, and it is the free G-graded Lie algebra, freely generated by XG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Moreover, UG(L) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' A graded version of Ado’s Theorem In this section, we assume that char F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If L is a G-graded Lie algebra, then it is known that its center is a graded ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Moreover, the following result due to Gordienko (also proved by Pagon, Repovˇs and Zaicev in [7]) is useful: Theorem 24 ([4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a finite- dimensional G-graded Lie algebra over a field of characteristic zero, and R its solvable radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' (1) The nilradical and the solvable radical of L are graded ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' (2) (Graded Levi’s decomposition) There exists a graded subalgebra L1 such that L1 ∩ R = 0 and L = R + L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We shall follow the classical ungraded version of the proof of Ado’s Theo- rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The main idea is to find a graded ideal I ⊆ UG(L) of finite codimension such that L ∩ I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded solvable Lie algebra, R its nilradical, and I ⊆ UG(L) a graded ideal of finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Assume that every x ∈ R is nilpotent modulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a graded ideal J ⊆ UG(L) such that: (1) J ⊆ I, (2) J has finite codimension in UG(L), (3) every x ∈ R is nilpotent modulo J, (4) every derivation D of L, that can be extended to UG(L), satisfies DJ ⊆ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 14 FELIPE YUKIHIDE YASUMURA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let I′ be the graded ideal of UG(L) generated by I and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then L/L ∩ I ∼= (L + I)/I is a G-graded Lie algebra, and I′/I is one of its graded ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since dim L/L ∩ I < ∞ and I′/I contains a basis constituted of nilpotent elements, then I′/I is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence there exists m ∈ N such that J := (I′)m ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Clearly J satisfies (1)–(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' On the other hand, since char F = 0, it is known that DL ⊆ R, for any derivation D of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence DUG(L) ⊆ I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus DJ ⊆ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ The following lemma is easy to deduce, and we include its statement for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a finite-dimensional G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, IDer(L), the set of inner derivations of L, is a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a finite-dimensional G-graded Lie algebra, and H ⊆ EndFL be a G-graded Lie algebra contained in Der(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then the Lie subal- gebra generated by H and IDer(L) is a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since IDer(L) is a Lie ideal of Der(L), the vector space H + IDer(L) is a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since both spaces are graded, their sum is a graded sub- space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As each one of H and IDer(L) is a graded Lie subalge- bras, it is enough to show the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If D ∈ H and x ∈ L are homogeneous, then [D, ad x] is either 0 or is homogeneous of degree degG D degG x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' This is clear, since [D, ad x] = ad(D(x)), and D(x) is either 0 or degG D(x) = degG D degG x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L and H be finite-dimensional G-graded Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let ρ: H → EndFL be a Lie homomorphism and a G-graded linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then Im ρ is a G-graded Lie algebra, and ρ is a graded Lie homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We know that Im ρ is a graded subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, it is enough to show that, given homogeneous u, v ∈ Im ρ, either [u, v] = 0 or [u, v] is homo- geneous of degree h := degG u degG v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let u0, v0 ∈ H be homogeneous elements such that u = ρ(u0) and v = ρ(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, [u, v] = ρ[u0, v0], so it is enough to show that this last one is a graded linear map of degree h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If [degg u, degg v] ̸= 1, then [u0, v0] = 0, and there is nothing to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Otherwise, if x ∈ L is homogeneous then uvx and vux are both homogeneous elements of degree h degG x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, [u, v] is homogeneous of degree h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G = ⟨α1, α2, α3⟩ be the free group, freely generated by {α1, α2, α3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be the 3-dimensional abelian Lie algebra, and assume that {x1, x2, x3} is a basis of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, imposing degG xi = αi, for i = 1, 2, 3, we obtain a (Lie algebra) G-grading on L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Note that IDer L = 0 and Der L = EndFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' However, Der L is not a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Indeed, let eij ∈ EndFL be the map defined by eijxℓ = δjℓxi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 15 Then eij is a homogeneous map of degree αiα−1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' On one hand, e12e23 = e13 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' On the other hand, [degG e12, degG e23] ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, by Lemma 4, it is not possible to have a structure of a G-graded Lie algebra on Der L, given by the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Note that Der L contains several G-graded Lie algebras, with respect to the commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' For instance, if 1 ≤ i < j ≤ 3, then Span{eii, eij, eji, ejj} is a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The proposition below is the main step in the proof of our main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proposition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L = R + L1 be a finite-dimensional G-graded Lie algebra, where R is a graded solvable ideal, and L1 is a graded subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Assume that J ⊆ UG(R) is a graded ideal satisfying (ii)–(iv) of Lemma 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a graded ideal J′ ⊆ UG(L) such that (i) R ∩ J′ ⊆ R ∩ J, (ii) if x ∈ n(R) and y ∈ L1 is such that adR y is nilpotent, then x + y is nilpotent modulo J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider the map IDer L → Der(R/R ∩ J), given by the restriction of the inner derivation to R, followed by the induced map on the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since R ∩ J is an ideal of R, the map is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since R is a graded ideal of L, it follows that the map is graded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, by Lemma 28, its image is a G-graded Lie algebra, say H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, consider the composition ρ: L → IDer L → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Therefore ρ is a G-graded homomorphism of Lie algebras, and it admits an extension ¯ρ: UG(L) → UG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let J′ = Ker ¯ρ ∩ ⟨J⟩, where ⟨J⟩ = UG(L)JUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Note that, by construction, R ∩ J′ ⊆ R ∩ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Moreover, if y ∈ L1 is such that adR y is nilpotent, then by construction ¯ρ(y) is nilpotent too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, y is nilpotent modulo J′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Now we proceed with the proof of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We divide the proof in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let z be a finite-dimensional G-graded abelian Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a graded ideal of finite codimension I ⊆ UG(z) such that (i) I ∩ z = 0, and (ii) every x ∈ z is nilpotent modulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since z has zero product, it is also an associative algebra with trivial product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So (z, z) is a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, the identity map z → z admits an extension ρ: UG(z) → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let I = Ker ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since dim z < ∞, then I has finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since ρ is a graded homomorphism, I is also a graded ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As ρ is 1–1 in z, we see that I ∩z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Finally, once z has the trivial product, ρ(UG(z)2) ⊆ z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 16 FELIPE YUKIHIDE YASUMURA In particular, z2 ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let n be a finite-dimensional G-graded nilpotent Lie algebra, and let z be its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a graded ideal of finite codimension I ⊆ UG(z) such that (i) I ∩ z = 0, and (ii) every x ∈ n is nilpotent modulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider a chain z = n0 ⊆ n1 ⊆ · · · ⊆ nt = n, where each ni is a graded ideal and dim ni = i + dim z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We shall prove the result by induction on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If i = 0, then we may apply Lemma 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' So, for some 0 ≤ i < i, assume that Ii ⊆ UG(ni) is a graded ideal satisfying (i), and such that each x ∈ ni is nilpotent modulo Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Write ni+1 = ni + Li, where Li is a graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By Proposition 29, there exists a graded ideal Ii+1 ⊆ UG(ni+1) satisfying (ii) and such that Ii+1 ∩ zi ⊆ Ii ∩ zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, Ii+1 ∩ z = Ii+1 ∩ z ∩ ni ⊆ Ii ∩ ni ∩ z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, Ii+1 satisfies (i) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Lemma 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let R be a finite-dimensional G-graded solvable Lie algebra, n its nilradical, and z its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a graded ideal of finite codimension I ⊆ UG(R) such that (i) I ∩ z = 0, and (ii) every x ∈ n is nilpotent modulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider a chain of graded subalgebras n = R0 ⊆ R1 ⊆ · · · ⊆ Rs = R, where Ri is an ideal of Ri+1, and dim Ri = i + dim n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, it is known that the nilradical of each Ri is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We may proceed by induction on i, where the validity for i = 0 holds thanks to Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, we may repeat the argument given in the proof of the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Corollary 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a finite-dimensional G-graded Lie algebra and z its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a graded ideal of finite codimension I ⊆ UG(L) such that I ∩ z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is enough to combine the graded Levi’s decomposition (Theorem 24), Lemma 32 and Proposition 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Now, we are in a position to prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Theorem 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a finite-dimensional G-graded Lie algebra over a field of characteristic zero, where G is a non-necessarily abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a finite-dimensional G-graded associative algebra A such that (L, A) is a G-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In other words L ⊆ A is a graded vector subspace, and L is a G-graded Lie algebra with respect to the commutator of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is enough to find a graded ideal I ⊆ UG(L) of finite codimension such that I ∩ L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If I is such an ideal then A = UG(L)/I is a finite- dimensional G-graded associative algebra, and the natural graded linear map L → A has L ∩ I = 0 as its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thu, (L, A) is a pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Consider the adjoint map L → IDer L, and its extension UG(L) → EndF(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let I1 be the kernel of the latter map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then it is known that I1 ∩ L = z, the center of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Since dim EndF L < ∞, we see that I1 has finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let I2 be the ideal of Corollary 33, and let I = I1 ∩ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, by construction, I is a graded ideal of finite codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Moreover, I ∩ L = (I1 ∩ L) ∩ I2 = z ∩ I2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' There is no guarantee that we have a strong pair (L, A) where A is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' This is because we cannot guarantee that (IDer L, EndF(L)) is a strong pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Group gradings on Lie algebras The following question was posed by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Shestakov: is any G-grading on a Lie algebra equivalent to an abelian group grading?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We shall use our constructions to provide a partial answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' To this end we need several technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' First we need a map that behaves like a homomorphism from a free abelian group to an arbitrary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' This construction is useful for finding an equivalence between an abelian grading and a G-grading, where G is not necessarily abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let G be a group, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm ∈ G, and Z = ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , am | [ai, aj] = 1, ∀i, j⟩ be the free abelian group of rank m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let S = {ai1 · · · air ∈ Z | {gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gir} g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then there exists a well-defined map α : S → G such that α(ai1 · · · air) = gi1 · · · gir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let F = {{i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , ir} ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , m} | {gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gir} g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If S1, S2 ∈ F, then S1 ∩ S2 ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' For each S0 = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , ir} ∈ F, let GS0 = ⟨gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' gir⟩ ⊆ G and ZS0 = ⟨ai1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , air⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then ZS0 is a free abelian group of rank r, and GS0 is an abelian subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus there is a well- defined group homomorphism αS0 : ZS0 → GS0 such that aij �→ gij, for each ij ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Note that S ⊆ � S0∈F ZS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now for each s ∈ S, let s ∈ S0 ∈ F, and set α(s) = αS0(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' 18 FELIPE YUKIHIDE YASUMURA If s ∈ S1 ∩ S2, then since the maps are uniquely defined by their values on the generators, we have αS1(s) = αS1∩S2(s) = αS2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Therefore α is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Let us consider once again the variety VG, defined in Section 4, that is, the variety of G-graded associative algebras satisfying the identities x(g) 1 x(h) 2 = 0, [g, h] ̸= 1, g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded Lie algebra and B = {ei | i ∈ N} a homogeneous vector space basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We let XB = {x(gi) i | i ∈ N, gi = degG ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Assume that {gi | i ∈ N} has m elements, and denote these by {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let Z = ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , am | [ai, aj] = 1, ∀i, j⟩ be the free abelian group of rank m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We consider the variables XZ = {x(zj) i | deg ei = gj, i ∈ N}, the “lifting” of the variables XB to Z-graded variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let F⟨XZ⟩ be the free associative Z-graded algebra, freely generated by XZ, and FG(XB) the relatively free algebra in VG, freely generated by XB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We have an algebra epimorphism ψ: F⟨XZ⟩ → FG(XB), where ψ(x(zj) i ) = x(gj) i , for each i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Following Lemma 35, let S = {zi1 · · · zir ∈ Z | {gi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , gir} g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' }, and let α: S → G be defined by α(zi1 · · · zir) = gi1 · · · gir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proposition 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The map ψ is α-graded, that is, ψ (F⟨XZ⟩)z ⊆ (FG(XL))α(z) , ∀z ∈ S, and ψ (F⟨XZ⟩)z = 0, for z ∈ Z \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let m = x (zi1) i1 · · x(zir ) ir ∈ F⟨XZ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It is enough to show that either ψ(m) = 0 or ψ(m) is homogeneous of degree α(zi1) · · · α(zir) in the G- grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The proof is by induction on r, with obvious basis the case r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let m0 = x (zi1) i1 · · x (zir−1) ir−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' There is nothing to do if ψ(m0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Otherwise, the subset {α(zi1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , α(zir−1)} generates an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If, for some j, [α(zij)α(zij+1) · · · α(zir−1), α(zir)] ̸= 1, then ψ(m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Otherwise, {α(zi1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' , α(zir−1), α(zir)} generates an abelian subgroup as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence, either ψ(m) = 0 or degG ψ(m) = α(degZ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ Corollary 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The grading of FG(XB) is a coarsening of a grading realized by the respective universal abelian grading group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' It follows from Proposition 36 that there is a fine abelian grading on FF(XZ) (obtained as a quotient of F⟨XZ⟩) that is a refinement of FG(XB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ UNIVERSAL ENVELOPING OF A GRADED LIE ALGEBRA 19 We do not know yet a proof for the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Conjecture 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let Γ be a group grading on a Lie algebra given by a coarsening of an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then Γ is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Conjecture 38 is false in a context other than of a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Indeed, we know that on any matrix algebra Mn(F) there is a unique fine ele- mentary grading, up to an equivalence, and it may be realized by an abelian group (see, for instance, [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Thus, any elementary grad- ing on a matrix algebra is a coarsening of an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' On the other hand, it is not hard to see that an abelian grading on a finite-dimensional algebra is equivalent to a finite group grading (see also [5, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' In [5, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='6], it is proved that there exists an elementary grading Γ on a matrix algebra Mn(F) that cannot be realized by a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' As a consequence, Γ cannot be realized by an abelian group, and it is a coarsening of an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Putting the pieces together (Corollary 37 and Conjecture 38), we obtain the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Lemma 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Then, if Conjecture 38 is true, the grading on SUG(L) is equivalent to an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' From Corollary 37 and Conjecture 38, we obtain that the grading on FG(XB) is equivalent to an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now the equivalence holds for the ideal JB and its quotient FG(XB)/JB ∼= SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ As a consequence, we obtain the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Theorem 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' If Conjecture 38 is true, then every G-grading on a Lie al- gebra is equivalent to an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Let L be a G-graded algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' By construction (Theorem 12), L is a graded subalgebra of SUG(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' From Lemma 39, the grading on SUG(L) is equivalent to an abelian grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Hence the grading on L is equivalent to an abelian grading as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ We summarize the results of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' We obtained the equivalence of the following two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Corollary 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' (1) Every group grading on a Lie algebra is equivalent to an abelian group grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' (2) Every coarsening of an abelian group grading on a Lie algebra is equivalent to an abelian group grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Clearly (1) implies (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Now, assuming that (2) holds valid, Theo- rem 40 proves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' □ 20 FELIPE YUKIHIDE YASUMURA Acknowledgements We are thankful to Professor Ivan Shestakov, Plamen Koshlukov, Lucia Ikemoto and Mikhail Kochetov for the useful discussion and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Drensky, Free algebras and PI-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Graduate course in algebra, Springer-Verlag Singapore, Singapore, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Elduque, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Kochetov, Gradings on simple Lie algebras, Mathematical Surveys and Monographs, 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' American Mathematical Society (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Gordienko, Amitsur’s conjecture for polynomial H-identities of H-module Lie alge- bras, Transactions of the American Mathematical Society 367 (2015), 313–354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Gordienko, Co-stability of radicals and its applications to PI-theory, Algebra Col- loquium 23 (2016), 481–492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Gordienko, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Schnabel, On weak equivalences of gradings, Journal of Algebra 501 (2018), 435–457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Jacobson, Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Republication of the 1962 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Dover Publications, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=', New York, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Pagon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Repovˇs, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Zaicev, Group gradings on finite dimensional Lie algebras, Algebra Colloquium 20 (2013), 573–578.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content=' Department of Mathematics, Instituto de Matem´atica e Estat´ıstica, Uni- versidade de S˜ao Paulo, SP, Brazil Email address: fyyasumura@ime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} +page_content='br' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFKT4oBgHgl3EQfyS7t/content/2301.11907v1.pdf'} diff --git a/pdE0T4oBgHgl3EQfaACR/vector_store/index.faiss b/pdE0T4oBgHgl3EQfaACR/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1d6dbbc6a7d41266d03371eaddf4314ddf82633b --- /dev/null +++ b/pdE0T4oBgHgl3EQfaACR/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1270d9d2e190b385b649818af401fea92eda5836cc3de9cdc4ece160e018b268 +size 4784173 diff --git a/qdFAT4oBgHgl3EQffB1m/content/tmp_files/2301.08579v1.pdf.txt b/qdFAT4oBgHgl3EQffB1m/content/tmp_files/2301.08579v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..175576be06adcaec58bd0d537311c08ab3ba9321 --- /dev/null +++ b/qdFAT4oBgHgl3EQffB1m/content/tmp_files/2301.08579v1.pdf.txt @@ -0,0 +1,8697 @@ +arXiv:2301.08579v1 [astro-ph.SR] 18 Jan 2023 +SAG: Stars and Galaxies , 1–??, ⟨publication date⟩ +© 2023. Center for Astronomy, University of Hyogo. +Secular Variation in the Interval of Outbursts in Z Cam-type Dwarf Novae +Tomohito Ohshima 1 +1Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo, +407–2 Nishigaichi, Sayo-cho, Hyogo 679–5313, Japan +ohshima@nhao.jp +(Received 2022 November 8; accepted 2022 December 19) +Abstract +The secular variation in the interval of outbursts in the following six Z Cam-type dwarf novae (including the +subtype IW And-type) is investigated: Z Cam, RX And, AH Her, HL CMa, SY Cnc, and WW Cet. An analysis using +the O − C diagram shows that the interval of outbursts is not steady in one system. The outburst properties before +standstill are the decrease in outburst interval, enhancement of the magnitude in quiescence, and disappearance of +the long outburst. Meanwhile, several objects have at least two typical intervals of outbursts. These characteristics +are difficult to be explained only by the variation in mass transfer from the secondary. +Key words: dwarf nova — variable star +1. +Introduction +Cataclysmic variables (CVs) are close binary systems con- +sisting of a white dwarf primary and a late-type secondary fill- +ing its Roche lobe. An accretion disk is formed around the +white dwarf by the material transferred from the secondary via +the inner Lagrangian point (L1). Dwarf novae (DNe) are a class +of CVs characterized by the presence of dwarf nova outbursts. +A dwarf nova outburst is a transient event with an amplitude of +several magnitudes generated on the accretion disk. The dwarf +nova outburst is considered to be caused by thermal instability +on the disk [for reviews, see Warner 1995; Osaki 1996); Hellier +2001]. +Z Cam-type dwarf nova is a subgroup of DNe. It is charac- +terized by an intermediate state called ”standstill” in the light +variations (Nijland 1930; de Roy 1932). +In general, the interval of outbursts of Z Cam-type DNe is +relatively short (10–30 d). This high frequency of outbursts +implies a high mass transfer rate onto the accretion disk from +the secondary in Z Cam-type systems. +The accretion disk of DNe shows cycles of a high (hot) state +and a low (cool) state. Meanwhile, disk instability does not +occur in cases where the column density is higher than a certain +borderline on the disk. In such a disk, the accretion disk is +permanently hot. A dense disk is permanently hot. Therefore, +such systems are called nova-like systems (NLs). +The mass transfer rate of Z Cam-type DNe is considered +to lie near the borderline of the mass transfer rate +˙Mcrit, i.e., +the intermediate systems between DNe and NLs. Therefore, +Z Cam-type systems have both phases showing a cycle of out- +bursts (outburst phase) and standstill. However, a certain type +of variation in mass transfer rate is required to transit two +states. +Meyer & Meyer-Hofmeister (1983) proposed that the irradi- +ation effect of the secondary star by the luminosity of the ac- +cretion disk enhances the mass transfer rate. Meanwhile, Ross +& Latter (2017) attempted to reproduce these without variation +in mass transfer rate. +There was a debate over the cause of DN outburst, disk in- +stability theory (DI; Osaki 1974), and enhanced mass transfer +theory (EMT; Bath 1973). The variation in disk radius as out- +bursts repeat was revealed to correspond to the prediction by +the DI theory. The debate was resolved observationally as well +(Osaki & Kato 2013a; Osaki & Kato 2013b; Ohshima et al. +2014). Nevertheless, the origin of the standstill phenomenon +remains unclear with respect to the DI theory. +The variation in outburst frequency is a noteworthy prob- +lem consideringthe role of mass transfer rate in the generation +of standstill. Shafter et al. (2005) investigated the outburst in- +terval in the Z Cam-type DN systems and demonstrated it to +be moderately consistent. However, the long-term variation in +outburst frequency has not been discussed fully. The variations +in the interval of outbursts (trec) in Z Cam-type DNe are dis- +cussed in this paper. +2. +Data set +The observation data used for this study was extracted from +the AAVSO database (Kafka 2021), VSNET data (Kato et +al. 2004), ASAS-SN Sky Patrol data (Shappee et al. 2014), +ASAS3 System data (Pojma´nski 2002), and Zwicky Transient +Facility data (Masci et al. +2019; Bellm et al. 2019). The +span of the data used is 2450000–2459884 (approximately 27 +years). +The target systems are selected from among the brightest +group of Z Cam-type DN systems. The number of known Z +Cam-type systems is 327. +These are listed in International +Variable Star Index (VSX) 1. Meanwhile, 44 are listed in the +1 +https://www.aavso.org/vsx/ + +2 +Center for Astronomy +[Vol. , +General Catalogue of Variable Stars (GCVS). This significant +difference is largely owing to the non-registration (in GCVS) +of objects newly identified by deep sky-survey. These objects +are significantly faint and thereby, are difficult to be monitored +adequately. +The brightest six objects are selected in this study: Z Cam, +RX And, WW Cet, HL CMa, AH Her, and SY Cnc. Although +IX Vel is the brightest Z Cam-type object in VSX, it had not +been considered as a DN object until recently (Kato 2020). The +fundamental property of the six objects is summarized in Table +1. +The outburst timings are determined based on the light +curve. Therefore, the start date of brightening should be used +if observed. However, the gap between observations inhibits +this clear estimation because the observations are generally +few in the long-term monitoring light curve. Thus, the date +on halfway of rising or the earliest date the main outburst is +used in many cases. Thereby, errors of a few days occur in the +determination of the start date of outbursts. Additionally, the +conjunction with the sun makes seasonal gap of observations, +which usually includes several outbursts, is inevitable. +Therefore, observed-minus-calculated (O−C) diagrams are +adopted in this research to investigate the long-term variations +in trec. +3. +Result and Analysis +3.1. +Z Cam +Z Cam is a prototype object of Z Cam-type systems. +Oppenheimer et al. (1998) analyzed AAVSO observation data +and indicated the long-term variation in trec. However, the in- +terval in this study is based on the moving average with the +1500-day window. Thus, the variations in trec in the shorter +term (i.e., in the timescale of months or a few years) cannot be +detected. +The light curve is shown in Figure 1. The Z Cam light vari- +ation broadly includes two phases, or standstill phases, and +outbursts occur consecutively (outburst phase). According to +the light curve, the light variation is divided into 13 outburst +phases and standstill phases between these. The corresponding +range for each outburst phase is plotted in Figure 2 (I–XIII). +Although the standstill phase is not plotted in this diagram, the +journal of standstill in Z Cam is summarized in Table 2. +Linear regression is used to estimate the mean trec with the +dates of the outbursts obtained. The period obtained is 27.61(6) +d. The ephemeris JD of the outburst start date (JDos) is cal- +culated based on this period by using the following ephemeris +equation: +JDos = 2449928.7 + 27.61 × E +The value of the O − C diagram with the ephemeris calcu- +lated using this equation is presented in Figure 2. +This diagram indicates the secular variation in trec. The O− +C diagram is divided by the standstill. The seasonal gap is +insignificant because Z Cam can be observed throughout the +year in the northern hemisphere. Rather, the O − C diagram +has a gap owing to a standstill. +Because a large increase is observed in the O−C diagram if +the number of outbursts is assumed as zero, the cycle number is +offset by a whole-number multiple of intervals near the length +of a standstill. +The characteristics seen outbursts before the start of stand- +still is widely seen; +(1) the minimum magnitude becomes brighter (typically 0.5 +mag); +(2) the duration of outbursts reduces; +(3) the interval of outbursts reduces. +(1) cannot be observed in certain cases (OP VIII, XII, and +XIII). This may be because of the data scattering. (2) includes +the most pronounced characteristics. +(3) is not observed in OP VI and VII. However, these OPs +are relatively peculiar. The outburst frequency in the earlier +stage of OP VI is low because of the circumvention of small +outbursts considering the difficulty of determining their start +timings. Thus, the estimated trec is significantly long. This +peculiar phase ends in the subsequent stage of OP VI. +OP VII is also atypical. The termination timing of the stand- +still between OP VI and VII is ambiguous because the standstill +(when the luminosity was almost constant) smoothly transited +the small oscillation. Such oscillation is generally observed +at the beginning or the final stage of the standstill (Szkody & +Mattei 1984). However, in this case, the amplitude of oscilla- +tions increased and showed a smooth transition to the outburst +of OP VII. Thus, the outbursts in the early and middle stages of +OP VII outbursts may be interpreted as oscillations. Only the +final three outbursts are considered to be ordinary outbursts. +The O − C diagram shows that the trec of Z Cam is incon- +sistent. In addition, the interval length varies abruptly (rather +than gradually) during the outburst phase. . +Oscillations are observed at the beginning (cf: SS IV) or the +final stage (cf: SS IX) of certain cases of standstill. This is +similar to the phenomenon reported in Kato (2001). +3.2. +RX And +RX And is one of the most popular Z Cam-type dwarf novae. +It is also known to have undergone an episode of a substantial +decline in 1997 (Kato et al. 2002). It showed a similar (albeit +significantly shorter decline) episode in 2000 2. These phenom- +ena imply that RX And has a certain mechanism that causes the +reduction in mass transfer rate to be nearly zero. Hence, this +system could play an important role in this study. +The entire light curve is presented in Figure 3. The data used +starts immediately after the standstill (JD 2449900–2450300) +and a substantial decline (JD 2450300–2450450). Ordinary +variations begin to be observed after JD 2450600. The cycle +number of outbursts is counted the standstill after that. +RX And can be observed in almost all the seasons in the +northern hemisphere. However, the observation condition is +less favorable than that for Z Cam because RX And is closer +to the zodiac than Z Cam. Thus, the seasonal gap is observed +more frequently than for Z Cam. The cycle number is esti- +mated by the extrapolation with the date and the preceding in- +terval. +The outburst phase is divided into 14 phases. Unlike One +outburst phase (OP V) is terminated by the long minimum in- +stead a standstill. +2 +[vsnet-alert 4863], http://www.kusastro.kyoto-u.ac.jp/vsnet//Mail +/alert4000/msg00863.html + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +3 +2450000 +2450200 +2450400 +2450600 +2450800 +15 +13 +11 +9 +−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−− +−−−− +−−− +−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−− +−−−− +−−−−−−−−−−−−−−−−−− +− +−−−− +−−−−−−−−− +−− +−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +−− −−−−−− +− −−− −−−− +− −− +−− − −− −− +−−− − +2451000 +2451200 +2451400 +2451600 +2451800 +15 +13 +11 +9 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+−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−− +− +−−− +−−− +− − +− +− +− +− +−− − − +2453000 +2453200 +2453400 +2453600 +2453800 +15 +13 +11 +9 +−− +− +−−− −−−− − − − +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− +−−− −−−−−−−−−−−−−−−−− +− +−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−− +− +−−− +−−−−−− +− +−−−−−− +− +−−−−−−−−−−−−−−−−−− +− +− +− +−− +2454000 +2454200 +2454400 +2454600 +2454800 +15 +13 +11 +9 +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−− +−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−− +−− +−−−−−−−−−−−−−−− +−−−− +− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−− +−−−−−−− −−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−− − +− +− +− +− +2455000 +2455200 +2455400 +2455600 +2455800 +15 +13 +11 +9 +− − −−−−−−−−−−−−−− +− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−− +−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−− +−−−−−−−−−−− +−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− +− +−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−− +− +−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +2456000 +2456200 +2456400 +2456600 +2456800 +15 +13 +11 +9 +−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−− +−−−−−−−−−−−−−−− +−−−−−− +−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +−−−−−−−−−−−−−− +− +−−− +− +−−− +− +−−−−−−−−−−−−−−−− +− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−− +− +−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +− +− +−− +2457000 +2457200 +2457400 +2457600 +2457800 +15 +13 +11 +9 +−−−−−−−−−−−−−−−−−−−− +− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +− +− +− +−−−−− +− +−− +−−−−−− +− +− +−−−−−−−−−− +−−−−− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +−− +−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−− −− +−−−−−−−−−−−−−−−−−−−−−−− +− +−−− +− +−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−− +−−−−−−− +− +−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−− +− +−−− − − +− +2458000 +2458200 +2458400 +2458600 +2458800 +15 +13 +11 +9 +−− − +−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−− +− +−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −− −−−−−−− +−−−− −−−−− +− − −−− +− − +− +2459000 +2459200 +2459400 +JD +2459600 +2459800 +15 +13 +11 +9 +−−−−−−−− +−− −− +− − −− −− +−−−−− +− − +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−− +−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− − +−−−− +Fig. 1. Z Cam light curve of the entire span. The y-axis represents the magnitude. The color bands are as follows: the visual data (black circle), the V +(green circle) or CV (blue circle) band on the CCD data, and the cG (red circle) band of digital camera photometry. The bar represents the upper limit +when not observed. + +4 +Center for Astronomy +[Vol. , +Table 1. Target List +Object +Mmax +Mmin +Orbital Period (d) +Remarks +Z Cam +10.0 +14.5 +0.289841 +RX And +10.3 +14.8 +0.209893 +WW Cet +10.4 +15.8 +0.17578 +HL CMa +10.7 +15.0 +0.216787 +IW And type +AH Her +10.8 +14.9 +0.258116 +IW And type +SY Cnc +11.0 +14.0 +0.3823753 +The data of the range of variation in magnitude (maximum Mmax, minimum Mmin) of variations is retrieved +from GCVS 5.1. The data of the orbital period is retrieved from RKCat Ritter & Kolb (2003). +0 +50 +100 +150 +200 +250 +300 +350 +−300 +−200 +−100 +0 +100 +200 +E +O−C +I II III +IV +V +VI VII VIII +IX +X +XI +XII +XIII +Fig. 2. Complete O − C diagram of Z Cam. The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the +ephemeris equation described in this paper. The Roman numbers I–XIII represent the outburst phases (divided with solid lines). +Table 2. Z Cam Standstill List +Period (JD–2400000) +Duration (d) +SS I +50249–50275 +26 +SS II +20349–50372 +23 +SS III +50830–50868 +38 +SS IV +52012–52123 +111 +SS V +52899–53379 +480 +SS VI +54321–54380 +59 +SS VII +54552–54572 +20 +SS VIII +55393–55532 +139 +SS IX +56270–56443 +163 +SS X +57131–57217 +86 +SS XI +57840–78096 +156 +SS XII +58407–9453 +1046 +SS XIII +9650– +- +trec is determined to be 15.239(13) d by performing linear +regression on the data. Szkody & Mattei (1984) reported the +mean period of RX And as 13 d (scattering 5–20 d). +The following is the ephemeris equation: +JDos = 2450765 + 15.239 × E +The O − C diagram calculated with this equation is pre- +sented in figure 4. OP VIII appears to have an irregularly long +interval. However, this case is similar to OP VI of Z Cam. That +is, small short outbursts are observed frequently, and their start +timings are difficult to determine. The light curve and O − C +diagram reveal that the characteristics identified in Z Cam (1)– +(3) are observed in RX And as well. +Long-continued outburst phases such as OP VI, X, and XII +show a cyclic variation in the O − C value. +This more or +less corresponds to the cyclic variation in luminosity in quies- +cence. However, the cyclic variation in O − C occurs abruptly. +Meanwhile, the variation in luminosity in quiescence occurs +more gradually. Figure 4 shows two types of gradients: the +downward-sloping curve and the upward-sloping curve. This +implies that the O − C curve is composed of two periods: 14 d +and 17 d. + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +5 +2450000 +2450200 +2450400 +2450600 +2450800 +16 +14 +12 +10 +−− +− +−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−− +−−−− +−− +− +−−−−−−−− +− +−−− +−−−−−−−−−−− +− +−−−−− +− +− +− +−− +− +−−−−−− +−−−−−−−−− +−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−− +−− +−−−−−−− +−−−− +− +−−−−−−−−−−−−−−−−−−−−−−− +− +− +− +− +−− +−−−−−−−−−−−−−−− +− +−−−− +−− +− +−−−−− +− +−−−−− +−− +−−−−−− +−− +−−−−−−−−−−− +− +−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−− +− +−−−−−−− +− +−−− +− +− +−−−− +−−−− +−−−− +−− +− +− +−−−−−−−−−− +− +− +−− +−− +− +−−−−−−−−−−− +−− +−−−−−−− +−−−− +− +− +−− +− +− +− +− +−−−−−−−− +− +−−− +− +− +−−− +− +−−−−− +− +−−−−−−−−−−− +− +−−−−−−−−−− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−− +−−− +− +−−− +−−−− +− +− +− +− +−−−− +− +−−−−− +−−−−−−−−−−−−−−−−−−−−− +−−−−− +− +−−− +−−−−−−−−−−−−−−− +− +−−−−−− +− +− +− +− +− +− +− +−−−−− +− +−−−−−− +−−−−−−− +−− 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+−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−− +− +−−−−−− +− +−−−−−− +−−−−−−−−−−−− +− +− +− +−−−− +− +−− +−−−−−− +−−− +− +−−− +−−−−−− +− +− +− +−− +− +− +− +−− +−−− +−− +−− +− +−− +−−−−−−−−−−−− −−− +− +−−−−−−−−−−−−−−−−− +− +− − +− +−−−− +− +−−−−−−−−− +− +−−−− +2459000 +2459200 +2459400 +JD +2459600 +2459800 +16 +14 +12 +10 +−−−−−− +−−−−− +− +− +−−−−−− +−−−−−− +−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−− +−− +−− +−−− +−−− +−−−−−−−− +− +−−−−−− +− +− +− +− +−− +−−− +− +−−−−− +−−−−− +−−−−−−−−−−−−−− +−−− +−−− +−− +− +− +−−− +−− +−−−−−−−− +− +−−−− +− +−−−−−−−−−− +− +−−−−−−− +−−−−− +−− +−−−−−−−−−−−−− +−− +−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−− −−−−−−−−−−−−−−− +−−−−−− +−−−−−−− +−−−− +−−−−−− +−−−−− +− +− +−−−−−− +−−−−−− +−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−− +−− +−− +−−− +−−− +−−−−−−−− +− +−−−−−− +− +− +− +− +−− +−−− +− +−−−−− +−−−−− +−−−−−−−−−−−−−− +−−− +−−− +−− +− +− +−−− +−− +−−−−−−−− +− +−−−− +− +−−−−−−−−−− +− +−−−−−−− +−−−−− +−− +−−−−−−−−−−−−− +−− +−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−− −−−−−−−−−−−−−−− +−−−−−− +−−−−−−− +−−−− +− +− −−− −− +− +− +− − +Fig. 3. RX And light curve of the entire span. The y-axis represents the magnitude. The color bands are as follows: the visual data (black circle), the V +(green circle) or CV (blue circle) band on the CCD data, and the cG (red circle) band of digital camera photometry. The bar represents the upper limit +when not observed. The bar represents the upper limit when not observed. + +6 +Center for Astronomy +[Vol. , +0 +100 +200 +300 +400 +500 +600 +−150 +−100 +−50 +0 +50 +100 +Cycle Number +O−C (d) +I II +III +IVV +VI +VIII +IX +X +XI +XII +XIII +XIV +XV +Fig. 4. The complete O − C diagram of RX And. The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with +the ephemeris equation described in the paper. The Roman numbers I–XV represent the outburst phases (divided with solid lines). +Table 3. RX And Standstill List +Period (JD - 2400000) +Duration (d) +SS I +50826 – 50965 +139 +SS II +51013 – 51068 +55 +SS III +51135 – 51186 +51 +SS IV +51384 – 51431 +47 +SS V +53265 – 53312 +47 +SS VI +53670 – 53710 +40 +SS VII +54203 – 54297 +94 +SS VIII +54366 – 54485 +119 +SS IX +56179 – 56245 +66 +SS X +56305 – 56375* +70 +SS XI +57788 – 57860* +82 +SS XII +58649 – 58681 +32 +* includes errors because of observational gap. +OP V and VI are distinguished by a declining state rather than a standstill. +3.3. +WW Cet +WW Cet is the most complex case of the six objects. +This object was observed as 2.1937 Ceti by Luyten (1962). +Paczynski (1963) indicated that it is a member of the Z Cam +type. +However, Warner (1987) and Ringwald et al. (1996) +indicated the unavailability of evidence to demonstrate that +WW Cet shows a standstill. However, Simonsen & Stubbings +(2011) reported the first observation of WW Cet in a standstill +state. This object is a ”true” Z Cam-type DN. +However, this ”standstill” phenomenon is highly peculiar. +The entire light curve is shown in Figure 5, including this +”standstill”. Although the variations of WW Cet in the 2010 +season was similar to a typical standstill, this object declined +gradually with oscillations in the 2011 season. This is unlike +the behavior called standstill. This oscillation had a period of +approximately 80 ds and an amplitude of 2 magn. This be- +havior is also atypical of the general DN outburst. Although +standstills with oscillations are known, these are not accompa- +nied by a gradual decline. +In addition, gradual variations during 13 mag–15 mag were +observed in the 2012 and 2013 seasons. This is also atypical +of dwarf novae. After the typical standstill again appeared in +the 2014 season, typical outbursts began to be observed, and +the general DN outburst phase started. This uncharacteristic +state continued during JD 2455400–2457150 (approximately 5 +years). Therefore, the case of WW Cet cannot be regarded as +the typical ”standstill”. Apart from this uncharacteristic state, +no standstill is detected in the WW Cet data used. Therefore, +the WW Cet light curve is divided into two phases: before and +after the ”standstill”. +trec is significantly long for a Z Cam-type system. Bateson +(1991) indicated that the mean trec of the WW Cet outburst is +30.70 d. However, trec is not identical (ranges from 18 to 44 +d). +The period is determined to be 38.4(3) d by performing lin- +ear regression on the outburst dates. +With this period, the +ephemeris equation is as follows: +JDos = 2450603 + 38.4(3) × E +The complete O − C diagram of the WW Cet outburst is +shown in Figure 6. This diagram clearly shows the variation +in trec because of the atypical ”standstill”. The mean trec of +outbursts was 48 d before the ”standstill”. It varied to 31 d +after the ”standstill”. This is similar to the value in Bateson +(1991). +3.4. +HL CMa +HL CMa was identified as an Einstein X-ray source 1E +0643.0-1648 (Fuhrmann 1980). +This object was identified + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +7 +2450000 +2450200 +2450400 +2450600 +16 +14 +12 +10 +−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +−−−−−−−−− +− +−−−−−−−− +− +− +−−−−− +− +− +−−−−−−−− +−−−− +−−−− +−−− +− +−−−−−−−−−−−−−−−−−−− +−−−−−−−− +− +− +− +−−−− +− +− +− +− +−−−−−−−− +− +−−−−− +− +− +−−−− +−−−−−− +− +− +−−− +− +−−− +−−− +− +−−− +− +−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− +− +− +−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−− +− +−−−−−−−−−−− +−−−−−−−−−−−− +− +−−−− +− +−−− +−−−−−−−−−−−−−−−−−−− +− +−−−− +−− +−−−−−−−−−−−−−− +− +−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−− +− +− +−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−− +− +− +−−−−−−−−−−−−− +−− +− +− +−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−− +−− −−−−− +−−− +−− +− +−− +−−− +− +2451000 +2451200 +2451400 +2451600 +2451800 +16 +14 +12 +10 +−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−− +−−−−− +−−− +− +− +−−−−−−− +−−−− +− +−−− +−− +− +− +− +−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−− +− +−−−− +−−−−−−− +− +−− +− +−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−− +−−−−−−−−−−− +−−−− +−− +−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−− +−−−−−−−−−− +−−−−−−−−−−−−− +−− +−−−− +− +−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−− +−− +− +−−−−−−−−−−−−−−−− +− +−− +− +− −−−−−−−−−−−− +−−−−−− − +− −−−−−−−−−−−−−−−− −−− +−−−−−−− −−−−−−−−−−−−−−− +−− +−− +−−−−−−−−−−−−−−−− +− +− +−−−−−−−−−−−−− +2452000 +2452200 +2452400 +2452600 +2452800 +16 +14 +12 +10 +− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−− +−−−−−−−− +−−−−−−−−−−−−−−−−−−−− +−−− −−−−−−− +−− +−−−−−−−− +−−−−−− +−−− +− +− +−−− +−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−−−−−−−− +− +−−−− +−−−−−−− +−−−−−− +−−−−− +− +− +− +−−−−−−−−−−−−−−−−−−− +−−−−−− +− +− +−−− +− +−−−− +− +− +−−−−−−−−−−− +−− +−−−− +−−−−−−−− +−−−−− +−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−− +− +−−−−−−−− +− +−−− +−−−−−−−−−−−−−−−− +− +−−− +−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−− +− +− +−−−−−−−−−−−− +−−−−− +− +− +−−−−−−−−−−−−−−−−−−−−−− +−−−−− +−−−−−−−−−− +−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−− +−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−− −−−−−− +− +−−−−−−−−−−−− +−−− +−−−−−−−−−−−−−−−−−−−−− +− +− +−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− +−−−−− +− +−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− +−− +− +− +−− +−− +−− +− +− +−−−−−−−− +− +−−−− +−−−−−−−−−−− +−−−−−−−−−−−− +− +−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +2453000 +2453200 +2453400 +2453600 +2453800 +16 +14 +12 +10 +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−− +−−−−−−−− −−−−−−− +−−−−−−−−−−−−−−− +−−−−−−−− +−−−−− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−− +− +− +−− +−−−−−−−−−−−− +− +−−−− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− −−−−−−−−−−−−−−−−−−−−−− +−−− +−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−− −−−−−−−−−−−−−−−−−−−−−−−−−−−−−− −−−−−−−− +− −−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−− +−− +−−−−−−−−− +−−−− +2454000 +2454200 +2454400 +2454600 +2454800 +16 +14 +12 +10 +−−−−−−−−−−−−−−−−−−−− +−−−− +−− +−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−− +− +−−− +−−− +−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−− +−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−− +−−−−−−−− +− +−−−−−−−−−−−−−−−−− +− +−−−−−− +− +− −−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− −− +2455000 +2455200 +2455400 +2455600 +2455800 +16 +14 +12 +10 +−− +−−−−−− +−−−−−−−−−−−−−− +−− +−−− +−− −− −−−− −−− − +− +−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−− +−− +− +−−−−−−−−−−− −−−− +− +− +− +−− +− +−−−− +−−−− +−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−− +2456000 +2456200 +2456400 +2456600 +2456800 +16 +14 +12 +10 +− +−− +−−−−− +−− +−−−−−−−−−−−−−−−−−−−−− +−−− +− +− +−− +− +−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−− +−−−−−−−−−−− +−−− +− +− +− − +2457000 +2457200 +2457400 +2457600 +2457800 +16 +14 +12 +10 +−−−− +−−−−−−−−−−− +−−− +−−−−−− +− +− +−− +− +− +− +−− +−−−−−−−− +−− − −−−− +− +−−−−−− −−−−−−−−− +− +−−− −− +− +−−−−−−− −−−−−−−−− +− +2458000 +2458200 +2458400 +2458600 +2458800 +16 +14 +12 +10 +−−− − +− +− +−− +2459000 +2459200 +2459400 +JD +2459600 +2459800 +16 +14 +12 +10 +−−− +−− +− +−−− +−−−−− +−−−−−− +−−− +−−−− −−−−−− +− − +− +−− +− +−− +Fig. 5. The WW Cet light curve of the entire span. The y-axis represents the magnitude. The color bands are as follows: the visual data (black circle), the +V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN +data, and the cG (red circle) band of digital camera photometry. The bar represents the upper limit when not observed. The bar represents the upper limit +when not observed. + +8 +Center for Astronomy +[Vol. , +0 +50 +100 +150 +200 +250 +−600 +−400 +−200 +0 +200 +400 +Cycle Times +O−C (d) +I +II +Fig. 6. The complete O − C diagram of WW Cet. The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with +the ephemeris equation described in the paper. The Roman numbers I and II represent the outburst phases (divided with solid lines). +Table 4. WW Cet Standstill List +N +JD-2400000 +Duration (d) +SS 1 +55300* - 56900* +1600 +* includes errors because of observational gap. +with an optical counterpart on a Harvard archive plate, which +showed variability with a recurrence time of approximately +15 d. Therefore, this object was indicated as a dwarf nova +(Chlebowski et al. 1981). The classification as Z Cam-type was +recommended in the context of the orbital period and the inter- +val of outbursts (Cannizzo et al. 1988). The existence of stand- +still was reported by AAVSO observations in 1982 (Mache & +Raymond 1987). The definite long standstill was reported in +1999 (Watanabe et al. 2000). (Kato 2002) reported that HL +CMa shows the third state neither, i.e. neither outburst or nor +standstill (cyclic small variation). +HL CMa is recommended to be classified as a member of the +new group of DNe: IW And-type. The IW And-type object is +a new subgroup of DNe similar to the Z Cam-type. A charac- +teristic of this type is that the standstill of this subgroup system +terminates with the outburst (Kato 2019). Certain members of +this subgroup are classified as Z Cam-type. Simonsen et al. +(2014) reported certain objects displaying this behavior in Z +Cam-type (including HL CMa). +The entire light curve is shown in Figure 7. According to +the light curve, the light variations are divided into 11 outburst +phases. These are terminated with standstill phases. The jour- +nal of standstill in HL CMa is summarized in Table 5. Because +the quiescence of HL CMa is faint (15 mag), observations in +the minimum are highly sparse. Thus, it is challenging to dis- +cuss the variations in luminosity in quiescence as identified in +Z Cam and RX And. +The mean trec is estimated as 16.97(2) d by linear regression. +With this value, the ephemeris equation is as follows: +JDos = 2549958 + 16.97(2) +OP II–III are peculiar phases. These correspond to the pe- +riod reported by Kato (2002). Although a short (∼ 60 d) stand- +still divides this stage into two phases, it is unclear whether this +standstill (SS II) is real or not because of the sparsity of obser- +vations of the conjunction with the sun and small oscillation. +This may be an oscillation phase with a low amplitude. +The standstill between OP VI and VII (SS VI) includes two +standstills. However, only one outburst occurs between these +two standstills. Therefore, this is not regarded as an outburst +phase because an analysis using the O − C diagram cannot +be performed. This behavior wherein a few outbursts appear +during standstill is generally detected in the light curve of IW +And-type dwarf novae (Kato 2019). Although the light curve is +unclear because of the seasonal gap, the transition of OP VII– +VIII (SS VII) appears to be a similar case. This behavior of the +light curve reveals that HL CMa shows the characteristics of +IW And-type. +The O − C diagram implies that HL CMa also has plural in- +tervals of outbursts: 14–15 d and 19–20 d. These correspond to +the downward-sloping and upward-sloping parts, respectively, +in the O − C diagram. Simonsen et al. (2014) indicated that +this object also shows IW And-type behavior. The O − C dia- +gram implies that trec varies around E > 350. However, this is +only an appearance. +3.5. +AH Her +Similar to HL CMa, AH Her is reported to display the char- +acteristic wherein the termination of standstill accompanies an +outburst Simonsen et al. 2014. These two systems are classified +as IW And-type (UGIW) in the VSX database. +This object was reported as a newly identified variable star +in 1923 (Blaˇzko 1923). Jacchia (1937) indicated the relation- + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +9 +2450000 +2450200 +2450400 +2450600 +2450800 +15 +13 +11 +− +−−−−−−−−−− +− +−−−−−−−− +− +−−−−−−−− +− +−−−−−−−−− +−−−−−− +− +−−− +− +−−−−−−− +−−−−−−− +− +−−−− +−−−−−−−−−−−−−−−−−−−−−− +− +− +− +− +− +−−−−− +− +− +− +−−−− +−−−− +− +−−− +− +−− +− +− +− +−− +− +−−−−−−− +− +−−−− +−−−−−− +− +−− +− +−−− +−−−− +−−− +− +−−−− +− +− +−− +− +−−−−−−−−−− +− +− +−−−−−−−− +− +−−− +−− +−−−−− +−− +− +−−−−−− +− +−−− +−− +−−−− +−−−− +−−−−−− +−− +−−−−−− +−− +− +−− +−− +− +− +− +−−−− +− +−−− +− +− +− +−−−−− +−−−−−−−− +− +−−−− +−− +− +− +−− +− +−− +− +− +− +−− +−−− +− +−−−−−− +− +−− +− +−− +−−− +− +− +−− +−−− +−−− +−−− +−−−−−−− +− +−−−−− +− +− +− +−−− +−− +−−− +−− +− +−−−− +− +−−− +− +−−− +−− +−− +−−−−−− +−−−− +−−− +− +−−−−−−−−− +−−− +− +−−−− +−−− +− +−−−− +− +−−−− +− +−−−− +− +−−−−−−−−−−− +− +− +− +−−− +− +−−−−−−−−−−− +− +− +− +−−− −−−−−−−−−−−−−−−−− +−− +−−−−−−−−−−−−−− +− +−−−−−−−− +− +−−−−−−− +− +−−− +− +− +−−−− 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+2456000 +2456200 +2456400 +2456600 +2456800 +15 +13 +11 +−−−−−− +−− +−−−− +− +−−−−−−−−−−−− +− +−− +− +− +− +−−− +− +−−−−−− +− +− +− +−−− +− +−−−− +− +−− +−− +− +− +−−−− +−−− +− +− +−−−− +−−−−−−−− +− +−−− +− +−− +− +−−− +−− +−−−−−−− +−−−−− +−−−−−− +− +− +− +− +−−− +− +− +−− +− +−− +− +− +− +−−− +−−−− +−−−− +− +− +−−−−−−−−−−− +−−−−−−−−− +−−− +−−−− +− +− +− +− +−−−−−−− +−−−− +− +−−−− +−− +− +− +−−− +− +−− +− +− +−−−−−− +−−−− −− −−−−−− +− +−−−−−−−− +− +−−−−−− +−−−− +− +−− +−−− +−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−− +−−−−−−−− +−− +− +−−−−−−−−−−−− +− +−− +− +−−−− +− +−−− +− +−−−−−−− +−− +−− +− +− −−− −−−−−−−−−−−−−−−− +−−− +− +− +−− +− +− +−−− +− +−−−− +− +−−−−− +− +−−− +− +−− +− +−− +− +− +−−−−−−−−−−−−−−−−− +− +−− +− +− +−−−−−−−− +−−− +− +−− +− +− +−− +−−−−−−−−−−−−− +− +−−−−− +−− +−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−−−−−−−−−−− +− +− +− +− +− +−− +− +− +− +− +− +2457000 +2457200 +2457400 +2457600 +2457800 +15 +13 +11 +−−−−−−−−−−−−−−−−− +−− +−−−− +−−−−−−−−−−− +− +−−−−−−−−−− +− +−− +−− +−−−− +−−−−−−−−−−−−−−− +−−−− +−−−−− +−−−−−−−− +−−−−−− +−−−− +−−−− +−−−−−−− +−−− +−−−−−−−−−−−− +− +− +− +− +−−− −−−−−−−−−−−− +−−−−−−−−−−−−−− +−−−− +−−−−−−−−−−−−−−−−− − −− +− +−−− +− +−−−−−−−− +−−−−−−−−−−−−−−−− +−− +−−− +−−−−−−−−−− +−−−−−− +−−−− − +− +− +− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +− +−−−−−−−−−−−−−− +− +−−− +−−−−−− +− +− +− +−− +− +−−−−−−−− +−−−−−−− +−− +− +− +−−−−−−−−−− +−−−− +− +−−−−−−−−−−−−−−−−−− +−−−−−−−−−−−−−−−−−−−− +− +− +−−− +− +− +2458000 +2458200 +2458400 +2458600 +2458800 +15 +13 +11 +− +−−− +− −−− − +− +− − − +−−−−−−− +−−−−− +−− +− +− +− − − +−−−−−−−−−−− +− +− +−−−− +−−− +−−−− +− − +− +− +−−− +− +−−−−−−−−−−−−−− +− +− − +−− +− +− −−−−−−−−−−− +2459000 +2459200 +2459400 +JD +2459600 +2459800 +15 +13 +11 +− +− +−− +−−−−−−−−−−−−−−−−−−−−−− +−−−−−−− +− +− +−−−−− +− +−−−−−−−−−−−−−− +− −−− +−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +− +−−−−− +− +− +− +− +− +−− +− +− +−−−− +− +−−−−− +− +−−− +− +−−−−−−−−−−−−−− +− +− +− +− +−−−− +− +−−−−−− +− +−−−−−−−−−−−−− +−−−− +−− +−−−−−−−− +− +−−−−−− +− +−−−− +Fig. 7. The HL CMa light curve of the entire span. The y-axis represents the magnitude. The color bands are as follows: the visual data (black circle), the +V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN +data, and the cG (red circle) band of digital camera photometry. The bar represents the upper limit when not observed. + +10 +Center for Astronomy +[Vol. , +Table 5. HL CMa Standstill List +Period (JD–2400000) +Duration (d) +Remarks +SS I +51412– 51570 +158 +TO +SS II +52230*–52250* +20 +uncertain +SS III +52345–52415 +70 +TO +SS IV +53264–53310 +46 +SS V +55668–55690 +22 +SS VI +58110–58332* +222 +O 58246 +SS VII +58475–58565* +90 +O 58506 +SS VIII +58797–58925 +128 +SS IX +59310–59352 +42 +TO +SS X +59557–59603 +46 +*:large errors because of observational gap. +TO: outburst as the termination of standstill +O: outburst during standstill and its date +0 +100 +200 +300 +400 +500 +−150 +−100 +−50 +0 +50 +100 +150 +Cycle Number +O−C (d) +I +II +III +IV +V +VI +VII +VIII +IX +X +XI +Fig. 8. The complete O − C diagram of HL CMa. The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with +the ephemeris equation described in the paper. The Roman numbers I–XI represent the outburst phases (divided with solid lines). +ship with Z Cam.Jacchia (1937) reported a period of 19.5 d. +Szkody & Mattei (1984) estimated the interval of outbursts as +18 d (scattering 7–27 d). +The mean trec is determined as 17.57(2) d by linear regres- +sion. With this value, the ephemeris equation is as follows: +JDos = 2450003 + 17.57 × E +The O − C diagram based on this ephemeris is shown in +Figure 10. +The entire light curve is shown in figure 7. According to the +light curve, the light variation is divided into 15 outburst phases +(OP I–XV) and standstill phases between these. The journal of +standstill in AH Her is summarized in Table 6. +Similar to HL CMa, the case wherein outbursts occur tran- +siently during a standstill is observed. The standstill between +OP II and III includes one or two outbursts. The standstill be- +tween IV and V displays a similar behavior. The standstill be- +tween OP X and XI show two such transitions. These behaviors +show that AH Her is IW And-type. +The three characteristics indicated in the Z Cam section are +also identified in AH Her. In particular, a decrease in the out- +burst duration is observed in many cases of OP. The increase in +minimum magnitude is observable in this object. This may be +expressed as oscillations. +In AH Her, OP VII is a peculiar phase. In this phase, the +minimum magnitude is significantly high, frequently attaining +12.5 mag. This is almost as high as that in the standstill. The +profile of outbursts in this OP is atypical for the Z Cam type. +The increase and decrease are highly steep. Finally, the out- +burst frequency reduces, and the standstill starts. +The early phase in OP X shows the converse behavior. The +luminosity attains approximately 12.5 mag even with the max- +imum outburst at that time. The luminosity in the quiescence + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +11 +Table 6. AH Her Standstill List +Period (JD–2400000) +Duration (d) +Remarks +SS I +50722–50744 +22 +SS II +50830*–51024 +194 +O 50958 +SS III +52390–52433 +43 +SS IV +52480–54822 +342 +O 52778 +SS V +53425–53472 +27 +TO +SS VI +54391–54485 +94 +TO +SS VII +54910–55417 +507 +SS VIII +55860–56264 +404 +O 56018, 56107 +SS IX +57042–57169 +127 +TO suspected +SS X +57377–57608 +43 +O 57425 +SS XI +58016–58029 +13 +TO +SS XII +58100–58138 +38 +TO +SS XIII +59784–59816 +32 +TO +* includes errors because of observational gap. +TO: outburst as the termination of standstill +O: outburst during standstill and its date +is similar to the general OP (14 mag). +Notably, the quiescent magnitude of AH Her is generally +high (∼ 12.5 mag) when the next standstill is not being ap- +proached. +3.6. +SY Cnc +SY Cnc is classified as RW Aur-type (a subgroup of the pre- +main sequence star) in the first edition of GCVS. However, +Herbig (1950) revealed that the spectrum of SY Cnc is simi- +lar to those of DNe such as SS Cyg and Z Cam. +Although SY Cnc is classified as Z Cam-type DN, the ob- +servational report of standstill is highly sparse. Only two short +standstills are detected in the used data of approximately 25 +years. In addition, one of the standstills is suspect because it +may be a long outburst. Szkody & Mattei (1984) estimated the +mean trec as 27 d, and trec scatters during 22–35 d. +The entire light curve is shown in Figure 11. As described +above, this object shows a standstill infrequently. Meanwhile, +the seasonal gap is pronounced because this object is close to +the zodiac. trec is determined as 26.330(17) d by linear regres- +sion. This value is close to that presented in Szkody & Mattei +(1984). +The ephemeris equation is obtained as follows with this trec: +JDos = 2449887 + 26.330 × E +The O − C diagram obtained is shown in Figure 12. It indi- +cates that SY Cnc displays variations in trec regardless of the +sparsity of standstill. There are two types of intervals. The +shorter one is 23.5 d, and the longer one is 28 d. However, a +gradual variation in trec is observed in this object. The interme- +diate period is also observed. It is nearly equal to the ”average” +length. It is noteworthy that this variation is not related to the +standstill. +Table 7. SY Cnc Standstill List +Period (JD-2400000) +Duration (d) +Remarks +SS I +56259 - 56283 +25 +SS II +57110 - 57137 +27 +uncertain +4. +Discussion +The standstill frequency and durations differ among the six +objects. However, their O − C diagrams indicate that all the +systems show a secular variation in outburst frequency. +The trend that the It is notable that trec reduces as the next +standstill is approached in almost all the cases among four ob- +jects (Z Cam, RX And, AH Her, and HL CMa) except for the +cases in the highly peculiar states. Similarly, the enhanced lu- +minosity in the quiescence state is also interpreted as the en- +hancement of the disk luminosity. +These two characteristics are explained by the variation in +mass transfer rate. Meanwhile, the long outbursts extinct be- +fore the occurrence of standstill causes a decrease in the mass +accretion from the disk. This is considered to result in an in- +crease in the column density of the disk and the start of the +standstill. +It is noteworthy that the luminosity in quiescence is occa- +sionally not enhanced. Meanwhile, the disappearance of the +long outburst is observed in almost all cases. This disappear- +ance can be interpreted as an enhanced mass transfer rate. +Another problem is the variation trend of trec. +Certain +systems (RX And and AH Her) show a cyclic variation in +the quiescent luminosity during the long-term outburst phase. +Because trec shortens during bright quiescence, this can be ex- +plained by the increased column density of the disk. However, +the O−C diagrams indicate an abrupt variation in trec although +the cyclic variation in quiescent luminosity is gradual. +SY Cnc shows the variation in trec during an outburst phase, +particularly OP I. The abrupt variation in period occurs for E ∼ +120 (JD sim 2455300) and 190 (JD ∼ 2454800). Both abrupt +shortening of the interval. The light curves do not show pecu- +liarity in these timings. +In addition, the trec in a system tends to display two values. +This is difficult to explain. SY Cnc has two typical intervals: +23.5 d and 28 d. HL CMa has two types of intervals: 14–15 d +and 19–20 d. The intervals for RX And are 13 d and 17 d, and +those for Z Cam are 22d and 33d. AH Her shows more than +two typical outburst intervals. +Finally, WW Cet is the most puzzling object in this research. +The irregular state observed during the early 10s may have been +a true standstill. However, this may be related to the variation +in mass transfer rate at any rate. trec varied after this irregular +state. +4.1. +Acknowledgement +I acknowledge with gratitude the variable star observations +obtained from the AAVSO International Database and VSNET +International Database contributed by observers worldwide +and used in this research. +I also acknowledge with grat- +itude the data obtained by ASAS-3, ASASSN, and The + +12 +Center for Astronomy +[Vol. , +2450000 +2450200 +2450400 +2450600 +2450800 +15 +13 +11 +− +−− +− +−− +− +−−− +− +−−−− +−−−−−−− +− +−−−− +− +− +−−− +−− +− −−− +−−−−−−− +− +−−−−− +−−− +− +− +−− +− +−−−− +− +−−− +−− +− +−−− +−−−− +− +−− +−−−−−−−−−−−−−−−−−−−−−− +− +−−−− +−− +− +−− +−− +−−−−−− +− +−−−−−−−−− +−− +− +−− +−−−− +−− +−−−− +−− +− +− +−− +− +−−−−− +−− +− +−−− +−− +− +−−− +− +−−− +− +− +−−−−−−− +− +− +− +− +−−−−−−− +− +−−−−−−−−−−− +− +−−− +− +− +−−−− +− +−−−− +− +−−−− +−− +−− +− +− +− +− +− +− +−−−− +− +− +−−−−−−−−−− +− +−−− +− +−−−−−−−−−−−− +− +−− +− +−− +− +−− +− +− +−− +− +−−−−−−− +− +−− +− +− +−−− +− +− +− +− +−− +− +−−−−− +− +− +− +−−−−−− +− +− +− +− +−−−−− +− +− +−−−−−−− +−−−−− +−− − − +− +− +− +−−−−− +− +− −− +− +− +− +−−− +−−−−−−−−−−− +−−−−− +−−− +−−−−− +−− +−− +− +− +− +−−−− +−− +−−−−− +− +−−−−− +−− +−−− +− +− +−−−−−−−− +− +−−−−− +−− +− +−−−−− +−−− +− +−− +− +−−−− +−−− +−− +− +− +− +− +− +− +− +− +− +− +−−−−−−−−−− +− +− +− +−−− +−− +− +−−− +−−− +− − +− +−− −− +−− +− +− +−−− +− +−−− +− +2451000 +2451200 +2451400 +2451600 +2451800 +15 +13 +11 +−−− +− +−−− +−−−− +−−−−−− +−−− +−−−−−−−−−−−−−−−−−−−−−− +−− +−−− +−−−−−−−−−−−−− +−−−−−− +− +− +− +−−− +− +−−−− +− +−− −−− +−− +− +−−−−− +−−−−−−−− +−−− +− +− +− +−−−−−−−−− +− +−−−−−− +− +−−−−−−−−−−−−−− +− +−− +− +−−−−−−−−−−− +−−− +−−−−−− +− +−− +−−− +−−−−− +−−−−− +−−−−−−−− +− +− +−− +− +− +−−−− +− +−−−−−−−−− +− − +− +−−−− +−−−−−− +−−−− +−−−−− +−−−−−−−−−−−− +−− +− +− +− +−− +− +−−−− +−−−−−−−−−− +−− +−− +−− +−− +− +− +− +−−− +− +−− +− +− +− +− +− +−−− −−− − +− +−−−−−− +−−−−−−−−−−−− −− +− −−−−−−− +−−−−− +−−− +−−−− +2452000 +2452200 +2452400 +2452600 +2452800 +15 +13 +11 +−− +− +− +− +−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−−−−−− +−−−−−− +− +− +−−−−− +− +−−− +−−−−−−−−− +− +−− +−−− +−−−−−−−−−−− +− +− +− +−−−−−−−− +− +− +−−−−− +−−−−−−−−−− +− +− +−− +− +−− +− +−−−−− +−−−−− +−−−−−−−−− +− +− −−−− +− +−−− +−−− −−−−− +− +− +− +−−− +− +− +−−−−−−− +−− +− +−−− +−− +− +−−−−−−−−−−−−−− +− −−−−− +− +−−−−−− +− +−−−−−−−−−− +− +−−−−− +−−−−−−−−−−−−−−−−−−− +−−−−− +2453000 +2453200 +2453400 +2453600 +2453800 +15 +13 +11 +− +−− +− +−−− +− +−−−−−−−−−−−−−− +− +− +−−−−−−−−−−−−− +− +−− +−−−−−−−−−−−−−−− +−−−−−−−−−−−− +−−−− +−−−− +− +−− +− +− +−− +− +−− −− − +− +−−− +−−−−−−−− +−−−−−− +−−−− +−−−−−−−− +− +−−−−−−−−−−−− − +−−−−− +− +−− +−−−−−−− +− +−−− +−−− +− +−− +−−−−−−−−−−− +−−−− +− +−− +− +−−−−−−−−−−−−−− +−−−−−−−−−−−−− +−−−−−−−− +−− +−−−−−−−−−− +−− +−−− +−− +− +− +− +− +−−− +−− +− +− +− +− +−−− −−−− +−− +− +− +−−− +−−−−−−− +−−− +− +− −− +− +−−− − +− +−−− +− +− +− +−− +−−−− +−− +− +− +−−−−−−−−−−− −−− +−− − +2454000 +2454200 +2454400 +2454600 +2454800 +15 +13 +11 +−−− +−− +− +− +− +− +−−− +− +−−−−−−−−−−− +− +− +−−−−− +−−−−−−− +− −−−−−−−−−−−−− +−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +−−− +−−−−−−−−−−− +−−− +−− +−−− +−−−−−− +− +−− +−−−−−− − +−− +−−−− −−−− +− −−− +−−−−−−−−−−−−−−−−− +−−−−−−−−−−−− +−−−−−− +− +−−−− +−−−−−−− − +− −−− +− − +2455000 +2455200 +2455400 +2455600 +2455800 +15 +13 +11 +−−− − +− +−−− +− +− +− +− +− +− +− +−−−− +−−−−−−−−−−−−−−−− +− +−−−−−−− +− +− +− +−−−−− +−−−−−−−−−−− +− +−−−−−− − − +−− +−− +−−−− +−− +−−−−− +− +− +− +−−−−−−− +−−− +− +− +− +− +−−− +− +−− +−−−−−− +−− +−− +−− +− +− +−−−−−−− +−−−−− +− +−−− +−− +−− +− +−− +− +−−−−−−− +−−−−−− +− +−−− +−− −−−−− +− − +− − +−− +− +− +− +2456000 +2456200 +2456400 +2456600 +2456800 +15 +13 +11 +− − +− +−−−−−−−−−−− +− +− +− +−−−−− +−−−−−−−−−−−−−−−−−− +− +−−−−−−− +− +− +− −−−−−−−−−−−−−−−−− +−− +− +−−− +−−−− +−− +− +− +− +−−−−−− +−−−− +− +−−− +−−− +−−−−−−− +− +− +− +−−−−−−−−− +−−− +− +− +−− +−−−−−−−−− +−− +− +−−−−−−−−−−−−− +− +−−− − +−− −−− −−−−−−−−−−−−−−−−−−−−−−−−− +−−−−− +−− +− +−−−−−−−−−−−− +− +− +−− +−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +−−−−−−−− +− +−−−−−−−−− +−− +−−−−−−−−−− +− +− +−−−−− +− +2457000 +2457200 +2457400 +2457600 +2457800 +15 +13 +11 +− +−−−−−−−−−−− +− +−−−−−−−− +− +−−−−− +−− +−− +−−−−− +−−−−−−−−− − +−− +− +−−−− −−−−−−−−−−−−−−−−−−−−−−−−− +−− +−−−−− +− +−−−−− +− +−−−−−− +− +−−− −−−−−−−−−−−−−−−−−−−−−−−−− +− +−−−− +−−−− +− +−−−− +−−− +− +− − − +− +− +−−−− +2458000 +2458200 +2458400 +2458600 +2458800 +15 +13 +11 +−− +−−−− +− +−−−−−− +−−−− +−−−−−−−− +−− +−−− −−−−−− +− +− +− −−−−−−−−−−−−−−−−−−−− +− +−−−−−−−− +− +− +− +−− −−− +− +−−−−−−−−−−−−−− +−−− −−−−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +− +− +− − +−− +− +− +− +− +−−−−− +−−− +− +2459000 +2459200 +2459400 +JD +2459600 +2459800 +15 +13 +11 +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +− +−−−−−−−−−− +− +−−− +−−−− +− +− +− +−−−−−− +− +−−−−−− +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− +− +− +−−−− +− +−−−−−−− +− +−−− +− +− +−−−− +− +− +− +−− − +− +−−−−−−−−−−− +−−−− +Fig. 9. The AH Her light curve of the entire span. The y-axis represents the magnitude. The color bands are as follows: the visual data (black circle), the +V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN +data, and the cG (red circle) band of digital camera photometry. The bar represents the upper limit when not observed. + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +13 +0 +100 +200 +300 +400 +500 +−200 +−100 +0 +100 +200 +Cycle Number +O−C (d) +I +II +III +IV V VI +VII +VIII +IX +X +XI +XII XIII +XIV +XV +XVI +Fig. 10. The complete O − C diagram of AH Her. The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with +the ephemeris equation described in the paper. The Roman numbers I–XV represent the outburst phases (divided with solid lines). +Zwicky Transient Facility. I also would like to thank Editage +(www.editage.com) for English language editing. +References +Bath, G. T. 1973, Nature. Physical Science, 246, 84 +Bellm, E. C. et al. 2019, Publications of the Astronomical Society of +the Pacific, 131, 018002 +Bateson, F. M. and McIntosh, R. 1991, Royal Astronomical Society +of New Zealand Publications of Variable Star Section, 16, 70 +Blaˇzko, S. 1923, Astron. Nachr., 219, 372 +Cannizzo, +J. K., +Shafter, +A. W., +and Wheeler, +J. C. 1988, +Astrophysical Journal, 333, 227 +Chlebowski, T., Halpem, J. P., and Steiner, J. E. 1981, Astrophysical +Journal, 247, L35 +de Roy, F. 1932, Gazette Astronomique, 19, 1 +Fuhrmann, B. 1980, Information Bulletin on Variable Stars, 1883 +Jacchia, L. 1937, Astron. Nachr., 261, 212 +Herbig, G. H. 1950, Publications of the Astronomical Society of the +Pacific, 62, 211 +Hellier, C. 2001, Cataclysmic Variable Stars: How and why they +vary(Berlin: Springer-Verlag) +Jayasinghe, T. et al. 2021, MNRAS, 503, 200 +Kafka, +S. 2021, +Observations from the AAVSO International +Database, https://www.aavso.org +Kato, T. 2001, Information Bulletin on Variable Stars, 5093 +Kato, T. 2002, Information Bulletin on Variable Stars, 5243 +Kato, T., Nogami, D., and Masuda, S. 2002, Publications of the +Astronomical Society of Japan, 54, 575 +Kato, T., Uemura, M., Ishioka, R., Nogami, D., Kunjaya, C., Baba, H., +and Yamaoka, H. 2004, Publications of the Astronomical Society +of Japan, 56, S1 +Kato, T. 2019, Publications of the Astronomical Society of Japan, 71, +20 +Kato, T. 2021, VSOLJ Bulletin, 87, 1 +Luyten, W. J. 1962, Harvard Observatory Announcement Card, 1574 +Mauche, C. W. and Raymond, J. C. 1987, Astrophysics & Space +Science, 130, 269 +Masci, F. J. et al. 2019, Publications of the Astronomical Society of +the Pacific, 131, 018003 +Meyer, F. and Meyer-Hofmeister, E. 1983, 121, 29 +Nijland, A. A. 1930, Bulletin of the Astronomical Institute of the +Netherlands, 5, 243 +,Ohshima, T. et al. 2014, Publications of the Astronomical Society of +Japan, 66, 63 +Oppenheimer, B. D., Kenyon, S. J., and Mattei, J. A. 1998, +Astronomical Journal, 115, 1175 +Osaki, Y. 1974, Publications of the Astronomical Society of Japan, +26, 429 +Osaki, Y. 1996, Publications of the Astronomical Society of the +Pacific, 108, 39 +Osaki, Y. and Kato, T. 2013, Publications of the Astronomical Society +of Japan, 65, 50 +Osaki, Y. and Kato, T. 2013, Publications of the Astronomical Society +of Japan, 64, 95 +Paczynski, B. 1963, Publications of the Astronomical Society of the +Pacific, 75, 278 +Pojma´nski, G. 2002, Acta Astronomica, 52, 397 +Ringwald, F. A., Thorstensen, J. R., Honeycutt, R. K., and Smith, R. +C. 1996, Astronomical Journal, 111, 2077 +Ritter, H. and Kolb, U. 2003, Astronomy & Astrophysics, 404, 301 +Ross, J. and Latter, H. N. 2017, MNRAS, 470, 34 +Shafter, A. W., Cannizzo, J. K., and Waagen, E. O. 2005, Publications +of the Astronomical Society of the Pacific, 117, 931 +Shappee, B. J. et al. 2014, Astrophysical Journal, 788, 48 +Simonsen, M. and Stubbings, R. 2011, J. American Assoc. Variable +Star Obs., 39, 73 +Simonsen, M. et al. 2014, JAAVSO, 42, 177 +Szkody, P. and Mattei, J. A. 1984, Publications of the Astronomical +Society of the Pacific, 96, 988 +Warner, B. 1987, Monthly Notices of the Royal Astronomical Society, +227, 23 +Warner, B. 1995, Cataclysmic Variable Stars (Cambridge: Cambridge +University Press) + +14 +Center for Astronomy +[Vol. , +2450000 +2450200 +2450400 +2450600 +2450800 +14 +12 +10 +−− +−−−−−− +−− +−−−− +−−−−− +− +−−−−−−− +− +−− +− +−−−− +− +−−−−−−−−− +− −− − − +−−−−−− +− +−− −− +−−−− +−−−−− −−− +−−−− −− +−−−− −− +− +− +−−−− −−−−− −− +−−− − +2451000 +2451200 +2451400 +2451600 +2451800 +14 +12 +10 +− +− +− +− +−−− +− +−−−−−− +−−− +− +− +−−− +− +−− +− +− +−−−−− +− +−−−− −−−− +2452000 +2452200 +2452400 +2452600 +2452800 +14 +12 +10 +−−− +− +− +− +− −−− +− +−− −− +−− +− +− − +−−−− +− −−− +− +− +− +− +−−−− +−− − +−− +− +2453000 +2453200 +2453400 +2453600 +2453800 +14 +12 +10 +− − +− +− +− +− +−−−−− −− +− +−− +− +− +− −− +−− +− +− +− −−−−−− +− +− +− +−− +− +− +− +− +−− +− +−− − +−−− +− − +−−−−− +−− +− − +2454000 +2454200 +2454400 +2454600 +2454800 +14 +12 +10 +−− +− +−−−− +−− +− +−− +−−−−−−− +− +− +− +−−−−−−−−− +− +− −−−−−−−−− −− +− +− −− +− +− +− +− +− − +− +− +− +2455000 +2455200 +2455400 +2455600 +2455800 +14 +12 +10 +−− +− −−−−− +− −− −−− +− −−−−−−− +− +− +− +−−−−− +− +−− +− +−−−− +− +−− − +− +−− +− − +− +−− −− +− +−−−− +− +−−−−−−−−−−−−−−−−−− +− +−−−−−−−−−−−−−−− +2456000 +2456200 +2456400 +2456600 +2456800 +14 +12 +10 +−−− +−−−− +−− +− +− +−− − +− +− +− +−− +− +−−−−−−−−−− +− +−−−− +−− +−−−−−−−−−−−− −−−− +− +−−−−−−−−−−−−−−−−−−−−−−−−−− +−−−−−−−− +− +2457000 +2457200 +2457400 +2457600 +2457800 +14 +12 +10 +− +−− +−− +− +−−−−−−−− +−−− +−− +− +−−−−−− +− +−− +− +−− +− +− +− +− − +− +− +− − +− − − +2458000 +2458200 +2458400 +2458600 +2458800 +14 +12 +10 +−−− −− +− +− +− +− +− +−−− +− +− +− +2459000 +2459200 +2459400 +JD +2459600 +2459800 +14 +12 +10 +− +−−−−−−−−− − +− +− − −−− +− − −−− +−−−−−− −− +Fig. 11. The SY Cnc light curve of the entire span. The y-axis represents the magnitude. The color bands are as follows: the visual data (black circle), the +V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN +data, and the cG (red circle) band of digital camera photometry. The bar represents the upper limit when not observed. + +No. ] +Outburst Variation of Z Cam-type Dwarf Novae +15 +0 +50 +100 +150 +200 +250 +300 +350 +−50 +0 +50 +100 +Cycle Times +O−C (d) +I +II +III +Fig. 12. The complete O − C diagram of SY Cnc. The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with +the ephemeris equation described in the paper. The Roman numbers I–III represent the outburst phases (divided with solid lines). +Watanabe, T. and Stubbings, R., and Maehara, H. 2000, VSOLJ +Bulletin, 35-36, 7 + diff --git a/qdFAT4oBgHgl3EQffB1m/content/tmp_files/load_file.txt b/qdFAT4oBgHgl3EQffB1m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d745096356363f0ca33111b632e05ca06480829 --- /dev/null +++ b/qdFAT4oBgHgl3EQffB1m/content/tmp_files/load_file.txt @@ -0,0 +1,8294 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf,len=8293 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='08579v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='SR] 18 Jan 2023 SAG: Stars and Galaxies , 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=', ⟨publication date⟩ © 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Center for Astronomy, University of Hyogo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Secular Variation in the Interval of Outbursts in Z Cam-type Dwarf Novae Tomohito Ohshima 1 1Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo, 407–2 Nishigaichi, Sayo-cho, Hyogo 679–5313, Japan ohshima@nhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='jp (Received 2022 November 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' accepted 2022 December 19) Abstract The secular variation in the interval of outbursts in the following six Z Cam-type dwarf novae (including the subtype IW And-type) is investigated: Z Cam, RX And, AH Her, HL CMa, SY Cnc, and WW Cet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' An analysis using the O − C diagram shows that the interval of outbursts is not steady in one system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The outburst properties before standstill are the decrease in outburst interval, enhancement of the magnitude in quiescence, and disappearance of the long outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, several objects have at least two typical intervals of outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These characteristics are difficult to be explained only by the variation in mass transfer from the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Key words: dwarf nova — variable star 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Introduction Cataclysmic variables (CVs) are close binary systems con- sisting of a white dwarf primary and a late-type secondary fill- ing its Roche lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' An accretion disk is formed around the white dwarf by the material transferred from the secondary via the inner Lagrangian point (L1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Dwarf novae (DNe) are a class of CVs characterized by the presence of dwarf nova outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' A dwarf nova outburst is a transient event with an amplitude of several magnitudes generated on the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The dwarf nova outburst is considered to be caused by thermal instability on the disk [for reviews, see Warner 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Osaki 1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Hellier 2001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Z Cam-type dwarf nova is a subgroup of DNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It is charac- terized by an intermediate state called ”standstill” in the light variations (Nijland 1930;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' de Roy 1932).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In general, the interval of outbursts of Z Cam-type DNe is relatively short (10–30 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This high frequency of outbursts implies a high mass transfer rate onto the accretion disk from the secondary in Z Cam-type systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The accretion disk of DNe shows cycles of a high (hot) state and a low (cool) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, disk instability does not occur in cases where the column density is higher than a certain borderline on the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In such a disk, the accretion disk is permanently hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' A dense disk is permanently hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, such systems are called nova-like systems (NLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The mass transfer rate of Z Cam-type DNe is considered to lie near the borderline of the mass transfer rate ˙Mcrit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=', the intermediate systems between DNe and NLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, Z Cam-type systems have both phases showing a cycle of out- bursts (outburst phase) and standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, a certain type of variation in mass transfer rate is required to transit two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meyer & Meyer-Hofmeister (1983) proposed that the irradi- ation effect of the secondary star by the luminosity of the ac- cretion disk enhances the mass transfer rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, Ross & Latter (2017) attempted to reproduce these without variation in mass transfer rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' There was a debate over the cause of DN outburst, disk in- stability theory (DI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Osaki 1974), and enhanced mass transfer theory (EMT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Bath 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The variation in disk radius as out- bursts repeat was revealed to correspond to the prediction by the DI theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The debate was resolved observationally as well (Osaki & Kato 2013a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Osaki & Kato 2013b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Ohshima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Nevertheless, the origin of the standstill phenomenon remains unclear with respect to the DI theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The variation in outburst frequency is a noteworthy prob- lem consideringthe role of mass transfer rate in the generation of standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Shafter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (2005) investigated the outburst in- terval in the Z Cam-type DN systems and demonstrated it to be moderately consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, the long-term variation in outburst frequency has not been discussed fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The variations in the interval of outbursts (trec) in Z Cam-type DNe are dis- cussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Data set The observation data used for this study was extracted from the AAVSO database (Kafka 2021), VSNET data (Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2004), ASAS-SN Sky Patrol data (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2014), ASAS3 System data (Pojma´nski 2002), and Zwicky Transient Facility data (Masci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Bellm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The span of the data used is 2450000–2459884 (approximately 27 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The target systems are selected from among the brightest group of Z Cam-type DN systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The number of known Z Cam-type systems is 327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These are listed in International Variable Star Index (VSX) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, 44 are listed in the 1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='aavso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='org/vsx/ 2 Center for Astronomy [Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' , General Catalogue of Variable Stars (GCVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This significant difference is largely owing to the non-registration (in GCVS) of objects newly identified by deep sky-survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These objects are significantly faint and thereby, are difficult to be monitored adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The brightest six objects are selected in this study: Z Cam, RX And, WW Cet, HL CMa, AH Her, and SY Cnc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although IX Vel is the brightest Z Cam-type object in VSX, it had not been considered as a DN object until recently (Kato 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The fundamental property of the six objects is summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The outburst timings are determined based on the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, the start date of brightening should be used if observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, the gap between observations inhibits this clear estimation because the observations are generally few in the long-term monitoring light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thus, the date on halfway of rising or the earliest date the main outburst is used in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thereby, errors of a few days occur in the determination of the start date of outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Additionally, the conjunction with the sun makes seasonal gap of observations, which usually includes several outbursts, is inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, observed-minus-calculated (O−C) diagrams are adopted in this research to investigate the long-term variations in trec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Result and Analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Z Cam Z Cam is a prototype object of Z Cam-type systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Oppenheimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (1998) analyzed AAVSO observation data and indicated the long-term variation in trec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, the in- terval in this study is based on the moving average with the 1500-day window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thus, the variations in trec in the shorter term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=', in the timescale of months or a few years) cannot be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The light curve is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Z Cam light vari- ation broadly includes two phases, or standstill phases, and outbursts occur consecutively (outburst phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' According to the light curve, the light variation is divided into 13 outburst phases and standstill phases between these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The corresponding range for each outburst phase is plotted in Figure 2 (I–XIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although the standstill phase is not plotted in this diagram, the journal of standstill in Z Cam is summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Linear regression is used to estimate the mean trec with the dates of the outbursts obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The period obtained is 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='61(6) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The ephemeris JD of the outburst start date (JDos) is cal- culated based on this period by using the following ephemeris equation: JDos = 2449928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='7 + 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='61 × E The value of the O − C diagram with the ephemeris calcu- lated using this equation is presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This diagram indicates the secular variation in trec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The O− C diagram is divided by the standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The seasonal gap is insignificant because Z Cam can be observed throughout the year in the northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Rather, the O − C diagram has a gap owing to a standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Because a large increase is observed in the O−C diagram if the number of outbursts is assumed as zero, the cycle number is offset by a whole-number multiple of intervals near the length of a standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The characteristics seen outbursts before the start of stand- still is widely seen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (1) the minimum magnitude becomes brighter (typically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 mag);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (2) the duration of outbursts reduces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (3) the interval of outbursts reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (1) cannot be observed in certain cases (OP VIII, XII, and XIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This may be because of the data scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (2) includes the most pronounced characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (3) is not observed in OP VI and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, these OPs are relatively peculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The outburst frequency in the earlier stage of OP VI is low because of the circumvention of small outbursts considering the difficulty of determining their start timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thus, the estimated trec is significantly long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This peculiar phase ends in the subsequent stage of OP VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' OP VII is also atypical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The termination timing of the stand- still between OP VI and VII is ambiguous because the standstill (when the luminosity was almost constant) smoothly transited the small oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Such oscillation is generally observed at the beginning or the final stage of the standstill (Szkody & Mattei 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, in this case, the amplitude of oscilla- tions increased and showed a smooth transition to the outburst of OP VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thus, the outbursts in the early and middle stages of OP VII outbursts may be interpreted as oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Only the final three outbursts are considered to be ordinary outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The O − C diagram shows that the trec of Z Cam is incon- sistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In addition, the interval length varies abruptly (rather than gradually) during the outburst phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Oscillations are observed at the beginning (cf: SS IV) or the final stage (cf: SS IX) of certain cases of standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is similar to the phenomenon reported in Kato (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' RX And RX And is one of the most popular Z Cam-type dwarf novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It is also known to have undergone an episode of a substantial decline in 1997 (Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It showed a similar (albeit significantly shorter decline) episode in 2000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These phenom- ena imply that RX And has a certain mechanism that causes the reduction in mass transfer rate to be nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Hence, this system could play an important role in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The entire light curve is presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The data used starts immediately after the standstill (JD 2449900–2450300) and a substantial decline (JD 2450300–2450450).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Ordinary variations begin to be observed after JD 2450600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The cycle number of outbursts is counted the standstill after that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' RX And can be observed in almost all the seasons in the northern hemisphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, the observation condition is less favorable than that for Z Cam because RX And is closer to the zodiac than Z Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thus, the seasonal gap is observed more frequently than for Z Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The cycle number is esti- mated by the extrapolation with the date and the preceding in- terval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The outburst phase is divided into 14 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Unlike One outburst phase (OP V) is terminated by the long minimum in- stead a standstill.' metadata={'source': 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+page_content='−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Z Cam light curve of the entire span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The y-axis represents the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The color bands are as follows: the visual data (black circle), the V (green circle) or CV (blue circle) band on the CCD data, and the cG (red circle) band of digital camera photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 4 Center for Astronomy [Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' , Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Target List Object Mmax Mmin Orbital Period (d) Remarks Z Cam 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='289841 RX And 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='209893 WW Cet 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='17578 HL CMa 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='216787 IW And type AH Her 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='258116 IW And type SY Cnc 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='3823753 The data of the range of variation in magnitude (maximum Mmax, minimum Mmin) of variations is retrieved from GCVS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The data of the orbital period is retrieved from RKCat Ritter & Kolb (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 0 50 100 150 200 250 300 350 −300 −200 −100 0 100 200 E O−C I II III IV V VI VII VIII IX X XI XII XIII Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Complete O − C diagram of Z Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the ephemeris equation described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Roman numbers I–XIII represent the outburst phases (divided with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Z Cam Standstill List Period (JD–2400000) Duration (d) SS I 50249–50275 26 SS II 20349–50372 23 SS III 50830–50868 38 SS IV 52012–52123 111 SS V 52899–53379 480 SS VI 54321–54380 59 SS VII 54552–54572 20 SS VIII 55393–55532 139 SS IX 56270–56443 163 SS X 57131–57217 86 SS XI 57840–78096 156 SS XII 58407–9453 1046 SS XIII 9650– trec is determined to be 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='239(13) d by performing linear regression on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Szkody & Mattei (1984) reported the mean period of RX And as 13 d (scattering 5–20 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The following is the ephemeris equation: JDos = 2450765 + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='239 × E The O − C diagram calculated with this equation is pre- sented in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' OP VIII appears to have an irregularly long interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, this case is similar to OP VI of Z Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' That is, small short outbursts are observed frequently, and their start timings are difficult to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The light curve and O − C diagram reveal that the characteristics identified in Z Cam (1)– (3) are observed in RX And as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Long-continued outburst phases such as OP VI, X, and XII show a cyclic variation in the O − C value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This more or less corresponds to the cyclic variation in luminosity in quies- cence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, the cyclic variation in O − C occurs abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, the variation in luminosity in quiescence occurs more gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Figure 4 shows two types of gradients: the downward-sloping curve and the upward-sloping curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This implies that the O − C curve is composed of two periods: 14 d and 17 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ] ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='− − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' RX And light curve of the entire span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The y-axis represents the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The color bands are as follows: the visual data (black circle), the V (green circle) or CV (blue circle) band on the CCD data, and the cG (red circle) band of digital camera photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 6 Center for Astronomy [Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' , 0 100 200 300 400 500 600 −150 −100 −50 0 50 100 Cycle Number O−C (d) I II III IVV VI VIII IX X XI XII XIII XIV XV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The complete O − C diagram of RX And.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the ephemeris equation described in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Roman numbers I–XV represent the outburst phases (divided with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' RX And Standstill List Period (JD - 2400000) Duration (d) SS I 50826 – 50965 139 SS II 51013 – 51068 55 SS III 51135 – 51186 51 SS IV 51384 – 51431 47 SS V 53265 – 53312 47 SS VI 53670 – 53710 40 SS VII 54203 – 54297 94 SS VIII 54366 – 54485 119 SS IX 56179 – 56245 66 SS X 56305 – 56375* 70 SS XI 57788 – 57860* 82 SS XII 58649 – 58681 32 includes errors because of observational gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' OP V and VI are distinguished by a declining state rather than a standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' WW Cet WW Cet is the most complex case of the six objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This object was observed as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='1937 Ceti by Luyten (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Paczynski (1963) indicated that it is a member of the Z Cam type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, Warner (1987) and Ringwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (1996) indicated the unavailability of evidence to demonstrate that WW Cet shows a standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, Simonsen & Stubbings (2011) reported the first observation of WW Cet in a standstill state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This object is a ”true” Z Cam-type DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, this ”standstill” phenomenon is highly peculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The entire light curve is shown in Figure 5, including this ”standstill”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although the variations of WW Cet in the 2010 season was similar to a typical standstill, this object declined gradually with oscillations in the 2011 season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is unlike the behavior called standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This oscillation had a period of approximately 80 ds and an amplitude of 2 magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This be- havior is also atypical of the general DN outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although standstills with oscillations are known, these are not accompa- nied by a gradual decline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In addition, gradual variations during 13 mag–15 mag were observed in the 2012 and 2013 seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is also atypical of dwarf novae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' After the typical standstill again appeared in the 2014 season, typical outbursts began to be observed, and the general DN outburst phase started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This uncharacteristic state continued during JD 2455400–2457150 (approximately 5 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, the case of WW Cet cannot be regarded as the typical ”standstill”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Apart from this uncharacteristic state, no standstill is detected in the WW Cet data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, the WW Cet light curve is divided into two phases: before and after the ”standstill”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' trec is significantly long for a Z Cam-type system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Bateson (1991) indicated that the mean trec of the WW Cet outburst is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='70 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, trec is not identical (ranges from 18 to 44 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The period is determined to be 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='4(3) d by performing lin- ear regression on the outburst dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' With this period, the ephemeris equation is as follows: JDos = 2450603 + 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='4(3) × E The complete O − C diagram of the WW Cet outburst is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This diagram clearly shows the variation in trec because of the atypical ”standstill”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The mean trec of outbursts was 48 d before the ”standstill”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It varied to 31 d after the ”standstill”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is similar to the value in Bateson (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' HL CMa HL CMa was identified as an Einstein X-ray source 1E 0643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='0-1648 (Fuhrmann 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This object was identified No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='Outburst Variation of Z Cam-type Dwarf Novae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='−−−−−− ' metadata={'source': 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+page_content=' The color bands are as follows: the visual data (black circle), the V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN data, and the cG (red circle) band of digital camera photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 8 Center for Astronomy [Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' , 0 50 100 150 200 250 −600 −400 −200 0 200 400 Cycle Times O−C (d) I II Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The complete O − C diagram of WW Cet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the ephemeris equation described in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Roman numbers I and II represent the outburst phases (divided with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' WW Cet Standstill List N JD-2400000 Duration (d) SS 1 55300* - 56900* 1600 includes errors because of observational gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' with an optical counterpart on a Harvard archive plate, which showed variability with a recurrence time of approximately 15 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, this object was indicated as a dwarf nova (Chlebowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The classification as Z Cam-type was recommended in the context of the orbital period and the inter- val of outbursts (Cannizzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The existence of stand- still was reported by AAVSO observations in 1982 (Mache & Raymond 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The definite long standstill was reported in 1999 (Watanabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (Kato 2002) reported that HL CMa shows the third state neither, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' neither outburst or nor standstill (cyclic small variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' HL CMa is recommended to be classified as a member of the new group of DNe: IW And-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The IW And-type object is a new subgroup of DNe similar to the Z Cam-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' A charac- teristic of this type is that the standstill of this subgroup system terminates with the outburst (Kato 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Certain members of this subgroup are classified as Z Cam-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Simonsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (2014) reported certain objects displaying this behavior in Z Cam-type (including HL CMa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The entire light curve is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' According to the light curve, the light variations are divided into 11 outburst phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These are terminated with standstill phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The jour- nal of standstill in HL CMa is summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Because the quiescence of HL CMa is faint (15 mag), observations in the minimum are highly sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Thus, it is challenging to dis- cuss the variations in luminosity in quiescence as identified in Z Cam and RX And.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The mean trec is estimated as 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='97(2) d by linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' With this value, the ephemeris equation is as follows: JDos = 2549958 + 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='97(2) OP II–III are peculiar phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These correspond to the pe- riod reported by Kato (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although a short (∼ 60 d) stand- still divides this stage into two phases, it is unclear whether this standstill (SS II) is real or not because of the sparsity of obser- vations of the conjunction with the sun and small oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This may be an oscillation phase with a low amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The standstill between OP VI and VII (SS VI) includes two standstills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, only one outburst occurs between these two standstills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Therefore, this is not regarded as an outburst phase because an analysis using the O − C diagram cannot be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This behavior wherein a few outbursts appear during standstill is generally detected in the light curve of IW And-type dwarf novae (Kato 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although the light curve is unclear because of the seasonal gap, the transition of OP VII– VIII (SS VII) appears to be a similar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This behavior of the light curve reveals that HL CMa shows the characteristics of IW And-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The O − C diagram implies that HL CMa also has plural in- tervals of outbursts: 14–15 d and 19–20 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These correspond to the downward-sloping and upward-sloping parts, respectively, in the O − C diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Simonsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' (2014) indicated that this object also shows IW And-type behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The O − C dia- gram implies that trec varies around E > 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, this is only an appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' AH Her Similar to HL CMa, AH Her is reported to display the char- acteristic wherein the termination of standstill accompanies an outburst Simonsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These two systems are classified as IW And-type (UGIW) in the VSX database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This object was reported as a newly identified variable star in 1923 (Blaˇzko 1923).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Jacchia (1937) indicated the relation- No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='Outburst Variation of Z Cam-type Dwarf Novae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='9 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='−−−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='−−−− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The HL CMa light curve of the entire span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The y-axis represents the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The color bands are as follows: the visual data (black circle), the V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN data, and the cG (red circle) band of digital camera photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 10 Center for Astronomy [Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' , Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' HL CMa Standstill List Period (JD–2400000) Duration (d) Remarks SS I 51412– 51570 158 TO SS II 52230*–52250* 20 uncertain SS III 52345–52415 70 TO SS IV 53264–53310 46 SS V 55668–55690 22 SS VI 58110–58332* 222 O 58246 SS VII 58475–58565* 90 O 58506 SS VIII 58797–58925 128 SS IX 59310–59352 42 TO SS X 59557–59603 46 :large errors because of observational gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' TO: outburst as the termination of standstill O: outburst during standstill and its date 0 100 200 300 400 500 −150 −100 −50 0 50 100 150 Cycle Number O−C (d) I II III IV V VI VII VIII IX X XI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The complete O − C diagram of HL CMa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the ephemeris equation described in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Roman numbers I–XI represent the outburst phases (divided with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ship with Z Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='Jacchia (1937) reported a period of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Szkody & Mattei (1984) estimated the interval of outbursts as 18 d (scattering 7–27 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The mean trec is determined as 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='57(2) d by linear regres- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' With this value, the ephemeris equation is as follows: JDos = 2450003 + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='57 × E The O − C diagram based on this ephemeris is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The entire light curve is shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' According to the light curve, the light variation is divided into 15 outburst phases (OP I–XV) and standstill phases between these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The journal of standstill in AH Her is summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Similar to HL CMa, the case wherein outbursts occur tran- siently during a standstill is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The standstill between OP II and III includes one or two outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The standstill be- tween IV and V displays a similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The standstill be- tween OP X and XI show two such transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These behaviors show that AH Her is IW And-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The three characteristics indicated in the Z Cam section are also identified in AH Her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In particular, a decrease in the out- burst duration is observed in many cases of OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The increase in minimum magnitude is observable in this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This may be expressed as oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In AH Her, OP VII is a peculiar phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In this phase, the minimum magnitude is significantly high, frequently attaining 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is almost as high as that in the standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The profile of outbursts in this OP is atypical for the Z Cam type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The increase and decrease are highly steep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Finally, the out- burst frequency reduces, and the standstill starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The early phase in OP X shows the converse behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The luminosity attains approximately 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 mag even with the max- imum outburst at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The luminosity in the quiescence No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ] Outburst Variation of Z Cam-type Dwarf Novae 11 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' AH Her Standstill List Period (JD–2400000) Duration (d) Remarks SS I 50722–50744 22 SS II 50830*–51024 194 O 50958 SS III 52390–52433 43 SS IV 52480–54822 342 O 52778 SS V 53425–53472 27 TO SS VI 54391–54485 94 TO SS VII 54910–55417 507 SS VIII 55860–56264 404 O 56018,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 56107 SS IX 57042–57169 127 TO suspected SS X 57377–57608 43 O 57425 SS XI 58016–58029 13 TO SS XII 58100–58138 38 TO SS XIII 59784–59816 32 TO includes errors because of observational gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' TO: outburst as the termination of standstill O: outburst during standstill and its date is similar to the general OP (14 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Notably, the quiescent magnitude of AH Her is generally high (∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 mag) when the next standstill is not being ap- proached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' SY Cnc SY Cnc is classified as RW Aur-type (a subgroup of the pre- main sequence star) in the first edition of GCVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, Herbig (1950) revealed that the spectrum of SY Cnc is simi- lar to those of DNe such as SS Cyg and Z Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Although SY Cnc is classified as Z Cam-type DN, the ob- servational report of standstill is highly sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Only two short standstills are detected in the used data of approximately 25 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In addition, one of the standstills is suspect because it may be a long outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Szkody & Mattei (1984) estimated the mean trec as 27 d, and trec scatters during 22–35 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The entire light curve is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' As described above, this object shows a standstill infrequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, the seasonal gap is pronounced because this object is close to the zodiac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' trec is determined as 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='330(17) d by linear regres- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This value is close to that presented in Szkody & Mattei (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The ephemeris equation is obtained as follows with this trec: JDos = 2449887 + 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='330 × E The O − C diagram obtained is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It indi- cates that SY Cnc displays variations in trec regardless of the sparsity of standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' There are two types of intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The shorter one is 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 d, and the longer one is 28 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, a gradual variation in trec is observed in this object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The interme- diate period is also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It is nearly equal to the ”average” length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It is noteworthy that this variation is not related to the standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' SY Cnc Standstill List Period (JD-2400000) Duration (d) Remarks SS I 56259 - 56283 25 SS II 57110 - 57137 27 uncertain 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Discussion The standstill frequency and durations differ among the six objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, their O − C diagrams indicate that all the systems show a secular variation in outburst frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The trend that the It is notable that trec reduces as the next standstill is approached in almost all the cases among four ob- jects (Z Cam, RX And, AH Her, and HL CMa) except for the cases in the highly peculiar states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Similarly, the enhanced lu- minosity in the quiescence state is also interpreted as the en- hancement of the disk luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' These two characteristics are explained by the variation in mass transfer rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, the long outbursts extinct be- fore the occurrence of standstill causes a decrease in the mass accretion from the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is considered to result in an in- crease in the column density of the disk and the start of the standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' It is noteworthy that the luminosity in quiescence is occa- sionally not enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Meanwhile, the disappearance of the long outburst is observed in almost all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This disappear- ance can be interpreted as an enhanced mass transfer rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Another problem is the variation trend of trec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Certain systems (RX And and AH Her) show a cyclic variation in the quiescent luminosity during the long-term outburst phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Because trec shortens during bright quiescence, this can be ex- plained by the increased column density of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, the O−C diagrams indicate an abrupt variation in trec although the cyclic variation in quiescent luminosity is gradual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' SY Cnc shows the variation in trec during an outburst phase, particularly OP I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The abrupt variation in period occurs for E ∼ 120 (JD sim 2455300) and 190 (JD ∼ 2454800).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Both abrupt shortening of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The light curves do not show pecu- liarity in these timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' In addition, the trec in a system tends to display two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' This is difficult to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' SY Cnc has two typical intervals: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='5 d and 28 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' HL CMa has two types of intervals: 14–15 d and 19–20 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The intervals for RX And are 13 d and 17 d, and those for Z Cam are 22d and 33d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' AH Her shows more than two typical outburst intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Finally, WW Cet is the most puzzling object in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The irregular state observed during the early 10s may have been a true standstill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' However, this may be related to the variation in mass transfer rate at any rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' trec varied after this irregular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Acknowledgement I acknowledge with gratitude the variable star observations obtained from the AAVSO International Database and VSNET International Database contributed by observers worldwide and used in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' I also acknowledge with grat- itude the data obtained by ASAS-3, ASASSN, and The 12 Center for Astronomy [Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='2450800 ' metadata={'source': 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data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN data, and the cG (red circle) band of digital camera photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ] Outburst Variation of Z Cam-type Dwarf Novae 13 0 100 200 300 400 500 −200 −100 0 100 200 Cycle Number O−C (d) I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The complete O − C diagram of AH Her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the ephemeris equation described in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Roman numbers I–XV represent the outburst phases (divided with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Zwicky Transient Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' I also would like to thank Editage (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='editage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content='com) for English language editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' References Bath, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 1973, Nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Physical Science, 246, 84 Bellm, E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 1950, Publications of the Astronomical Society of the Pacific, 62, 211 Hellier, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2001, Cataclysmic Variable Stars: How and why they vary(Berlin: Springer-Verlag) Jayasinghe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2021, MNRAS, 503, 200 Kafka, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The SY Cnc light curve of the entire span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The y-axis represents the magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The color bands are as follows: the visual data (black circle), the V (green circle) or CV (blue circle) band on the CCD data, the zg (dark green circle) band on the Zwichy data, the g (cyan circle) band on the ASASSN data, and the cG (red circle) band of digital camera photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The bar represents the upper limit when not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' ] Outburst Variation of Z Cam-type Dwarf Novae 15 0 50 100 150 200 250 300 350 −50 0 50 100 Cycle Times O−C (d) I II III Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The complete O − C diagram of SY Cnc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The x-axis represents the number of cycles, and the y-axis represents the O − C value calculated with the ephemeris equation described in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' The Roman numbers I–III represent the outburst phases (divided with solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' and Stubbings, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=', and Maehara, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} +page_content=' 2000, VSOLJ Bulletin, 35-36, 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdFAT4oBgHgl3EQffB1m/content/2301.08579v1.pdf'} diff --git a/qtE1T4oBgHgl3EQf2gXF/vector_store/index.faiss 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Properties of resolved UC H�� regions +C.Zhang,1,2Feng-Yao Zhu,3¢ Tie Liu,4† Z.-Y. Ren, 5 H.-L. Liu,6 Ke Wang,7 J.-W. Wu,8,5 Y.Zhang,9 +J.-W. Zhou,5 K.Tatematsu,10 Guido Garay,11 Anandmayee Tej,12 Shanghuo Li,13 W.F.Xu,7 +Chang Won Lee,13 Leonardo Bronfman,11 Archana Soam,14 and D. Li5,15 +1Department of Physics, Taiyuan Normal University, Jinzhong 030619,China +2Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong 030619,China +3 Research Center for Intelligent Computing Platforms, Zhejiang Laboratory, Hangzhou, 311100, China +4Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai 200030, China +5National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, People’s Republic of China +6Department of Astronomy, Yunnan University, Kunming,650091, PR China +7Kavli Institute for Astronomy and Astrophysics, Peking University, 5 Yiheyuan Road, Haidian District, Beijing 100871, People’s Republic of China +8University of Chinese Academy of Sciences, Beijing 100049, PR China +9School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Road, Zhuhai, Guangdong, 519082, People’s Republic of China +10Nobeyama Radio Observatory, National Astronomical Observatory of Japan, National Institutes of Natural Sciences, Nobeyama, Minamimaki, Minamisaku, Nagano 384-1305, Japan +11Departamento de Astronomıa, Universidad de Chile, Las Condes, Santiago, Chile +12Indian Institute of Space Science and Technology, Thiruvananthapuram 695 547, Kerala, India +13Korea Astronomy and Space Science Institute, 776 Daedeokdaero, Yuseong-gu, Daejeon 34055, Republic of Korea +14SOFIA Science Centre, USRA, NASA Ames Research Centre, MS-12, N232, Mo�ett Field, CA 94035, USA +15NAOC-UKZN Computational Astrophysics Centre, University of KwaZulu-Natal, Durban 4000, South Africa +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Hydrogen recombination lines (RRLs) are one of the major diagnostics of the physical properties of H�� regions. We use RRL +H40U, He40U and 3 mm continuum emission to investigate the properties of a large sample of resolved UC H�� regions identified +in the ATOMS survey. In total, we identify 94 UC H�� regions from H40U emission. The basic parameters for these UC H�� +regions such as electron density, emission measure, electron temperature, ionic abundance ratio (nHe+/nH+), and line width are +derived. The median electron density and the median nHe+/nH+ ratio of these UC H�� regions derived from RRLs are ⇠9000 +cm�3 and 0.11, respectively. Within UC H�� regions, the nHe+/nH+ ratios derived from the intensity ratio of the He40U and H40U +lines seems to be higher in the boundary region than in the center. The H40U line width is mainly broadened by thermal motion +and microturbulence. The electron temperature of these UC H�� regions has a median value of ⇠6700 K, and its dependence on +galactocentric distance is weak. +Key words: ISM: clouds, (ISM:) H�� regions, stars: formation, radio lines: ISM +1 INTRODUCTION +Massive stars can significantly influence their local environments +through powerful feedback such as winds, radiation and supernova +explosions. Ionizing radiation from massive stars produces pro- +nounced H�� regions around them (e.g., Krumholz et al. 2014). Thus, +observations of H�� regions provide important insight into massive +star formation (Habing & Israel 1979; Churchwell 2002; Zinnecker +& Yorke 2007). +H�� regions in the Galaxy show low nHe+/nH+ number density +ratios (less than 0.1 which is the actual He/H abundance; Draine +2011) from RRL observations, and this ratio seems to decrease with +¢ E-mail: zhufy@zhejianglab.com +† E-mail: liutie@shao.ac.cn +increasing distance from the H�� region beyond the photodissociation +region (PDR; Luisi et al. 2016; Roshi et al. 2017). Here nHe+ is the +number density of singly ionized helium (He+), and nH+ is that of +ionized hydrogen (H+). The observed low nHe+/nH+ number density +ratios outside the PDR regions may indicate that there will be fewer +photons with enough energy to ionize He compared to H there or +part of radiation field is attenuated by interstellar dust (Roshi et al. +2012). Therefore, the =He+/=H+ ratio is very sensitive to the strength +of radiation field and dust properties. The nHe+/nH+ ratios inside UC +H�� regions, however, have not been systematically investigated. +Electron temperatures, T4, of H�� regions were estimated from +the intensity ratios between radio recombination lines (RRLs) and +continuum emission on the LTE approximation in previous observa- +tions (e.g. Shaver et al. 1983; Caswell & Haynes 1987). However, +Gordon & Sorochenko (2002) points out that using LTE approxi- +© 2015 The Authors + +2 +Chao Zhang +mation could cause significant deviation of estimated T4 in some +conditions. Previous studies indicate that T4 of H�� regions tends to +be positively correlated with the distance from the Galactic Center +(RGC) (Churchwell et al. 1978; Downes et al. 1980; Wink et al. 1983; +Shaver et al. 1983; Paladini et al. 2004). This is because that inner +Galaxy tends to have higher metallicities, resulting in greater cooling +rates and lower T4 of H�� region (Shaver et al. 1983, Quireza et al. +2006, Balser et al. 2011). However, most of these studies were con- +ducted with single-dishes, which cannot resolve distant H�� regions. +These relations need to be tested with high resolution data. +Observations of hydrogen recombination lines at radio and +(sub)millimetre (submm/mm) frequencies are routinely used by ob- +servers to infer densities, temperatures and velocity structures inside +H�� regions (Gordon & Sorochenko 2002). For young Ultra-Compact +(UC) H�� regions that are still embedded in their natal gas clumps +and are not visible at optical and infrared wavelengths, submm/mm +RRLs are among the best tracers for probing their physical proper- +ties because they are much less a�ected by the pressure broadening, +and brighter than the centimeter RRLs with larger principal quantum +numbers (Keto et al. 2008). +Previous statistical observations of submm/mm RRLs toward UC +H�� regions were mostly conducted with single dishes (Churchwell +et al. 2010; Kim et al. 2017, 2018; Nguyen-Luong et al. 2017). +These observations, however, cannot resolve the structures as well as +gas properties of UC H�� regions. In this work, we present the first +systematic investigation of the properties of a large sample of resolved +UC H�� regions with high resolution (⇠2.500) RRLs H40U and He40U +as well as the 3mm continuum emission that was obtained from the +ATOMS (ALMA Three-millimeter Observations of Massive Star +forming regions) survey. In particular, we focus on the measurements +of nHe+/nH+ number density ratios and electron temperatures of these +UC H�� regions. +In section 2, we describe the observations. Section 3 presents the +results, focusing on statistics of parameters such as electron density +(n4), emission measure (EM), T4, nHe+/nH+ ratio and line width. A +discussion of the results is given in Section 4. We summarize our +main conclusions in Section 5. +2 OBSERVATIONS +2.1 The sample +We made the ALMA observations for 146 active Galactic star form- +ing regions through the ATOMS (ALMA Three-millimeter Obser- +vations of Massive Star forming regions) survey. These 146 sources +were selected from the CS J = 2-1 survey of Bronfman et al. (1996), +a complete and homogeneous molecular line survey of UC H�� re- +gion candidates in the Galactic plane. The sample of 146 targets is +complete for proto-clusters with bright CS J = 2-1 emission (T1 >2 +K), indicative of rather dense gas. The overall properties of these 146 +targets are shown in Table A of Liu et al. (2020a). Majority (139) of +the targets are located in the first and fourth Galactic Quadrants of +the inner Galactic Plane. The distances of the sample clouds range +from 0.4 kpc to 13.0 kpc with a mean value of 4.5 kpc. The sam- +ple includes 27 distant (d>7 kpc) sources that are either close to +the Galactic center or mini-starbursts, representing extreme environ- +ments for star formation. The properties of these sources have been +described in detail in Liu et al. (2016, 2020a,b) and Liu et al. (2021). +2.2 ALMA observations +The ATOMS data consist of 146 sources observed in both ALMA- +12m and ACA-7m arrays. The spectral setup consists of 8 spectral +windows (spw), where spws 1-6 are narrow bands centered on par- +ticular lines, and spws 7-8 are broad band (1.875 GHz), low spectral +resolution for continuum and lines. Table 2 of Liu et al. 2020a sum- +merizes the details of receiver setups. +Fully calibrated visibility data are generated by running the ALMA +Pipeline, which applies calibration tables obtained from the QA2 to +the raw data. Images are made from the combined 12m and ACA vis- +ibility data. Continuum images are constructed from line-free chan- +nels in spw 7 and 8 centered at ⇠ 99.4 GHz (or 3 mm). Line images +are made for each spw with native spectral resolution. More details of +data reduction are described in Liu et al. (2022), Zhou et al. (2021) +and Wang Ke et al. (in prep). Uncertainties in flux calibration are +estimated to be ⇠ 10%. All images are primary-beam corrected. The +typical rms noise (1 sigma) for the continuum images is ⇠0.2 mJy +for a synthesized beam of ⇠ 200. In this work, we also use RRLs +H40U and He40U line data with spectral resolution of 1.5 km s�1. +The typical beam size and channel rms noise level for RRLs are ⇠ 2.5 +00 and 0.02 Jy beam�1, respectively. The rest frequencies of H40U +and He40U line are 99.022952 and 99.063305 GHz, respectively. +3 RESULT +3.1 Resolved UC H�� regions identified with RRL H40U +The raw data of RRL and 3mm observations are cubes and continuum +images, respectively. We integrate the RRL cube along the velocity +axis to get intensity maps. Then we extracted compact objects (UC H�� +regions) from the integrated intensity maps of the H40U and He40U. +Figure 1 shows the integrated intensity maps for two exemplar sources +I15567-5236 and I09002-4732. In each source, the left and middle +panels show the integrated intensity maps of H40U and He40U, +respectively. The 3 mm continuum emission is shown as contours +in the left panel. The 3 mm continuum emission coincides with the +RRLs emission very well, indicating that the 3 mm continuum is +dominated by free-free emission, as also suggested in Keto et al. +(2008). +The compact objects in H40U and He40U emission can be easily +identified by eyes. In total, we detected 94 and 44 compact sources +(UC H�� regions) in emission from H40U and He40U, respectively. +The number of the UC H�� regions agrees with that by Liu et al. +(2021). Table C1 summarized the parameters for H40U, He40U and +3mm emission. The majority (⇠90%) of UC H�� regions are resolved. +In further analysis, we are neglecting unresolved H�� regions (with +sizes smaller than one beam), which are labeled ⇤ after the IRAS +name in Table 1 and Table C1 . +The majority (⇠86%) of targeted sources contain only one UC H�� +region in RRLs line emission as illustrated for I09002-4732 in the +bottom panel of Figure 1. For the case of ‘mutiple sources’ which are +very close to each other and hard to divide, like I15567-5236 in the +top panel of Figure 1, we treated them as one single UC H�� region. +However, we identified multiple UC H�� regions in 10 sources from +H40U maps. We derive parameters for individual UC H�� regions +and listed them in Table 1. The e�ective radius ('e�) of each UC +H�� region is defined as 'e� = +p +(/c, where S is the area of the +UC H�� region within the 5f contours. 'e� are listed in the third +column of Table 1. The median 'e� is ⇠0.1 pc. For resolved ob- +jects, the uncertainties of 'e� are mainly determined by distances. +The uncertainties of 'e� caused by measured area (S) are negligible. +MNRAS 000, 1–13 (2015) + +ATOMS-XIV: H�� regions +3 +Figure 1. The source of I15567-5236 (top) and I09002-4732(bottom). The value of di�erent colors show in the colorbar which in the top of each panel. For each +source, the left and middle panels are H40U and He40U intensity map over 5f, respectively. The 3 mm continuum emission is shown in contours in intensity +map of H40U. Contours are from 20% to 80% in step of 20% of peak values. The right panel shows the ionic abundance ration of the source. The black ellipses +represent the synthesized beam sizes. +However, the uncertainties of distances are not well known. There- +fore, we did not consider the uncertainties of 'e�. We note that this +should a�ect the error analysis of some derived parameteres (such as +emission measure and electron density) in following analysis, which +can be better dealt with from more accurate distance measurements +in future. +Figure 2 presents the correlation between distance and 'e� of +the sample. There is a noticeable increasing trend between the dis- +tance and 'e�. The correlation coe�cient is 0.45 with p-value ⌧ +0.001. The increasing trend indicates that in distant sources only +large ('e� > 0.1 pc) UC H�� regions were resolved in ATOMS ob- +servations. +3.2 Properties of ionized Gas within UC H�� regions +3.2.1 Emission measure and electron density derived from H40U +emission +To derive emission measure (EM) and electron density (n4) from +an RRL, we assume that the total rate of recombinations can be +computed from 'rec = =4=?+UB, where n? is the proton density. + +is the volume and UB is the total recombination coe�cient excluding +captures to the n=1 levels (i.e., case B). The ionization of helium and +stimulation of RRLs are also assumed to be negligible. Following +Zhang et al. (2022), we define the line luminosity (in CGS units) as: +!a(H40U) = =4=?+n, +(1) +MNRAS 000, 1–13 (2015) + +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +lonic Abundance Ratio (Y+)5 +10 +15 +20 +H40α +-52°45'15" +18″" +Dec +21" +24" +16h00m37.0s +36.5s +36.0s +RA0.25 +0.75 +1.00 +1.25 +1.50 +1.75 +0.50 +He40α2 +4 +6 +8 +10 +12 +H40α +-4744'30" +33" +Dec +36" +39" +9h01m58.258.0s +57.8s +57.2s +57.6s +57.4s +57.0s +RA0.2 +0.4 +0.6 +0.8 +1.00.10 +0.15 +0.20 +0.25 +0.30 +0.05 +lonic Abundance Ratio (Y+)4 +Chao Zhang +Table 1. Gas clumps identified in H40U. +IRAS +ID +'e� +RGC +vlsr +EM +n4 +EM +n4 +Y+ +Temperature +(pc) +(kpc) +(km/s) +(106 cm�6 pc) +(104 cm�3) +(106 cm�6 pc) +(104 cm�3) +(K) +RRL +RRL +3mm +3mm +I09002-4732 +1 +0.03 +8.4 +3.1 +83.01 ± 8.31 +3.67 ± 0.18 +108.77 ± 9.75 +4.19 ± 0.19 +0.11 ± 0.01 +7570 ± 790 +I11298-6155 +1 +0.25 +10.1 +32.9 +3.04 ± 0.30 +0.25 ± 0.01 +4.94 ± 0.41 +0.32 ± 0.01 +0.00 ± 0.00 +7270 ± 690 +I12320-6122 +1 +0.06 +7.2 +-42.5 +27.93 ± 2.80 +1.54 ± 0.08 +34.67 ± 2.52 +1.71 ± 0.06 +0.07 ± 0.01 +7040 ± 600 +I12326-6245 +1 +0.06 +7.2 +-39.6 +150.57 ± 15.15 +3.56 ± 0.18 +285.82 ± 21.72 +4.92 ± 0.19 +0.09 ± 0.01 +9000 ± 850 +I12383-6128 +1 +0.07 +7.2 +-39.1 +3.79 ± 0.38 +0.52 ± 0.03 +4.75 ± 0.34 +0.58 ± 0.02 +0.00 ± 0.00 +6560 ± 560 +I12572-6316 +1 +0.16 +9.8 +30.9 +2.62 ± 0.26 +0.29 ± 0.01 +6.36 ± 0.52 +0.45 ± 0.02 +0.00 ± 0.00 +11060 ± 940 +I13080-6229 +1 +0.25 +6.9 +-35.6 +8.27 ± 0.83 +0.41 ± 0.02 +11.79 ± 1.31 +0.49 ± 0.03 +0.14 ± 0.02 +7480 ± 860 +I13111-6228 +1 +0.07 +6.9 +-38.8 +2.10 ± 0.21 +0.38 ± 0.02 +3.10 ± 0.41 +0.46 ± 0.03 +0.00 ± 0.00 +8420 ± 720 +I13291-6229 +1 +0.08 +7.0 +-36.5 +3.10 ± 0.31 +0.45 ± 0.02 +2.85 ± 0.22 +0.43 ± 0.02 +0.00 ± 0.00 +5000 ± 420 +I13291-6229 +2 +0.08 +7.0 +-36.5 +1.91 ± 0.19 +0.34 ± 0.02 +2.22 ± 0.17 +0.37 ± 0.01 +0.00 ± 0.00 +5810 ± 610 +I13291-6249 +1 +0.26 +7.1 +-34.7 +6.95 ± 0.70 +0.37 ± 0.02 +10.57 ± 0.91 +0.45 ± 0.02 +0.14 ± 0.02 +7300 ± 910 +I13471-6120* +1 +0.07 +6.4 +-56.7 +79.29 ± 7.95 +2.33 ± 0.12 +85.02 ± 7.37 +2.41 ± 0.10 +0.09 ± 0.01 +6500 ± 620 +I14013-6105 +1 +0.11 +6.4 +-48.1 +24.59 ± 2.46 +1.04 ± 0.05 +33.70 ± 3.58 +1.22 ± 0.06 +0.12 ± 0.01 +7290 ± 840 +I14050-6056 +1 +0.15 +6.6 +-47.1 +7.20 ± 0.72 +0.48 ± 0.02 +8.87 ± 1.05 +0.53 ± 0.03 +0.16 ± 0.02 +6310 ± 850 +I14382-6017 +1 +0.57 +6.0 +-60.7 +2.52 ± 0.25 +0.15 ± 0.01 +2.99 ± 0.33 +0.16 ± 0.01 +0.00 ± 0.00 +6470 ± 620 +I14453-5912 +1 +0.19 +6.6 +-40.2 +1.31 ± 0.13 +0.19 ± 0.01 +1.41 ± 0.15 +0.19 ± 0.01 +0.00 ± 0.00 +5330 ± 510 +I15254-5621 +1 +0.06 +5.7 +-67.3 +98.39 ± 9.87 +2.88 ± 0.14 +111.57 ± 8.74 +3.05 ± 0.12 +0.09 ± 0.01 +7140 ± 680 +I15290-5546 +1 +0.11 +4.9 +-87.5 +27.35 ± 2.74 +1.09 ± 0.05 +43.16 ± 3.45 +1.38 ± 0.06 +0.10 ± 0.01 +7570 ± 790 +I15290-5546 +2 +0.21 +4.9 +-87.5 +23.78 ± 2.38 +0.74 ± 0.04 +27.35 ± 2.83 +0.80 ± 0.04 +0.11 ± 0.01 +6920 ± 730 +I15384-5348 +1 +0.10 +6.9 +-41.0 +7.04 ± 0.70 +0.60 ± 0.03 +9.17 ± 1.09 +0.69 ± 0.04 +0.14 ± 0.02 +6490 ± 880 +I15408-5356 +1 +0.14 +6.9 +-39.7 +6.10 ± 0.61 +0.47 ± 0.02 +7.16 ± 1.38 +0.51 ± 0.05 +0.18 ± 0.02 +7580 ± 1020 +I15411-5352 +1 +0.10 +6.9 +-41.5 +15.51 ± 1.55 +0.90 ± 0.05 +18.84 ± 2.11 +1.00 ± 0.06 +0.12 ± 0.01 +6640 ± 760 +I15439-5449 +1 +0.06 +5.9 +-54.6 +13.71 ± 1.37 +1.06 ± 0.05 +18.66 ± 1.39 +1.24 ± 0.05 +0.00 ± 0.00 +7040 ± 600 +I15502-5302 +1 +0.10 +4.6 +-91.4 +238.75 ± 23.93 +3.55 ± 0.18 +285.80 ± 23.75 +3.88 ± 0.16 +0.11 ± 0.01 +8640 ± 940 +I15520-5234 +1 +0.07 +6.2 +-41.3 +52.62 ± 5.27 +1.91 ± 0.10 +49.02 ± 3.67 +1.85 ± 0.07 +0.00 ± 0.00 +5840 ± 500 +I15567-5236 +1 +0.15 +4.4 +-107.1 +65.14 ± 6.52 +1.49 ± 0.07 +77.70 ± 6.76 +1.63 ± 0.07 +0.11 ± 0.01 +6650 ± 700 +I15570-5227 +1 +0.31 +4.4 +-101.5 +3.37 ± 0.34 +0.23 ± 0.01 +4.22 ± 0.35 +0.26 ± 0.01 +0.00 ± 0.00 +6430 ± 550 +I16037-5223* +1 +0.12 +4.9 +-80.0 +43.34 ± 4.36 +1.35 ± 0.07 +51.39 ± 4.25 +1.47 ± 0.06 +0.11 ± 0.01 +6430 ± 670 +I16037-5223* +2 +0.14 +4.9 +-80.0 +22.70 ± 2.28 +0.91 ± 0.05 +27.44 ± 2.34 +1.00 ± 0.04 +0.09 ± 0.02 +6590 ± 820 +I16037-5223* +3 +0.12 +4.9 +-80.0 +12.40 ± 1.25 +0.70 ± 0.04 +12.84 ± 1.19 +0.72 ± 0.03 +0.11 ± 0.02 +5860 ± 730 +I16060-5146 +1 +0.09 +4.5 +-91.6 +403.90 ± 40.48 +4.82 ± 0.24 +539.91 ± 43.10 +5.57 ± 0.22 +0.10 ± 0.01 +10060 ± 970 +I16065-5158 +1 +0.11 +5.2 +-63.3 +10.39 ± 1.04 +0.70 ± 0.04 +23.15 ± 1.74 +1.05 ± 0.04 +0.00 ± 0.00 +9250 ± 880 +I16065-5158 +2 +0.07 +5.2 +-63.3 +7.62 ± 0.77 +0.73 ± 0.04 +11.77 ± 0.70 +0.90 ± 0.03 +0.00 ± 0.00 +5990 ± 630 +I16071-5142 +1 +0.10 +4.5 +-87.0 +7.63 ± 0.77 +0.60 ± 0.03 +16.54 ± 1.23 +0.89 ± 0.03 +0.00 ± 0.00 +8930 ± 760 +I16132-5039 +1 +0.09 +5.8 +-47.5 +1.83 ± 0.18 +0.32 ± 0.02 +1.53 ± 0.07 +0.30 ± 0.01 +0.00 ± 0.00 +4230 ± 400 +I16158-5055 +1 +0.10 +5.4 +-49.2 +3.44 ± 0.34 +0.40 ± 0.02 +4.31 ± 0.69 +0.45 ± 0.04 +0.00 ± 0.00 +6500 ± 620 +I16164-5046 +1 +0.06 +5.4 +-57.3 +306.24 ± 30.72 +5.06 ± 0.25 +349.31 ± 27.77 +5.40 ± 0.21 +0.08 ± 0.01 +6970 ± 660 +I16172-5028 +1 +0.06 +5.4 +-51.9 +201.51 ± 20.19 +4.06 ± 0.20 +245.53 ± 20.05 +4.49 ± 0.18 +0.12 ± 0.01 +8400 ± 900 +I16177-5018 +1 +0.05 +5.4 +-50.2 +43.19 ± 4.35 +2.11 ± 0.11 +63.76 ± 3.59 +2.58 ± 0.07 +0.11 ± 0.02 +6610 ± 830 +I16177-5018 +2 +0.08 +5.4 +-50.2 +39.48 ± 3.95 +1.54 ± 0.08 +58.31 ± 2.59 +1.87 ± 0.04 +0.10 ± 0.02 +6910 ± 860 +I16177-5018 +3 +0.03 +5.4 +-50.2 +43.76 ± 4.41 +2.76 ± 0.14 +66.06 ± 4.10 +3.37 ± 0.10 +0.11 ± 0.02 +6660 ± 900 +I16177-5018 +4 +0.08 +5.4 +-50.2 +15.15 ± 1.52 +0.95 ± 0.05 +21.75 ± 1.98 +1.14 ± 0.05 +0.15 ± 0.02 +6490 ± 880 +I16177-5018 +5 +0.12 +5.4 +-50.2 +19.52 ± 1.95 +0.92 ± 0.05 +36.00 ± 3.64 +1.25 ± 0.06 +0.10 ± 0.02 +6790 ± 780 +I16297-4757 +1 +0.20 +4.2 +-79.6 +3.26 ± 0.33 +0.28 ± 0.01 +3.55 ± 0.35 +0.29 ± 0.01 +0.00 ± 0.00 +6010 ± 570 +I16304-4710 +1 +0.27 +4.9 +-62.8 +5.95 ± 0.60 +0.33 ± 0.02 +6.72 ± 0.53 +0.35 ± 0.01 +0.00 ± 0.00 +6280 ± 600 +I16313-4729* +1 +0.06 +4.4 +-73.7 +8.70 ± 0.88 +0.88 ± 0.04 +13.21 ± 0.99 +1.09 ± 0.04 +0.00 ± 0.00 +7660 ± 730 +I16318-4724 +1 +0.19 +3.3 +-119.8 +3.84 ± 0.39 +0.32 ± 0.02 +4.18 ± 0.43 +0.33 ± 0.02 +0.00 ± 0.00 +5960 ± 620 +I16330-4725 +1 +0.18 +4.6 +-75.1 +22.73 ± 2.28 +0.80 ± 0.04 +32.39 ± 2.41 +0.95 ± 0.04 +0.00 ± 0.00 +6820 ± 650 +I16330-4725 +2 +0.21 +4.6 +-75.1 +11.65 ± 1.17 +0.53 ± 0.03 +18.56 ± 1.53 +0.66 ± 0.03 +0.00 ± 0.00 +7120 ± 750 +I16348-4654* +1 +0.15 +5.4 +-46.5 +40.44 ± 4.06 +1.18 ± 0.06 +63.09 ± 4.68 +1.47 ± 0.05 +0.00 ± 0.00 +8220 ± 780 +I16351-4722* +1 +0.03 +5.7 +-41.4 +24.24 ± 2.44 +2.01 ± 0.10 +49.37 ± 3.28 +2.87 ± 0.10 +0.00 ± 0.00 +8070 ± 770 +I16385-4619 +1 +0.13 +3.1 +-117.0 +10.32 ± 1.03 +0.62 ± 0.03 +11.15 ± 0.81 +0.65 ± 0.02 +0.00 ± 0.00 +5740 ± 540 +I16445-4459 +1 +0.14 +2.8 +-121.3 +6.82 ± 0.68 +0.50 ± 0.02 +8.51 ± 0.60 +0.56 ± 0.02 +0.00 ± 0.00 +6210 ± 590 +I16458-4512* +1 +0.05 +5.1 +-50.4 +20.52 ± 2.06 +1.47 ± 0.07 +21.07 ± 1.71 +1.50 ± 0.06 +0.00 ± 0.00 +6070 ± 580 +I16506-4512 +1 +0.21 +6.1 +-26.2 +3.54 ± 0.35 +0.29 ± 0.01 +4.06 ± 0.53 +0.31 ± 0.02 +0.00 ± 0.00 +6670 ± 630 +MNRAS 000, 1–13 (2015) + +ATOMS-XIV: H�� regions +5 +Table 1 – continued +IRAS +ID +'e� +RGC +vlsr +EM +n4 +EM +n4 +Y+ +Temperature +(pc) +(kpc) +(km/s) +(106 cm�6 pc) +(104 cm�3) +(106 cm�6 pc) +(104 cm�3) +(K) +RRL +RRL +3mm +3mm +I17006-4215 +1 +0.08 +6.3 +-23.2 +14.15 ± 1.42 +0.94 ± 0.05 +13.66 ± 1.38 +0.92 ± 0.05 +0.10 ± 0.01 +6120 ± 640 +I17016-4124* +1 +0.02 +7.0 +-27.1 +43.56 ± 4.37 +3.58 ± 0.18 +74.99 ± 6.22 +4.70 ± 0.19 +0.00 ± 0.00 +8010 ± 680 +I17136-3617 +1 +0.04 +7.0 +-10.6 +41.83 ± 4.19 +2.34 ± 0.12 +51.71 ± 3.94 +2.61 ± 0.10 +0.08 ± 0.01 +6880 ± 650 +I17143-3700 +1 +0.22 +4.7 +-31.1 +11.36 ± 1.14 +0.51 ± 0.03 +13.10 ± 1.13 +0.55 ± 0.02 +0.13 ± 0.01 +6210 ± 710 +I17160-3707 +1 +0.62 +2.7 +-69.5 +6.06 ± 0.61 +0.22 ± 0.01 +6.12 ± 0.72 +0.22 ± 0.01 +0.00 ± 0.00 +6180 ± 590 +I17175-3544 +1 +0.04 +7.0 +-5.7 +79.67 ± 7.97 +3.26 ± 0.16 +83.59 ± 7.04 +3.32 ± 0.14 +0.00 ± 0.00 +6780 ± 580 +I17204-3636 +1 +0.06 +5.1 +-18.2 +6.51 ± 0.65 +0.76 ± 0.04 +9.42 ± 0.79 +0.92 ± 0.04 +0.00 ± 0.00 +6410 ± 670 +I17220-3609 +1 +0.13 +1.3 +-93.7 +63.37 ± 6.35 +1.57 ± 0.08 +76.06 ± 6.50 +1.72 ± 0.07 +0.11 ± 0.01 +6720 ± 700 +I17233-3606 +1 +0.03 +7.0 +-2.7 +14.13 ± 1.42 +1.63 ± 0.08 +12.85 ± 0.91 +1.57 ± 0.06 +0.00 ± 0.00 +5060 ± 430 +I17244-3536 +1 +0.03 +7.0 +-10.2 +5.54 ± 0.55 +0.93 ± 0.05 +5.87 ± 0.53 +0.96 ± 0.04 +0.00 ± 0.00 +6610 ± 560 +I17258-3637 +1 +0.13 +5.8 +-11.9 +71.44 ± 7.15 +1.63 ± 0.08 +83.80 ± 7.84 +1.77 ± 0.08 +0.12 ± 0.01 +6720 ± 700 +I17271-3439 +1 +0.06 +5.3 +-18.2 +28.63 ± 2.87 +1.57 ± 0.08 +32.06 ± 2.67 +1.66 ± 0.07 +0.00 ± 0.00 +6380 ± 480 +I17441-2822* +1 +0.08 +0.2 +50.8 +759.17 ± 76.67 +6.95 ± 0.35 +1613.45 ± 118.76 +10.11 ± 0.37 +0.07 ± 0.01 +12110 ± 1330 +I17441-2822* +2 +0.10 +0.2 +50.8 +333.11 ± 33.60 +4.03 ± 0.20 +439.87 ± 31.63 +4.62 ± 0.17 +0.07 ± 0.01 +6850 ± 790 +I17455-2800 +1 +0.55 +1.7 +-15.6 +4.84 ± 0.48 +0.21 ± 0.01 +5.00 ± 0.69 +0.21 ± 0.01 +0.21 ± 0.03 +5610 ± 870 +I17545-2357* +1 +0.04 +5.4 +7.9 +15.64 ± 1.57 +1.47 ± 0.07 +16.69 ± 1.15 +1.52 ± 0.05 +0.00 ± 0.00 +6220 ± 530 +I17599-2148 +1 +0.05 +5.4 +18.6 +6.26 ± 0.63 +0.76 ± 0.04 +10.53 ± 0.94 +0.99 ± 0.04 +0.00 ± 0.00 +7450 ± 780 +I17599-2148 +2 +0.06 +5.4 +18.6 +10.55 ± 1.06 +0.93 ± 0.05 +17.26 ± 1.47 +1.19 ± 0.05 +0.00 ± 0.00 +7210 ± 760 +I17599-2148 +3 +0.06 +5.4 +18.6 +9.85 ± 0.99 +0.94 ± 0.05 +15.09 ± 1.40 +1.16 ± 0.05 +0.00 ± 0.00 +6660 ± 700 +I18032-2032* +1 +0.05 +3.4 +4.3 +16.17 ± 1.63 +1.23 ± 0.06 +24.12 ± 1.88 +1.51 ± 0.06 +0.00 ± 0.00 +6640 ± 560 +I18110-1854 +1 +0.10 +5.1 +37.0 +20.15 ± 2.02 +0.99 ± 0.05 +20.36 ± 2.02 +1.00 ± 0.05 +0.10 ± 0.01 +6350 ± 670 +I18116-1646 +1 +0.28 +4.6 +48.5 +5.32 ± 0.53 +0.31 ± 0.02 +6.14 ± 0.65 +0.33 ± 0.02 +0.00 ± 0.00 +6360 ± 540 +I18139-1842 +1 +0.06 +5.4 +39.8 +4.00 ± 0.40 +0.56 ± 0.03 +4.83 ± 0.34 +0.61 ± 0.02 +0.00 ± 0.00 +5900 ± 500 +I18228-1312 +1 +0.07 +5.4 +32.3 +9.31 ± 0.93 +0.83 ± 0.04 +11.98 ± 0.85 +0.94 ± 0.03 +0.00 ± 0.00 +6150 ± 580 +I18311-0809 +1 +0.16 +3.7 +113.0 +5.73 ± 0.57 +0.43 ± 0.02 +6.67 ± 0.62 +0.46 ± 0.02 +0.00 ± 0.00 +6280 ± 600 +I18314-0720 +1 +0.52 +3.9 +101.5 +1.96 ± 0.20 +0.14 ± 0.01 +1.19 ± 0.20 +0.11 ± 0.01 +0.00 ± 0.00 +5400 ± 460 +I18317-0757 +1 +0.25 +4.4 +80.7 +7.70 ± 0.77 +0.39 ± 0.02 +7.24 ± 0.71 +0.38 ± 0.02 +0.00 ± 0.00 +5600 ± 480 +I18434-0242 +1 +0.25 +4.7 +97.2 +20.78 ± 2.08 +0.65 ± 0.03 +20.69 ± 2.18 +0.65 ± 0.03 +0.13 ± 0.01 +6250 ± 720 +I18479-0005 +1 +0.16 +7.5 +14.6 +145.15 ± 14.58 +2.13 ± 0.11 +215.65 ± 18.35 +2.59 ± 0.11 +0.09 ± 0.01 +7720 ± 890 +I18479-0005 +2 +0.13 +7.5 +14.6 +70.49 ± 7.12 +1.65 ± 0.08 +117.27 ± 6.97 +2.12 ± 0.06 +0.11 ± 0.01 +7530 ± 940 +I18479-0005 +3 +0.19 +7.5 +14.6 +39.86 ± 4.00 +1.02 ± 0.05 +86.43 ± 8.12 +1.51 ± 0.07 +0.12 ± 0.02 +7580 ± 1020 +I18507p0110 +1 +0.02 +7.1 +57.2 +429.07 ± 42.99 +9.55 ± 0.48 +597.33 ± 50.43 +11.16 ± 0.47 +0.00 ± 0.00 +9270 ± 1080 +I18530p0215 +1 +0.12 +5.3 +74.1 +7.33 ± 0.73 +0.56 ± 0.03 +7.64 ± 0.68 +0.58 ± 0.03 +0.00 ± 0.00 +5560 ± 580 +I19078p0901 +1 +0.17 +7.6 +2.9 +252.93 ± 25.40 +2.71 ± 0.14 +440.42 ± 37.65 +3.58 ± 0.15 +0.09 ± 0.01 +10590 ± 1200 +I19078p0901 +2 +0.10 +7.6 +2.9 +236.05 ± 23.87 +3.41 ± 0.17 +473.53 ± 35.59 +4.84 ± 0.18 +0.08 ± 0.01 +9820 ± 1270 +I19078p0901 +3 +0.09 +7.6 +2.9 +188.82 ± 19.13 +3.32 ± 0.17 +343.64 ± 28.59 +4.47 ± 0.19 +0.08 ± 0.01 +9340 ± 990 +I19078p0901 +4 +0.14 +7.6 +2.9 +65.85 ± 6.61 +1.56 ± 0.08 +109.36 ± 9.88 +2.01 ± 0.09 +0.08 ± 0.01 +7880 ± 750 +I19095p0930 +1 +0.07 +5.8 +43.7 +17.60 ± 1.77 +1.10 ± 0.06 +36.83 ± 2.43 +1.59 ± 0.05 +0.00 ± 0.00 +9930 ± 840 +I19097p0847 +1 +0.28 +6.2 +58.0 +2.64 ± 0.26 +0.22 ± 0.01 +2.50 ± 0.26 +0.21 ± 0.01 +0.00 ± 0.00 +5280 ± 500 +The ⇤ symbol after the IRAS name indicates that the sources are unresolved. The uncertainties of EM, n4, Y+ and temperature are calculated by the law of +propagation of uncertainties from measured line intensities listed in Table C1. +where n is the e�ciency for producing H40U photons per recombi- +nation. Following Zhu et al. (2019), n = 1.99 ⇥ 10�32 at )4 = 104 K +and =4 = 104 cm�3, which are the typical temperature and density +of H�� regions. The variations of n is about 10 per cent for densities +as low as 100 cm3 and temperatures as low as 5000 K (Zhang et al. +2022). For the sources with detections of H40U, we can derive n4 in +unit of cm�3: +=4 = 3.582 + +!(H40U) +Jy km/s kpc2 +�0.5  'e� +pc +��1.5 +(2) +where !a(H40U) = !(H40U)a/2 is the line luminosity in obser- +vational units (Jy km s�1 kpc2). We assume an H�� region in our +sample to be a homogeneous medium with a thickness of 2Re�, then +the relation between n4 and EM is: +⇢" = 2'e�=4=? +(3) +⇢" = 25.66 +!(H40U) +Jy km/s kpc2 + 'e� +pc +��2 +(4) +where EM in unit cm�6 pc. +The median, mean and standard deviation of EM are 1.5⇥107, +5.7⇥107 and 1.1⇥108 cm�6 pc, respectively. The median, mean and +standard deviation of n4 are 9.2⇥103, 1.4⇥104 and 1.5⇥104 cm�3, +respectively. The EM and n4 are listed in the sixth and seventh column +of Table 1. +3.2.2 Ionic Abundance Ratio +Here we present measurements of the ionized helium-to-hydrogen +ratio (Y+ = =He+/=H+) toward UC H�� regions. We derive Y+ within +MNRAS 000, 1–13 (2015) + +6 +Chao Zhang +Figure 2. The correlation between distance and 'e�. The green line shows +the linear fitting. +UC H�� regions pixel-by-pixel using (Luisi et al. 2016; Roshi et al. +2017) : +.+ = +Ø +�He+3a +Ø +�H+3a +(5) +where +Ø +�He+3a and +Ø +�H+3a are the integrated intensity in +Jy beam�1 km s�1. Here we assume both helium and hydrogen +RRLs are in the LTE condition. The right panel of Figure 1 shows +the Y+ map of two exemplar sources. The Y+ distribution within the +UC H�� regions is not uniform. Y+ appears to higher in the boundary +region than in the center. More disscussion on Y+ maps are presented +in Section 4. +There are 44 sources showing He40U emission in our paper. The +source-averaged Y+ values of these 44 UC H�� regions are summa- +rized in the tenth column of Table 1. The left panel of Figure 3 +shows the distributions of source-averaged Y+. The median, mean +and standard deviation of source-averaged Y+ are 0.11, 0.11 and +0.03, respectively. +3.2.3 Electron temperature +As demonstrated in Keto et al. (2008), the observed 3 mm continuum +emission of UC H�� regions in high resolution interferometer obser- +vations should be mostly free-free emission rather than from dust. In +Figure 4, we compare the intensity maps of H40U and H13CO+ for +an example source, I09002-4732. From this figure, one can see that +the UC H�� region where H40U is bright shows very weak or even no +H13CO+ molecular line emission, indicating that the UC H�� region +contains negligible cold thermal dust radiation. Therefore, its 3 mm +continuum emission could mainly come from free-free emission. The +other UC H�� regions in our sample are similar to I09002-4732 in that +regard. However, we cannot rule out the possibility of the existence +of some warm/hot dust emission inside UC H�� regions that cannot +be traced by H13CO+ line emission. +There is the another way to show if 3 mm continuum emission +should be mostly free-free emission. The amount of continuum flux +in excess of the flux predicted can be used to calculate the mass of +dust causing the excess. This mass is given by (Pratap et al. 1992) +"dust = 1.91 ⇥ 10�2( _<< +0.2 )V+3(a[4G?( 14.4 +_<<)3 +)]32 +kpc"� +(6) +where Sa is the excess flux due to emission from the dust, V +is the dust emissivity (assumed = 1 Pratap et al. 1992), T3 is the +dust temperature, and d: ?2 is the distance. For the exemplar source, +I09002-4732, its observed flux densities (Sa) is 13.3 Jy and the +T3 and dkpc are 39 K and 1.2 kpc (Liu et al. 2020a), respectively. +If there are 10% 3mm continuum emission come from dust, the +corresponding gas mass derived from dust emission is ⇠ 250 M�, +comparable to that of its natal clump (⇠251 M� Liu et al. 2020a). +This also indicates that only a negligible fraction (less than 10%) of +3mm continuum emission from the UC H�� region is from thermal +dust radiation. The other UC H�� regions in our sample are similar +to I09002-4732 in that regard. We note that future higher frequency +continuum observations can help constrain the dust emission within +UC H�� regions in a better way. +Assuming that the 3 mm continuum emission is mainly from free- +free emission, the electron temperatures derived from the intensities +of the H40U RRL emission and the 3 mm continuum emission, using +the equations given in Appendix A in non-LTE conditions, are listed +in the last column of Table 1. The middle panel of Figure 3 shows the +electron temperature distributions of UC H�� regions. The median, +mean and standard deviation of electron temperature are 6662, 7026 +and 1348 K, respectively. The values of the electron temperatures +are typical of H�� regions, but somehow on the cool side for what’s +expected of UC H�� regions, indicating that 3 mm continuum emission +is mainly from free-free emission is reasonable. +3.2.4 Electron density derived from 3 mm radio continuum +With electron temperature information, electron density can also be +derived from 3 mm continuum emission with equations in Appendix +B. The electron densities derived from 3 mm continuum are listed in +the eighth column of Table 1. +The left panel in Figure 5 shows the correlation between EM (cal- +culated by 3mm continuum emission) and n4 (calculated by RRLs). +The two parameters n4 and EM are strongly correlated. The slope of +linear fitting is 1.55 ± 0.05. The correlation coe�cient is 0.94 with p- +value ⌧ 0.001. The right panel shows the correlation between n4 and +H�� region diameter. The black crosses represent the data from our +paper. The gray markers are the data from Garay & Lizano (1999), +which show similar trends as our data. The correlation coe�cient is +-0.63 with p-value ⌧ 0.001. Kim et al. (2017) also found a strong +negative correlation between n4 and H�� region diameter. +We present a comparison of the n4 derived independently from +the 3 mm radio continuum and H40U emission in Figure 6. There +is a strong correlation between these two measurements with a cor- +relation coe�cient of 0.97 and p-value ⌧ 0.001. The xy-bisector fit +to these data gives a slope of 0.95 ± 0.01, indicating that millimeter +RRLs are good probes of electron densities. +3.2.5 Line widths of RRL H40U +We calculate line widths (�+) of H40U from the Gaussian fit of +the source-averaged spectral line. The right panel in Figure 3 shows +the distribution of �+. The mean, median and standard deviation of +�+ are 28.06, 30.41 and 8.90 km/s, respectively. The �+ is weakly +correlated with Re�. The correlation coe�cient is -0.25 with p-value +⌧ 0.001. It indicates that the H40U line width decreases as the Re� +MNRAS 000, 1–13 (2015) + +ATOMS-XIV: H�� regions +7 +Figure 3. Left panel: The distribution of Y+ which calculated by Equation 5. Middle panel: The distribution of Temperature which is calculated from both 3mm +continuum emission and H40U emission. Right panel: The distribution of H40U emission line width. All the sources detected in the H40U line are used in the +middel and right panels, and the sources in the left panel are detected in both H40U and He40U lines. +Figure 4. The left and right panel are the intensity of H40U and H13CO+, +respectively. The contours in right panel is H40U. Countours are from 20% +to 80% in step of 20% of peak value. +increases. Liu et al. (2021) found similar results and proposed non- +thermal (e.g., dynamical) mechanisms dominate the line broadening +of H40U on small scales of UC H�� regions. +4 DISCUSSION +4.1 Variance of ionic abundance ratio (Y+) within UC H�� +regions +Previous studies such as Luisi et al. (2016) and Roshi et al. (2017) +found Y+ below 0.1 (around 0.08 and 0.06) inside H�� regions. While +the mean value of Y+ in our example is 0.11. Luisi et al. (2016) and +Roshi et al. (2017) also witnessed a decreasing trend in Y+ with in- +creasing distance from UC H�� region beyond the photodissociation +region. However, as shown in the right upper panel of Figure 1, we +found that Y+ within the resolved UC H�� region I09002-4732 and +I15567-5236 increase with increasing distance from the center of +UC H�� region. The other resolved UC H�� regions in our sample also +exhibit similar spatial variance in Y+ similar to the example sources +I09002-4732 and I15567-5236. Figure 7 shows how the Y+ of our +data changes over the distance from the center. We pick the 80%, +60%, 40%, 30%, 20% of the peak emission of H40U in H�� regions +as the dividing lines, and divide each source into five regions. For +most sources, their boundary regions clearly show higher Y+ than +their inner regions. The right panel of Figure 7 shows the distribu- +tion of Y+ within 80%-100% (black) and 20%-40% (red) regions, +respectively. The two distributions are significantly di�erent with a +p-value of ⌧ 0.001 from Kolmogorov-Smirnov test. This figure fur- +ther demonstrates that outer parts of UC H�� regions show higher +Y+. +Is the higher Y+ near the edges of UC H�� regions caused by +low signal-to-noise (S/N) He40U spectra? He40U emission shown in +Figure 1, however, is pretty strong with S/N ratios larger than 5 at +the boundary of the UC H�� region. To further check the data quality +of He40U emission, we plot the H40U and He40U spectra near the +boundary (marked by red box) and central region (marked by black +box) in Figure 8. The spectra of RRLs H40U and He40U are clearly +detected in the boundary region with high S/N. Therefore, we argue +that the large Y+ values near the edges of UC H�� regions are not +caused by noise in data. +The possible explanations for the increasing trend of Y+ near the +boundaries of UC H�� regions could be: (1)The He in the central +region has the secondary ionization which lead to a lower density +of singly ionized He and weaker He40U line emission. However, +the HeII64U line, which is produced by the He+ ion recombined +from He2+ and free electron, is too weak to be detected in our +data. (2) He40U emission which frequency is 99.063305 GHz and +typical width around 30 km/s is contaminated by C40U emission, +whose frequency is 99.07236032 GHz. Near the photodissociation +region (PDR), CO is dissociated and ionized, and produces Carbon +RRLs.The contamination of C40U emission in He40U could result +in “stronger” observed He40U emission near PDR or the boundary +of UC H�� region. +The variance of Y+ should be further studied through future higher +sensitivity and higher resolution observations with more transitions +of RRLs. +4.2 Line broadening mechenisms for RRLs +The line widths (�+) of RRLs could be caused by thermal and +microturbulent broadening as well as electron pressure broadening +(Brocklehurst & Seaton 1972). +In Figure 9, we investigate the correlation between the line width +(�+) of H40U and electron temperature (T4). A clear increasing +trend is seen in the relation between �+ and )4. It indicates that +higher electron temperature corresponds to larger line width. The +correlation coe�cient is 0.49 with p-value ⌧ 0.001. +We calculate the thermal and electron pressure broadening follow- +MNRAS 000, 1–13 (2015) + +8 +Chao Zhang +Figure 5. Left panel: The distribution of EM which calculated by 3mm continuum emission and n4 which calculated by RRL. Right panel: The correlation +between n4 which calculated by RRL and H�� region diameter. The black crosses represent the data from our paper. The gray markers and the line are the data +from Garay & Lizano (1999) and the corresponding linear fit, respectively. +Figure 6. n4 drived from 3 mm radio continuum emission versus n4 from +H40U. The black dash line is y=x. The slope of linear fitting is 0.95 which is +shown by black line. +ing Peters et al. (2012). The left panel in Figure 10 shows the e�ect of +thermal and pressure broadening on the spectral line widths of H40U +at di�erent electron density (n4) and electron temperature (T4). For +the UC H�� regions in our ATOMS sources, n4 are between 103 and +105 cm�3, and thus the pressure broadening has no significant e�ect +on the spectral line widths. +The +thermal +broadening, +�+th, +is +given +by +�+th += +(8;=2:B)4/ 8 kpc. +5 CONCLUSION +In this paper, we have studied 146 ATOMS sources using RRLs +H40U and He40U as well as the 3 mm continuum emission. We +derived basic parameters for these UC H�� regions. Our main results +are summarized as follows: +(i) We extracted 94 and 44 compact sources from H40U and +He40U, respectively. These compact sources are resolved UC H�� +regions. +(ii) Within UC H�� regions, the observed ionic abundance ratio +Y+ seems to be higher in the boundary region than in the center. The +reason of this phenomenon should be further studied through fu- +ture higher sensitivity and higher resolution observations with more +transitions of RRLs. +(iii) Pressure broadening has no significant e�ect on the line +widths of H40U. The microturbulent broadening decreasing with +increasing Re� indicate that dynamical mechanisms may contribute +more to the line broadening for smaller (Re� <0.1 pc) UC H�� regions. +(iv) The electron temperatures of UC H�� regions in our sample +seem to weakly depend on Galactocentric distance (RGC). +ACKNOWLEDGEMENTS +This +paper +makes +use +of +the +following +ALMA +data: +ADS/JAO.ALMA#2019.1.00685.S. ALMA is a partnership of ESO +(representing its member states), NSF (USA), and NINS (Japan), to- +gether with NRC (Canada), MOST and ASIAA (Taiwan), and KASI +(Republic of Korea), in cooperation with the Republic of Chile. The +Joint ALMA Observatory is operated by ESO, AUI/NRAO, and +NAOJ. +This work has been supported by the National Key R&D Pro- +gram of China (No. 2022YFA1603100). Tie Liu acknowledges +the supports by National Natural Science Foundation of China +(NSFC) through grants No.12073061 and No.12122307, the in- +ternational partnership program of Chinese Academy of Sciences +through grant No.114231KYSB20200009, Shanghai Pujiang Pro- +gram 20PJ1415500 and the science research grants from the China +Manned Space Project with no. CMS-CSST-2021-B06. Fengyao Zhu +acknowledges the supports by National Natural Science Foundation +of China (NSFC) through grants No.12003055 and Key Research +Project of Zhejiang Lab (No. 2021PE0AC03). H.-L. Liu is supported +by National Natural Science Foundation of China (NSFC) through +the grant No.12103045. G.G. and L.B. gratefully acknowledge sup- +port by the ANID BASAL projects ACE210002 and FB210003. +K.T. was supported by JSPS KAKENHI (Grant Number 20H05645). +L.B. gratefully acknowledges support by the ANID BASAL projects +ACE210002 and FB210003 +DATA AVAILABILITY +The rawdata are available in ALMA archive. The derived data un- +derlying this article are available in the article and in its online +supplementary material. +MNRAS 000, 1–13 (2015) + +10 +Chao Zhang +H40α +He40α +Y+ = 0.14 +Y+ = 0.098 +area = 0.39±0.019 Jy km/s +area = 1.27±0.019 Jy km/s +area = 12.9±0.22 Jy km/s +area = 2.8±0.22 Jy km/s +(a) +area=5.85±0.008 +area=2.49±0.008 +area=0.30±0.01 +area=0.38±0.01 +Y+ = 0.05 +Y+ = 0.15 +H40α +He40α +(b) +Figure 8. (a) The central panel is the Y+ map of source I15567-5236. (b) The central panel is the Y+ map of source I09002-4732. The black contour represents +the intensity of H40U. Countours are from 20% to 80% in step of 20% of peak value. The red and black box indicate the boundary and central area of source, +respectively. The spectrums in the left and right show the H40U and He40U. The Y+ of the boundary and centre are 0.14 and 0.098. +MNRAS 000, 1–13 (2015) + +A(0)= +0.44 ±0.27 +△x(0)= +-112 ± +12 +0.5 +g(0)= +17 ± +12 +0.4 +Flux (Jy/beam) +0.3 +0.2 +0.1 +-140 +-120 +-100 +-80 +Velocitv (km / s)A(0)= +0.12 ± +0.31 +0.150 +△x(0)= +-118 ± +38 +g(0)= +13 ± +38 +0.125 +am) +0.100 +0.075 +0.050 +0.025 +0.000 +-140 +-120 +-100 +-80 +Velocitv (km/ s)A(0)= 0.043 ± +0.27 +Ax(0)= -1.1E+2 ± 1.3E+2 +g(0)= +17 ± 1.3E+2 +0.05 +m +0.04 +Flux (Jy/bean +0.03 +0.02 +0.01 +-140 +-120 +-100 +-80 +Velocitv (km/ s)A(0)= 0.017 ± 0.32 +△x(0)= -1.2E+2 ± 2.7E+2 +g(0)= +12 ± 2.7E+2 +0.020 +m +0.015 +Jy/bear +0.010 +Flux +0.005 +0.000 +-140 +-120 +-100 +-80 +Velocitv (km/ s)0.1 +0.2 +0.3 +lonic Abundance Ratio (Y +)0.1 +0.2 +0.3 +lonic Abundance Ratio (Y+)0.30 +0.25 +0.20 +0.15 +xni +0.10 +0.05 +0.00 +-40 +-60 +-20 +0 +20 +40 +60 +80 +Velocitv (km / s)0.10 +0.08 +< (Jy/beam) +0.06 +0.04 +0.02 +0.00 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Velocitv (km / s)0.020 +0.015 +< (Jy/beam) +0.010 +0.005 +0.000 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Velocitv (km / s)0.03 +Flux (Jy/beam) +0.02 +0.01 +0.00 +-60 +-40 +-20 +0 +20 +40 +60 +80 +Velocitv (km / s)ATOMS-XIV: H�� regions +11 +Figure 9. 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This free-free radiation causes a continuum opac- +ity at frequency a with the Rayleigh-Jeans approximation as below +(Gordon & Sorochenko 2002). +^a,2 = 9.77 ⇥ 10�3 #4#8 +a2)3/2 [17.72 + ;=)3/2 +4 +a +] +(A1) +where the numerical version of this equation also in CGS units, which +gives ^2 in units of cm�1. The N4 and N8 is the electron and ion +number densities, respectively. Likewise, the plasma emits thermal +radiation in LTE conditions with an emissivity of +9a,2 = ⌫a())^a,2 +(A2) +where ⌫a()) denotes the intensity of a blackbody of temperature T +at frequency a. +For the hydrogen recombination lines from upper state m to low +state n. The corresponding line absorption coe�cient is +^a,! = ⌘a +4c qa(#=⌫=,< � #<⌫<,=) +(A3) +with the Planck constant h, the Einstein coe�cients B=,< and B<,= +for absorption and stimulated emission, respectively. The line profile +function including the thermal, turbulent and pressure broadenings +is provided by qa (Peters et al. 2012). And #: = 1: #!) ⇢ +: +is the +number densities of hydrogen atoms in state k (for k = m, n) with the +departure coe�cient 1: (Zhu et al. 2019, 2022). The line emissivity +is +9a,! = ⌘a +4c qa#<�<,= +(A4) +�<,= is the Einstein coe�cient for spontaneous emission. The de- +tail method of calculating �<,= and B=,< is given by Brocklehurst +(1971). We assume an H�� region in our sample to be a homogeneous +medium of thickness D= 2R. The total optical depth is given by +ga = ga,2 + ga,! = (^a,2 + ^a,!)⇡ +(A5) +Then, the source function is written as below (Peters et al. 2012) +(a = 9a,2 + 9a,! +^a,2 + ^a,! +. +(A6) +The total intensity at frequency a is +�a = (a(1 � 4�ga) +. +(A7) +The continuum intensity �a,! and the line intensity �a,2 are calcu- +lated as below +�a,⇠ = ⌫a())(1 � 4�ga,2) +, +(A8) +and +�a,! = (a(1 � 4�ga) � ⌫a())(1 � 4�ga,2) +. +(A9) +MNRAS 000, 1–13 (2015) + +12000 +11000 +10000 +Temperature +9000 +8000 +7000 +6000 +5000 +correlation coefficient : 0.49 +p-value << 0.001 +4000 +20 +25 +30 +35 +40 +45 +50 +55 +60 +△V (km/s)12 +Chao Zhang +Figure 10. Left panel: the e�ect of thermal and pressure broadening on the spectral line widths at di�erent n4 and temperature. Right panel: the correlation +between �+tur and Re�. +Figure 11. Electron temperatures versus 'GC. The yellow dots are the samples of our paper. The red solid line shows the linear fit to the data. The gray cross +dots and dashed line are the data from Balser et al. (2011). Left: The grey dots are the data from Paladini et al. (2004). The solid grey line shows least-squares +linear relationship (Equation 8) that found by Paladini et al. (2004). The dot dashed line that found by Shaver et al. (1983). Right: The blue triangles are the data +from A�erbach et al. (1996). +In our sample of H�� regions, there is no evidence for strong maser +of hydrogen recombination line. So the observed line profile should +be similar to the line profile function. Because the line and continuum +intensities are functions of )4, =4, qa and D, the values of electron +temperature and density can be estimated from the observed thermal- +emission line and continuum intensities when the thickness of an H�� +region and the line profile function are known. +The method of estimating T4 and n4 in H�� regions is similar with +that in Zhu et al. (2022). However, the observable parameters are +simulated values in that work, but are measured values in this work. +First, the continuum and frequency-integrated H40U line intensities +are measured in the observations. Second, a group of two random +numbers are generated from the normal distribution based on the +two measured intensities. Third, the simulated line and continuum +intensities are calculated for the given values of T4 and n4 by using +Eq. A8 and A9. Y+ is used to estimate the number ratio of He+ to +H+. We adjust the values of T4 and n4 until the simulated values +fit the random values generated from the normal distribution. The +least-squares method is used in the fitting process. The best fit values +of T4 and n4 are the estimates. After a large number of groups of +two random numbers are generated and fitted, the distributions of the +estimated T4 and n4 are created. The mean values of the distributions +are written as the estimates of T4 and n4. +MNRAS 000, 1–13 (2015) + +40 +T =5000 K +e +T =10000 K +35 +e +T =15000K +FWHM of H40α line [km s' +30 +25 +20 +15 +10 +102 +103 +104 +105 +106 +107 +10860 +correlation coefficient: -0.22 +p-value << 0.001 +50 +40 +△Vtur (km/s) +30 +20 +10 +0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Reff (pc)1.4×104 +1.2×10 +. +1.0×104 +8 +人。 +8 +6.0x10 +p D +2.0×10 +0 +0 +2 +4 +6 +8 +10 +12 +14 +R (kpc)ATOMS-XIV: H�� regions +13 +APPENDIX B: DERIVATION OF ELECTRON DENSITY +USING 3MM CONTINUUM +Electron density can also be estimated from the 3 mm continuum +emission. The intensity of continuum emission is computed by +�a,⇠ = ⌫a()) ga,2 = ⌫a()) ^a,2⇡ +(B1) +All quantities are in cgs units. Combining Equation B1 and A1 that +assume a Rayleigh-Jeans approximation, we get +�a,⇠ = 2.931 ⇥ 10�39)�0.5[17.72 + ;=)3/2 +4 +a +]⇢" +(B2) +where ⇢" = #4#8⇡ and the numerical version of this equation also +in CGS units. Then switching units, we get +⇢" = +�a,2 +879.3 )�0.5[17.72 + ;=) 3/2 +4 +a ] +(B3) +where EM in unit cm�6 pc, �a,2 in unit Jy, a in unit Hz After getting +the value of EM, we use =4 = +q +⇢" +⇡ +to get electron density. +APPENDIX C: OBVERVATION PARAMETERS +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2015) + +14 +Chao Zhang +Table C1. The derived parameters of RRLs and 3mm continuum. +IRAS +ID +Peak Intensity +vlsr +�+H40U +peak intensity +vlsr +�+He40U +Intensity +H40U +H40U +H40U +He40U +He40U +He40U +3mm +(Jy beam�1) +(km/s) +(km/s) +(Jy beam�1) +(km/s) +(km/s) +(Jy) +I09002-4732 +1 +0.132 ± 0.054 +-0.3 ± 1.2 +32.0 ± 1.2 +0.012 ± 0.011 +-1.6 ± 1.2 +32.3 ± 1.2 +13.37 ± 0.01 +I11298-6155 +1 +0.012 ± 0.009 +39.5 ± 1.2 +21.1 ± 1.2 +- +- +- +0.57 ± 0.00 +I12320-6122 +1 +0.155 ± 0.070 +-41.4 ± 1.2 +39.0 ± 1.2 +0.006 ± 0.014 +-44.9 ± 1.2 +44.2 ± 1.2 +1.92 ± 0.00 +I12326-6245 +1 +0.425 ± 0.815 +-63.4 ± 1.2 +58.4 ± 1.2 +0.032 ± 0.158 +-60.5 ± 1.2 +72.6 ± 1.2 +8.26 ± 0.02 +I12383-6128 +1 +0.026 ± 0.015 +-39.2 ± 1.2 +24.7 ± 1.2 +- +- +- +0.41 ± 0.00 +I12572-6316 +1 +0.013 ± 0.012 +24.2 ± 1.2 +26.1 ± 1.2 +- +- +- +0.20 ± 0.00 +I13080-6229 +1 +0.055 ± 0.015 +-36.5 ± 1.2 +28.9 ± 1.2 +0.007 ± 0.003 +-36.8 ± 1.2 +25.2 ± 1.2 +9.12 ± 0.00 +I13111-6228 +1 +0.009 ± 0.011 +-34.9 ± 1.2 +33.5 ± 1.2 +- +- +- +0.21 ± 0.00 +I13291-6229 +1 +0.015 ± 0.014 +-36.7 ± 1.2 +26.6 ± 1.2 +- +- +- +0.42 ± 0.00 +I13291-6229 +2 +0.011 ± 0.012 +-31.7 ± 1.2 +23.9 ± 1.2 +- +- +- +0.35 ± 0.00 +I13291-6249 +1 +0.046 ± 0.018 +-38.7 ± 1.2 +34.6 ± 1.2 +0.005 ± 0.007 +-40.6 ± 1.2 +22.0 ± 1.2 +2.27 ± 0.00 +I13471-6120* +1 +0.335 ± 0.098 +-57.1 ± 1.2 +34.0 ± 1.2 +0.023 ± 0.021 +-57.8 ± 1.2 +36.4 ± 1.2 +2.90 ± 0.01 +I14013-6105 +1 +0.109 ± 0.062 +-57.9 ± 1.2 +37.0 ± 1.2 +0.010 ± 0.011 +-59.7 ± 1.2 +34.3 ± 1.2 +4.71 ± 0.01 +I14050-6056 +1 +0.039 ± 0.010 +-49.1 ± 1.2 +40.7 ± 1.2 +0.003 ± 0.007 +-47.8 ± 1.2 +43.6 ± 1.2 +3.50 ± 0.00 +I14382-6017 +1 +0.011 ± 0.004 +-58.3 ± 1.2 +30.9 ± 1.2 +- +- +- +3.09 ± 0.00 +I14453-5912 +1 +0.005 ± 0.004 +-40.8 ± 1.2 +36.7 ± 1.2 +- +- +- +1.29 ± 0.00 +I15254-5621* +1 +0.364 ± 0.191 +-70.8 ± 1.2 +47.9 ± 1.2 +0.027 ± 0.023 +-73.0 ± 1.2 +54.4 ± 1.2 +4.69 ± 0.02 +I15290-5546 +1 +0.148 ± 0.021 +-89.4 ± 1.2 +25.2 ± 1.2 +0.013 ± 0.014 +-88.9 ± 1.2 +19.3 ± 1.2 +2.26 ± 0.00 +I15290-5546 +2 +0.155 ± 0.040 +-88.4 ± 1.2 +29.0 ± 1.2 +0.013 ± 0.012 +-90.6 ± 1.2 +32.7 ± 1.2 +5.16 ± 0.01 +I15384-5348 +1 +0.029 ± 0.008 +-41.5 ± 1.2 +22.0 ± 1.2 +0.004 ± 0.006 +-40.7 ± 1.2 +22.9 ± 1.2 +4.87 ± 0.00 +I15408-5356 +1 +0.027 ± 0.004 +-42.8 ± 1.2 +23.0 ± 1.2 +0.002 ± 0.003 +-48.4 ± 1.2 +51.5 ± 1.2 +7.80 ± 0.00 +I15411-5352 +1 +0.076 ± 0.024 +-41.0 ± 1.2 +30.3 ± 1.2 +0.007 ± 0.007 +-43.4 ± 1.2 +28.8 ± 1.2 +9.75 ± 0.00 +I15439-5449 +1 +0.058 ± 0.023 +-52.5 ± 1.2 +26.6 ± 1.2 +- +- +- +1.20 ± 0.00 +I15502-5302 +1 +0.953 ± 0.344 +-94.8 ± 1.2 +35.2 ± 1.2 +0.082 ± 0.045 +-95.8 ± 1.2 +33.7 ± 1.2 +13.67 ± 0.03 +I15520-5234 +1 +0.300 ± 0.061 +-38.9 ± 1.2 +32.6 ± 1.2 +- +- +- +7.08 ± 0.01 +I15567-5236 +1 +0.230 ± 0.124 +-112.0 ± 1.2 +40.1 ± 1.2 +0.021 ± 0.019 +-113.6 ± 1.2 +39.5 ± 1.2 +8.88 ± 0.01 +I15570-5227 +1 +0.010 ± 0.007 +-103.2 ± 1.2 +33.6 ± 1.2 +- +- +- +2.11 ± 0.00 +I16037-5223* +1 +0.138 ± 0.078 +-81.9 ± 1.2 +34.4 ± 1.2 +0.010 ± 0.024 +-83.7 ± 1.2 +36.6 ± 1.2 +1.44 ± 0.00 +I16037-5223* +2 +0.094 ± 0.021 +-81.2 ± 1.2 +22.3 ± 1.2 +0.011 ± 0.020 +-81.9 ± 1.2 +15.9 ± 1.2 +1.01 ± 0.00 +I16037-5223* +3 +0.043 ± 0.025 +-78.9 ± 1.2 +27.5 ± 1.2 +0.002 ± 0.011 +-92.2 ± 1.2 +71.0 ± 1.2 +0.40 ± 0.00 +I16060-5146 +1 +1.439 ± 0.583 +-90.8 ± 1.2 +37.9 ± 1.2 +0.112 ± 0.062 +-91.3 ± 1.2 +39.2 ± 1.2 +24.95 ± 0.05 +I16065-5158 +1 +0.030 ± 0.036 +-57.0 ± 1.2 +32.6 ± 1.2 +- +- +- +2.88 ± 0.00 +I16065-5158 +2 +0.017 ± 0.049 +-49.6 ± 1.2 +35.7 ± 1.2 +- +- +- +0.75 ± 0.00 +I16071-5142 +1 +0.032 ± 0.015 +-82.1 ± 1.2 +21.9 ± 1.2 +- +- +- +1.15 ± 0.00 +I16132-5039 +1 +0.008 ± 0.007 +-48.3 ± 1.2 +18.7 ± 1.2 +- +- +- +0.25 ± 0.00 +I16158-5055 +1 +0.015 ± 0.010 +-51.7 ± 1.2 +20.1 ± 1.2 +- +- +- +0.71 ± 0.00 +I16164-5046 +1 +1.093 ± 0.529 +-65.0 ± 1.2 +30.2 ± 1.2 +0.082 ± 0.073 +-67.1 ± 1.2 +29.4 ± 1.2 +18.53 ± 0.03 +I16172-5028 +1 +0.798 ± 0.241 +-53.3 ± 1.2 +35.4 ± 1.2 +0.079 ± 0.042 +-52.1 ± 1.2 +37.2 ± 1.2 +12.87 ± 0.03 +I16177-5018 +1 +0.090 ± 0.114 +-55.7 ± 1.2 +29.0 ± 1.2 +0.010 ± 0.036 +-55.8 ± 1.2 +35.4 ± 1.2 +2.19 ± 0.00 +I16177-5018 +2 +0.124 ± 0.043 +-49.4 ± 1.2 +24.3 ± 1.2 +0.014 ± 0.034 +-49.9 ± 1.2 +22.9 ± 1.2 +5.93 ± 0.00 +I16177-5018 +3 +0.089 ± 0.059 +-51.6 ± 1.2 +26.3 ± 1.2 +0.008 ± 0.030 +-50.2 ± 1.2 +28.1 ± 1.2 +0.83 ± 0.00 +I16177-5018 +4 +0.043 ± 0.024 +-48.5 ± 1.2 +25.4 ± 1.2 +0.005 ± 0.018 +-48.8 ± 1.2 +34.1 ± 1.2 +2.30 ± 0.00 +I16177-5018 +5 +0.076 ± 0.015 +-50.1 ± 1.2 +23.1 ± 1.2 +0.009 ± 0.015 +-50.0 ± 1.2 +22.0 ± 1.2 +7.18 ± 0.00 +I16297-4757 +1 +0.010 ± 0.007 +-79.9 ± 1.2 +26.3 ± 1.2 +- +- +- +1.15 ± 0.00 +I16304-4710 +1 +0.022 ± 0.011 +-60.6 ± 1.2 +24.8 ± 1.2 +- +- +- +0.74 ± 0.00 +I16313-4729* +1 +0.023 ± 0.026 +-78.6 ± 1.2 +40.4 ± 1.2 +- +- +- +0.34 ± 0.00 +I16318-4724 +1 +0.011 ± 0.012 +-111.4 ± 1.2 +25.6 ± 1.2 +- +- +- +0.50 ± 0.00 +I16330-4725 +1 +0.088 ± 0.036 +-75.5 ± 1.2 +28.5 ± 1.2 +- +- +- +1.60 ± 0.00 +I16330-4725 +2 +0.032 ± 0.024 +-78.3 ± 1.2 +25.1 ± 1.2 +- +- +- +1.27 ± 0.00 +I16348-4654* +1 +0.158 ± 0.036 +-48.6 ± 1.2 +30.9 ± 1.2 +- +- +- +1.66 ± 0.01 +I16351-4722* +1 +0.070 ± 0.030 +-31.6 ± 1.2 +28.0 ± 1.2 +- +- +- +0.88 ± 0.00 +I16385-4619 +1 +0.048 ± 0.010 +-113.3 ± 1.2 +24.8 ± 1.2 +- +- +- +0.78 ± 0.00 +I16445-4459 +1 +0.030 ± 0.011 +-128.1 ± 1.2 +19.5 ± 1.2 +- +- +- +0.49 ± 0.00 +I16458-4512* +1 +0.089 ± 0.020 +-54.6 ± 1.2 +29.1 ± 1.2 +- +- +- +0.71 ± 0.00 +I16506-4512 +1 +0.013 ± 0.002 +-27.9 ± 1.2 +25.5 ± 1.2 +- +- +- +5.85 ± 0.00 +MNRAS 000, 1–13 (2015) + +ATOMS-XIV: H�� regions +15 +Table C1 – continued +IRAS +ID +Peak Intensity +vlsr +�+H40U +peak intensity +vlsr +�+He40U +Intensity +H40U +H40U +H40U +He40U +He40U +He40U +3mm +(Jy beam�1) +(km/s) +(km/s) +(Jy beam�1) +(km/s) +(km/s) +(Jy) +I17006-4215 +1 +0.113 ± 0.019 +-24.1 ± 1.2 +30.1 ± 1.2 +0.007 ± 0.008 +-27.1 ± 1.2 +38.7 ± 1.2 +3.46 ± 0.00 +I17016-4124* +1 +0.166 ± 0.058 +-32.2 ± 1.2 +35.5 ± 1.2 +- +- +- +2.10 ± 0.01 +I17136-3617 +1 +0.123 ± 0.073 +-6.6 ± 1.2 +35.2 ± 1.2 +0.009 ± 0.012 +-5.4 ± 1.2 +36.5 ± 1.2 +7.83 ± 0.00 +I17143-3700 +1 +0.050 ± 0.014 +-34.6 ± 1.2 +31.4 ± 1.2 +0.001 ± 0.011 +-40.0 ± 1.2 +37.1 ± 1.2 +0.76 ± 0.00 +I17160-3707 +1 +0.029 ± 0.004 +-72.2 ± 1.2 +25.2 ± 1.2 +- +- +- +4.06 ± 0.00 +I17175-3544 +1 +0.394 ± 0.103 +-4.5 ± 1.2 +26.0 ± 1.2 +- +- +- +12.71 ± 0.01 +I17204-3636 +1 +0.023 ± 0.020 +-5.5 ± 1.2 +22.1 ± 1.2 +- +- +- +0.51 ± 0.00 +I17220-3609 +1 +0.325 ± 0.041 +-92.8 ± 1.2 +26.5 ± 1.2 +0.022 ± 0.017 +-93.6 ± 1.2 +21.5 ± 1.2 +3.68 ± 0.01 +I17233-3606 +1 +0.069 ± 0.012 +-1.8 ± 1.2 +22.9 ± 1.2 +- +- +- +0.98 ± 0.00 +I17244-3536 +1 +0.020 ± 0.010 +-12.5 ± 1.2 +48.7 ± 1.2 +- +- +- +0.62 ± 0.00 +I17258-3637 +1 +0.410 ± 0.047 +-12.6 ± 1.2 +28.1 ± 1.2 +0.048 ± 0.011 +-13.3 ± 1.2 +25.3 ± 1.2 +42.49 ± 0.01 +I17271-3439 +1 +0.109 ± 0.028 +-13.8 ± 1.2 +31.2 ± 1.2 +- +- +- +2.15 ± 0.00 +I17441-2822* +1 +0.889 ± 1.443 +62.3 ± 1.2 +49.4 ± 1.2 +0.074 ± 0.151 +62.5 ± 1.2 +51.7 ± 1.2 +25.06 ± 0.04 +I17441-2822* +2 +0.507 ± 0.626 +59.7 ± 1.2 +32.7 ± 1.2 +0.039 ± 0.110 +59.4 ± 1.2 +32.9 ± 1.2 +13.41 ± 0.01 +I17455-2800 +1 +0.014 ± 0.003 +-20.9 ± 1.2 +20.8 ± 1.2 +0.003 ± 0.003 +-22.8 ± 1.2 +18.8 ± 1.2 +2.95 ± 0.00 +I17545-2357* +1 +0.079 ± 0.012 +7.7 ± 1.2 +24.5 ± 1.2 +- +- +- +0.49 ± 0.00 +I17599-2148 +1 +0.012 ± 0.022 +28.2 ± 1.2 +24.9 ± 1.2 +- +- +- +0.63 ± 0.00 +I17599-2148 +2 +0.023 ± 0.024 +21.6 ± 1.2 +32.5 ± 1.2 +- +- +- +1.34 ± 0.00 +I17599-2148 +3 +0.026 ± 0.014 +21.1 ± 1.2 +27.4 ± 1.2 +- +- +- +1.00 ± 0.00 +I18032-2032* +1 +0.054 ± 0.019 +11.5 ± 1.2 +26.9 ± 1.2 +- +- +- +0.49 ± 0.00 +I18110-1854 +1 +0.148 ± 0.015 +41.7 ± 1.2 +24.6 ± 1.2 +0.008 ± 0.006 +42.3 ± 1.2 +25.1 ± 1.2 +3.58 ± 0.00 +I18116-1646 +1 +0.019 ± 0.003 +53.0 ± 1.2 +21.2 ± 1.2 +- +- +- +6.16 ± 0.00 +I18139-1842 +1 +0.020 ± 0.007 +36.2 ± 1.2 +19.5 ± 1.2 +- +- +- +0.42 ± 0.00 +I18228-1312 +1 +0.021 ± 0.020 +29.8 ± 1.2 +31.5 ± 1.2 +- +- +- +1.04 ± 0.00 +I18311-0809 +1 +0.022 ± 0.008 +108.6 ± 1.2 +23.3 ± 1.2 +- +- +- +0.85 ± 0.00 +I18314-0720 +1 +0.006 ± 0.002 +105.8 ± 1.2 +26.0 ± 1.2 +- +- +- +1.86 ± 0.00 +I18317-0757 +1 +0.024 ± 0.008 +78.7 ± 1.2 +27.9 ± 1.2 +- +- +- +3.96 ± 0.00 +I18434-0242 +1 +0.204 ± 0.018 +99.0 ± 1.2 +26.8 ± 1.2 +0.019 ± 0.005 +97.7 ± 1.2 +26.7 ± 1.2 +9.20 ± 0.01 +I18479-0005 +1 +0.436 ± 0.126 +13.9 ± 1.2 +27.7 ± 1.2 +0.044 ± 0.088 +13.5 ± 1.2 +23.0 ± 1.2 +6.10 ± 0.01 +I18479-0005 +2 +0.129 ± 0.130 +19.4 ± 1.2 +32.5 ± 1.2 +0.014 ± 0.042 +19.3 ± 1.2 +30.1 ± 1.2 +2.18 ± 0.00 +I18479-0005 +3 +0.104 ± 0.027 +14.8 ± 1.2 +26.4 ± 1.2 +0.014 ± 0.034 +14.9 ± 1.2 +22.7 ± 1.2 +3.42 ± 0.00 +I18507+0110 +1 +1.263 ± 0.551 +49.1 ± 1.2 +54.5 ± 1.2 +- +- +- +24.76 ± 0.05 +I18530+0215 +1 +0.026 ± 0.010 +80.8 ± 1.2 +26.0 ± 1.2 +- +- +- +0.92 ± 0.00 +I19078+0901 +1 +0.582 ± 0.560 +10.4 ± 1.2 +42.8 ± 1.2 +0.054 ± 0.117 +10.1 ± 1.2 +35.0 ± 1.2 +17.86 ± 0.03 +I19078+0901 +2 +0.339 ± 0.611 +12.9 ± 1.2 +46.8 ± 1.2 +0.027 ± 0.073 +13.6 ± 1.2 +49.4 ± 1.2 +6.75 ± 0.01 +I19078+0901 +3 +0.220 ± 0.444 +13.5 ± 1.2 +52.8 ± 1.2 +0.018 ± 0.049 +24.0 ± 1.2 +79.1 ± 1.2 +3.60 ± 0.02 +I19078+0901 +4 +0.178 ± 0.083 +-2.2 ± 1.2 +34.5 ± 1.2 +0.017 ± 0.024 +-2.3 ± 1.2 +29.7 ± 1.2 +2.94 ± 0.01 +I19095+0930 +1 +0.050 ± 0.043 +64.9 ± 1.2 +50.6 ± 1.2 +- +- +- +0.93 ± 0.00 +I19097+0847 +1 +0.010 ± 0.006 +64.3 ± 1.2 +22.0 ± 1.2 +- +- +- +0.53 ± 0.00 +The ⇤ symbol after the IRAS name indicates that the sources are unresolved. The peak intensity, velocity (+lsr) and line widths (�+ ) of H40U and He40U are +derived from the Gaussian fitting of the spectra of each line. The uncertainties of �+ is derived by considering the velocity resolution and the uncertainties +given by gaussian fitting. The uncertainties of the other parameters are derived from gaussian fitting. +MNRAS 000, 1–13 (2015) + diff --git a/s9AzT4oBgHgl3EQf6f6J/content/tmp_files/load_file.txt b/s9AzT4oBgHgl3EQf6f6J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd5ed34461d6eb9a4616cef98d91a0828d84c11d --- /dev/null +++ b/s9AzT4oBgHgl3EQf6f6J/content/tmp_files/load_file.txt @@ -0,0 +1,3210 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf,len=3209 +page_content='MNRAS 000, 1–13 (2015) Preprint 4 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0 ATOMS: ALMA Three-millimeter Observations of Massive Star-forming regions -XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Properties of resolved UC H�� regions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='Zhang,1,2Feng-Yao Zhu,3¢ Tie Liu,4† Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Ren, 5 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Liu,6 Ke Wang,7 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='-W.' metadata={'source': 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+page_content='Xu,7 Chang Won Lee,13 Leonardo Bronfman,11 Archana Soam,14 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Li5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Taiyuan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Jinzhong 030619,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='China 2Institute of Computational and Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Taiyuan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Jinzhong 030619,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='China 3 Research Center for Intelligent Computing Platforms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Zhejiang Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 311100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' China 4Shanghai Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 80 Nandan Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Shanghai 200030,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' China 5National Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Beijing,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 5 Yiheyuan Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Haidian District,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' People’s Republic of China 8University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' PR China 9School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Sun Yat-sen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2 Daxue Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Zhuhai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Guangdong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 519082,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Thiruvananthapuram 695 547,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Kerala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' India 13Korea Astronomy and Space Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 776 Daedeokdaero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Yuseong-gu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Daejeon 34055,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Hydrogen recombination lines (RRLs) are one of the major diagnostics of the physical properties of H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We use RRL H40U, He40U and 3 mm continuum emission to investigate the properties of a large sample of resolved UC H�� regions identified in the ATOMS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In total, we identify 94 UC H�� regions from H40U emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The basic parameters for these UC H�� regions such as electron density, emission measure, electron temperature, ionic abundance ratio (nHe+/nH+), and line width are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The median electron density and the median nHe+/nH+ ratio of these UC H�� regions derived from RRLs are ⇠9000 cm�3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Within UC H�� regions, the nHe+/nH+ ratios derived from the intensity ratio of the He40U and H40U lines seems to be higher in the boundary region than in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The H40U line width is mainly broadened by thermal motion and microturbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The electron temperature of these UC H�� regions has a median value of ⇠6700 K, and its dependence on galactocentric distance is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Key words: ISM: clouds, (ISM:) H�� regions, stars: formation, radio lines: ISM 1 INTRODUCTION Massive stars can significantly influence their local environments through powerful feedback such as winds, radiation and supernova explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Ionizing radiation from massive stars produces pro- nounced H�� regions around them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=', Krumholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Thus, observations of H�� regions provide important insight into massive star formation (Habing & Israel 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Churchwell 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Zinnecker & Yorke 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' H�� regions in the Galaxy show low nHe+/nH+ number density ratios (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 which is the actual He/H abundance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Draine 2011) from RRL observations, and this ratio seems to decrease with ¢ E-mail: zhufy@zhejianglab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='com † E-mail: liutie@shao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='cn increasing distance from the H�� region beyond the photodissociation region (PDR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Luisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Roshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Here nHe+ is the number density of singly ionized helium (He+), and nH+ is that of ionized hydrogen (H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The observed low nHe+/nH+ number density ratios outside the PDR regions may indicate that there will be fewer photons with enough energy to ionize He compared to H there or part of radiation field is attenuated by interstellar dust (Roshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Therefore, the =He+/=H+ ratio is very sensitive to the strength of radiation field and dust properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The nHe+/nH+ ratios inside UC H�� regions, however, have not been systematically investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Electron temperatures, T4, of H�� regions were estimated from the intensity ratios between radio recombination lines (RRLs) and continuum emission on the LTE approximation in previous observa- tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Shaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Caswell & Haynes 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, Gordon & Sorochenko (2002) points out that using LTE approxi- © 2015 The Authors 2 Chao Zhang mation could cause significant deviation of estimated T4 in some conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Previous studies indicate that T4 of H�� regions tends to be positively correlated with the distance from the Galactic Center (RGC) (Churchwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Downes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Wink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Shaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Paladini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' This is because that inner Galaxy tends to have higher metallicities, resulting in greater cooling rates and lower T4 of H�� region (Shaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1983, Quireza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2006, Balser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, most of these studies were con- ducted with single-dishes, which cannot resolve distant H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' These relations need to be tested with high resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Observations of hydrogen recombination lines at radio and (sub)millimetre (submm/mm) frequencies are routinely used by ob- servers to infer densities, temperatures and velocity structures inside H�� regions (Gordon & Sorochenko 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For young Ultra-Compact (UC) H�� regions that are still embedded in their natal gas clumps and are not visible at optical and infrared wavelengths, submm/mm RRLs are among the best tracers for probing their physical proper- ties because they are much less a�ected by the pressure broadening, and brighter than the centimeter RRLs with larger principal quantum numbers (Keto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Previous statistical observations of submm/mm RRLs toward UC H�� regions were mostly conducted with single dishes (Churchwell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Nguyen-Luong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' These observations, however, cannot resolve the structures as well as gas properties of UC H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In this work, we present the first systematic investigation of the properties of a large sample of resolved UC H�� regions with high resolution (⇠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='500) RRLs H40U and He40U as well as the 3mm continuum emission that was obtained from the ATOMS (ALMA Three-millimeter Observations of Massive Star forming regions) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In particular, we focus on the measurements of nHe+/nH+ number density ratios and electron temperatures of these UC H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In section 2, we describe the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Section 3 presents the results, focusing on statistics of parameters such as electron density (n4), emission measure (EM), T4, nHe+/nH+ ratio and line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' A discussion of the results is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We summarize our main conclusions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2 OBSERVATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 The sample We made the ALMA observations for 146 active Galactic star form- ing regions through the ATOMS (ALMA Three-millimeter Obser- vations of Massive Star forming regions) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' These 146 sources were selected from the CS J = 2-1 survey of Bronfman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (1996), a complete and homogeneous molecular line survey of UC H�� re- gion candidates in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The sample of 146 targets is complete for proto-clusters with bright CS J = 2-1 emission (T1 >2 K), indicative of rather dense gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The overall properties of these 146 targets are shown in Table A of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Majority (139) of the targets are located in the first and fourth Galactic Quadrants of the inner Galactic Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The distances of the sample clouds range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 kpc to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0 kpc with a mean value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The sam- ple includes 27 distant (d>7 kpc) sources that are either close to the Galactic center or mini-starbursts, representing extreme environ- ments for star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The properties of these sources have been described in detail in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2016, 2020a,b) and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 ALMA observations The ATOMS data consist of 146 sources observed in both ALMA- 12m and ACA-7m arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The spectral setup consists of 8 spectral windows (spw), where spws 1-6 are narrow bands centered on par- ticular lines, and spws 7-8 are broad band (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='875 GHz), low spectral resolution for continuum and lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Table 2 of Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2020a sum- merizes the details of receiver setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Fully calibrated visibility data are generated by running the ALMA Pipeline, which applies calibration tables obtained from the QA2 to the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Images are made from the combined 12m and ACA vis- ibility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Continuum images are constructed from line-free chan- nels in spw 7 and 8 centered at ⇠ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 GHz (or 3 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Line images are made for each spw with native spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' More details of data reduction are described in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2022), Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2021) and Wang Ke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Uncertainties in flux calibration are estimated to be ⇠ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' All images are primary-beam corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The typical rms noise (1 sigma) for the continuum images is ⇠0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 mJy for a synthesized beam of ⇠ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In this work, we also use RRLs H40U and He40U line data with spectral resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 km s�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The typical beam size and channel rms noise level for RRLs are ⇠ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 00 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 Jy beam�1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The rest frequencies of H40U and He40U line are 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='022952 and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='063305 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 3 RESULT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 Resolved UC H�� regions identified with RRL H40U The raw data of RRL and 3mm observations are cubes and continuum images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We integrate the RRL cube along the velocity axis to get intensity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Then we extracted compact objects (UC H�� regions) from the integrated intensity maps of the H40U and He40U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Figure 1 shows the integrated intensity maps for two exemplar sources I15567-5236 and I09002-4732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In each source, the left and middle panels show the integrated intensity maps of H40U and He40U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The 3 mm continuum emission is shown as contours in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The 3 mm continuum emission coincides with the RRLs emission very well, indicating that the 3 mm continuum is dominated by free-free emission, as also suggested in Keto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The compact objects in H40U and He40U emission can be easily identified by eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In total, we detected 94 and 44 compact sources (UC H�� regions) in emission from H40U and He40U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The number of the UC H�� regions agrees with that by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Table C1 summarized the parameters for H40U, He40U and 3mm emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The majority (⇠90%) of UC H�� regions are resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In further analysis, we are neglecting unresolved H�� regions (with sizes smaller than one beam), which are labeled ⇤ after the IRAS name in Table 1 and Table C1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The majority (⇠86%) of targeted sources contain only one UC H�� region in RRLs line emission as illustrated for I09002-4732 in the bottom panel of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For the case of ‘mutiple sources’ which are very close to each other and hard to divide, like I15567-5236 in the top panel of Figure 1, we treated them as one single UC H�� region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, we identified multiple UC H�� regions in 10 sources from H40U maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We derive parameters for individual UC H�� regions and listed them in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" The e�ective radius ('e�) of each UC H�� region is defined as 'e� = p (/c, where S is the area of the UC H�� region within the 5f contours." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" 'e� are listed in the third column of Table 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" The median 'e� is ⇠0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" For resolved ob- jects, the uncertainties of 'e� are mainly determined by distances." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" The uncertainties of 'e� caused by measured area (S) are negligible." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' MNRAS 000, 1–13 (2015) ATOMS-XIV: H�� regions 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The source of I15567-5236 (top) and I09002-4732(bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The value of di�erent colors show in the colorbar which in the top of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For each source, the left and middle panels are H40U and He40U intensity map over 5f, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The 3 mm continuum emission is shown in contours in intensity map of H40U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Contours are from 20% to 80% in step of 20% of peak values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The right panel shows the ionic abundance ration of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The black ellipses represent the synthesized beam sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, the uncertainties of distances are not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" There- fore, we did not consider the uncertainties of 'e�." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We note that this should a�ect the error analysis of some derived parameteres (such as emission measure and electron density) in following analysis, which can be better dealt with from more accurate distance measurements in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" Figure 2 presents the correlation between distance and 'e� of the sample." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" There is a noticeable increasing trend between the dis- tance and 'e�." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The correlation coe�cient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='45 with p-value ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" The increasing trend indicates that in distant sources only large ('e� > 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 pc) UC H�� regions were resolved in ATOMS ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 Properties of ionized Gas within UC H�� regions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content="1 Emission measure and electron density derived from H40U emission To derive emission measure (EM) and electron density (n4) from an RRL, we assume that the total rate of recombinations can be computed from 'rec = =4=?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='+UB, where n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' is the proton density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' + is the volume and UB is the total recombination coe�cient excluding captures to the n=1 levels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=', case B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The ionization of helium and stimulation of RRLs are also assumed to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Following Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2022), we define the line luminosity (in CGS units) as: !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='a(H40U) = =4=?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='+n, (1) MNRAS 000, 1–13 (2015) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='30 lonic Abundance Ratio (Y+)5 10 15 20 H40α 52°45\'15" 18″" Dec 21" 24" 16h00m37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0s 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5s 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0s RA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='50 He40α2 4 6 8 10 12 H40α 4744\'30" 33" Dec 36" 39" 9h01m58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0s 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='8s 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2s 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6s 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4s 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0s RA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 lonic Abundance Ratio (Y+)4 Chao Zhang Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Gas clumps identified in H40U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" IRAS ID 'e� RGC vlsr EM n4 EM n4 Y+ Temperature (pc) (kpc) (km/s) (106 cm�6 pc) (104 cm�3) (106 cm�6 pc) (104 cm�3) (K) RRL RRL 3mm 3mm I09002-4732 1 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='18 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 7570 ± 790 I11298-6155 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 7270 ± 690 I12320-6122 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='93 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='67 ± 2.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='78 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='35 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='80 ± 0.' 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='71 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='66 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 7040 ± 600 I15502-5302 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 8640 ± 940 I15520-5234 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 5840 ± 500 I15567-5236 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 6650 ± 700 I15570-5227 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 6430 ± 550 I16037-5223* 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='34 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='39 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 6430 ± 670 I16037-5223* 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='70 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='44 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 6590 ± 820 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='84 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 5860 ± 730 I16060-5146 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 10060 ± 970 I16065-5158 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 9250 ± 880 I16065-5158 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 5990 ± 630 I16071-5142 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='54 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 4230 ± 400 I16158-5055 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='40 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='3 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='24 ± 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='72 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='31 ± 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 6970 ± 660 I16172-5028 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='51 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='20 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='53 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 8400 ± 900 I16177-5018 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 6610 ± 830 I16177-5018 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='48 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='31 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 6910 ± 860 I16177-5018 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='76 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='41 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='14 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 6660 ± 900 I16177-5018 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='75 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 6490 ± 880 I16177-5018 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='52 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 6790 ± 780 I16297-4757 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 6010 ± 570 I16304-4710 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='27 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 6280 ± 600 I16313-4729* 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 7660 ± 730 I16318-4724 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='3 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='8 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='80 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 6720 ± 700 I17271-3439 1 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 6380 ± 480 I17441-2822* 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='76 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 12110 ± 1330 I17441-2822* 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 6850 ± 790 I17455-2800 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='21 ± 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='69 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='78 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='69 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 6250 ± 720 I18479-0005 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='58 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='65 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 7720 ± 890 I18479-0005 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='49 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='27 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 7530 ± 940 I18479-0005 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='86 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='43 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 7580 ± 1020 I18507p0110 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 ± 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='99 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='48 597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='33 ± 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 9270 ± 1080 I18530p0215 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 5560 ± 580 I19078p0901 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='93 ± 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='14 440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='42 ± 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 10590 ± 1200 I19078p0901 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='17 473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='53 ± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 9820 ± 1270 I19078p0901 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='82 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='17 343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='64 ± 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 9340 ± 990 I19078p0901 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='14 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='85 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='36 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 7880 ± 750 I19095p0930 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='8 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='60 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='83 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 9930 ± 840 I19097p0847 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='00 5280 ± 500 The ⇤ symbol after the IRAS name indicates that the sources are unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The uncertainties of EM, n4, Y+ and temperature are calculated by the law of propagation of uncertainties from measured line intensities listed in Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' where n is the e�ciency for producing H40U photons per recombi- nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Following Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2019), n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='99 ⇥ 10�32 at )4 = 104 K and =4 = 104 cm�3, which are the typical temperature and density of H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The variations of n is about 10 per cent for densities as low as 100 cm3 and temperatures as low as 5000 K (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For the sources with detections of H40U, we can derive n4 in unit of cm�3: =4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='582 \uf8ff !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (H40U) Jy km/s kpc2 �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content="5 \uf8ff 'e� pc ��1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 (2) where !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='a(H40U) = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (H40U)a/2 is the line luminosity in obser- vational units (Jy km s�1 kpc2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We assume an H�� region in our sample to be a homogeneous medium with a thickness of 2Re�, then the relation between n4 and EM is: ⇢" = 2\'e�=4=?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (3) ⇢" = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='66 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" (H40U) Jy km/s kpc2 \uf8ff 'e� pc ��2 (4) where EM in unit cm�6 pc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The median, mean and standard deviation of EM are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5⇥107, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='7⇥107 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1⇥108 cm�6 pc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The median, mean and standard deviation of n4 are 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2⇥103, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4⇥104 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5⇥104 cm�3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The EM and n4 are listed in the sixth and seventh column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 Ionic Abundance Ratio Here we present measurements of the ionized helium-to-hydrogen ratio (Y+ = =He+/=H+) toward UC H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We derive Y+ within MNRAS 000, 1–13 (2015) 6 Chao Zhang Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=" The correlation between distance and 'e�." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The green line shows the linear fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' UC H�� regions pixel-by-pixel using (Luisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Roshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2017) : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='+ = Ø �He+3a Ø �H+3a (5) where Ø �He+3a and Ø �H+3a are the integrated intensity in Jy beam�1 km s�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Here we assume both helium and hydrogen RRLs are in the LTE condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The right panel of Figure 1 shows the Y+ map of two exemplar sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The Y+ distribution within the UC H�� regions is not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Y+ appears to higher in the boundary region than in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' More disscussion on Y+ maps are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' There are 44 sources showing He40U emission in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The source-averaged Y+ values of these 44 UC H�� regions are summa- rized in the tenth column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The left panel of Figure 3 shows the distributions of source-averaged Y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The median, mean and standard deviation of source-averaged Y+ are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='03, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='3 Electron temperature As demonstrated in Keto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2008), the observed 3 mm continuum emission of UC H�� regions in high resolution interferometer obser- vations should be mostly free-free emission rather than from dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In Figure 4, we compare the intensity maps of H40U and H13CO+ for an example source, I09002-4732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' From this figure, one can see that the UC H�� region where H40U is bright shows very weak or even no H13CO+ molecular line emission, indicating that the UC H�� region contains negligible cold thermal dust radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Therefore, its 3 mm continuum emission could mainly come from free-free emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The other UC H�� regions in our sample are similar to I09002-4732 in that regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, we cannot rule out the possibility of the existence of some warm/hot dust emission inside UC H�� regions that cannot be traced by H13CO+ line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' There is the another way to show if 3 mm continuum emission should be mostly free-free emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The amount of continuum flux in excess of the flux predicted can be used to calculate the mass of dust causing the excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' This mass is given by (Pratap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1992) "dust = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='91 ⇥ 10�2( _<< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 )V+3(a[4G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' ( 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 _<<)3 )]32 kpc"� (6) where Sa is the excess flux due to emission from the dust, V is the dust emissivity (assumed = 1 Pratap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 1992), T3 is the dust temperature, and d: ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 is the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For the exemplar source, I09002-4732, its observed flux densities (Sa) is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='3 Jy and the T3 and dkpc are 39 K and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 kpc (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2020a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' If there are 10% 3mm continuum emission come from dust, the corresponding gas mass derived from dust emission is ⇠ 250 M�, comparable to that of its natal clump (⇠251 M� Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' This also indicates that only a negligible fraction (less than 10%) of 3mm continuum emission from the UC H�� region is from thermal dust radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The other UC H�� regions in our sample are similar to I09002-4732 in that regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We note that future higher frequency continuum observations can help constrain the dust emission within UC H�� regions in a better way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Assuming that the 3 mm continuum emission is mainly from free- free emission, the electron temperatures derived from the intensities of the H40U RRL emission and the 3 mm continuum emission, using the equations given in Appendix A in non-LTE conditions, are listed in the last column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The middle panel of Figure 3 shows the electron temperature distributions of UC H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The median, mean and standard deviation of electron temperature are 6662, 7026 and 1348 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The values of the electron temperatures are typical of H�� regions, but somehow on the cool side for what’s expected of UC H�� regions, indicating that 3 mm continuum emission is mainly from free-free emission is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='4 Electron density derived from 3 mm radio continuum With electron temperature information, electron density can also be derived from 3 mm continuum emission with equations in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The electron densities derived from 3 mm continuum are listed in the eighth column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The left panel in Figure 5 shows the correlation between EM (cal- culated by 3mm continuum emission) and n4 (calculated by RRLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The two parameters n4 and EM are strongly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The slope of linear fitting is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The correlation coe�cient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='94 with p- value ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The right panel shows the correlation between n4 and H�� region diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The black crosses represent the data from our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The gray markers are the data from Garay & Lizano (1999), which show similar trends as our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The correlation coe�cient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='63 with p-value ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2017) also found a strong negative correlation between n4 and H�� region diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We present a comparison of the n4 derived independently from the 3 mm radio continuum and H40U emission in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' There is a strong correlation between these two measurements with a cor- relation coe�cient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='97 and p-value ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The xy-bisector fit to these data gives a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='01, indicating that millimeter RRLs are good probes of electron densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='5 Line widths of RRL H40U We calculate line widths (�+) of H40U from the Gaussian fit of the source-averaged spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The right panel in Figure 3 shows the distribution of �+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The mean, median and standard deviation of �+ are 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='41 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='90 km/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The �+ is weakly correlated with Re�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The correlation coe�cient is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='25 with p-value ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' It indicates that the H40U line width decreases as the Re� MNRAS 000, 1–13 (2015) ATOMS-XIV: H�� regions 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Left panel: The distribution of Y+ which calculated by Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Middle panel: The distribution of Temperature which is calculated from both 3mm continuum emission and H40U emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Right panel: The distribution of H40U emission line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' All the sources detected in the H40U line are used in the middel and right panels, and the sources in the left panel are detected in both H40U and He40U lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The left and right panel are the intensity of H40U and H13CO+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The contours in right panel is H40U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Countours are from 20% to 80% in step of 20% of peak value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2021) found similar results and proposed non- thermal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=', dynamical) mechanisms dominate the line broadening of H40U on small scales of UC H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 4 DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 Variance of ionic abundance ratio (Y+) within UC H�� regions Previous studies such as Luisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2016) and Roshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2017) found Y+ below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='1 (around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='06) inside H�� regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' While the mean value of Y+ in our example is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Luisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2016) and Roshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2017) also witnessed a decreasing trend in Y+ with in- creasing distance from UC H�� region beyond the photodissociation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, as shown in the right upper panel of Figure 1, we found that Y+ within the resolved UC H�� region I09002-4732 and I15567-5236 increase with increasing distance from the center of UC H�� region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The other resolved UC H�� regions in our sample also exhibit similar spatial variance in Y+ similar to the example sources I09002-4732 and I15567-5236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Figure 7 shows how the Y+ of our data changes over the distance from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We pick the 80%, 60%, 40%, 30%, 20% of the peak emission of H40U in H�� regions as the dividing lines, and divide each source into five regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For most sources, their boundary regions clearly show higher Y+ than their inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The right panel of Figure 7 shows the distribu- tion of Y+ within 80%-100% (black) and 20%-40% (red) regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The two distributions are significantly di�erent with a p-value of ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001 from Kolmogorov-Smirnov test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' This figure fur- ther demonstrates that outer parts of UC H�� regions show higher Y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Is the higher Y+ near the edges of UC H�� regions caused by low signal-to-noise (S/N) He40U spectra?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' He40U emission shown in Figure 1, however, is pretty strong with S/N ratios larger than 5 at the boundary of the UC H�� region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' To further check the data quality of He40U emission, we plot the H40U and He40U spectra near the boundary (marked by red box) and central region (marked by black box) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The spectra of RRLs H40U and He40U are clearly detected in the boundary region with high S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Therefore, we argue that the large Y+ values near the edges of UC H�� regions are not caused by noise in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The possible explanations for the increasing trend of Y+ near the boundaries of UC H�� regions could be: (1)The He in the central region has the secondary ionization which lead to a lower density of singly ionized He and weaker He40U line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' However, the HeII64U line, which is produced by the He+ ion recombined from He2+ and free electron, is too weak to be detected in our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2) He40U emission which frequency is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='063305 GHz and typical width around 30 km/s is contaminated by C40U emission, whose frequency is 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='07236032 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Near the photodissociation region (PDR), CO is dissociated and ionized, and produces Carbon RRLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='The contamination of C40U emission in He40U could result in “stronger” observed He40U emission near PDR or the boundary of UC H�� region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The variance of Y+ should be further studied through future higher sensitivity and higher resolution observations with more transitions of RRLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='2 Line broadening mechenisms for RRLs The line widths (�+) of RRLs could be caused by thermal and microturbulent broadening as well as electron pressure broadening (Brocklehurst & Seaton 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' In Figure 9, we investigate the correlation between the line width (�+) of H40U and electron temperature (T4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' A clear increasing trend is seen in the relation between �+ and )4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' It indicates that higher electron temperature corresponds to larger line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The correlation coe�cient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='49 with p-value ⌧ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' We calculate the thermal and electron pressure broadening follow- MNRAS 000, 1–13 (2015) 8 Chao Zhang Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Left panel: The distribution of EM which calculated by 3mm continuum emission and n4 which calculated by RRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Right panel: The correlation between n4 which calculated by RRL and H�� region diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The black crosses represent the data from our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The gray markers and the line are the data from Garay & Lizano (1999) and the corresponding linear fit, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' n4 drived from 3 mm radio continuum emission versus n4 from H40U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The black dash line is y=x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The slope of linear fitting is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='95 which is shown by black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' ing Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The left panel in Figure 10 shows the e�ect of thermal and pressure broadening on the spectral line widths of H40U at di�erent electron density (n4) and electron temperature (T4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' For the UC H�� regions in our ATOMS sources, n4 are between 103 and 105 cm�3, and thus the pressure broadening has no significant e�ect on the spectral line widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content=' The thermal broadening, �+th, is given by �+th = (8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9AzT4oBgHgl3EQf6f6J/content/2301.01876v1.pdf'} +page_content='=2:B)4/ 𝑛, 𝜋/2 for 𝑚 < 𝑛, as well as 2𝜋 for 𝑚 = 𝑛. +On this space, 𝑍 is a diagonal trace-less operator, a generalized +form of the Pauli 𝜎𝑧 applied to the subspace spanned by |𝑚⟩ and |𝑛⟩, +while 𝑌 is the generalized form of the Pauli 𝜎𝑦 applied to the sub- +space on the subspace spanned by |𝑚⟩ and |𝑛⟩. +𝑍𝑚,𝑛 = |𝑚⟩⟨𝑚| − |𝑛⟩⟨𝑛| +for 0 ≤ 𝑚 < 𝑛 ≤ 𝑑 − 1 +𝑌𝑚,𝑛 = −𝑖|𝑚⟩⟨𝑛| + 𝑖|𝑛⟩⟨𝑚| +for 0 ≤ 𝑚 < 𝑛 ≤ 𝑑 − 1 +This formulation comes with the promise of being expressible +for the single qudits and it has the minimum number of parameters +required for representing the special unitary group. +The second objective is an efficient use of user-specified entan- +gling gates in the ansatz compilation. The usage of multiple species +of qudit entangling gates inside the same ansatz is future research. +Although the user can specify the preferred choice for the entan- +gling primitive, here, we focus on using a single entangling gate, +choosing between two gates commonly used in the trapped-ion +platform: the Molmer-Sorenson (MS) gate in Eq. (6) [7] and the +generalized light-shift (LS) in Eq. (7), a genuine qudit entangling +𝑈1 +· · · +𝑈𝑓1 +𝑈2 +· · · +𝑈𝑓2 +𝐸 +𝑄0 +𝑄1 +Figure 2: Example of an ansatz. Each layer contains an en- +tangling gate, preceded by a generic unitary 𝑈 on each one +of the qudits and followed by two more generic unitaries; in +this way, every layer is universal. +gate [25], [39]. +𝑀𝑆(𝜃) = 𝑒−𝑖 𝜃 +4 ·(𝐼 ⊗𝐼+𝜎𝑥01 ⊗𝜎𝑥01) +(6) +𝐿𝑆(𝜃) = 𝑒−𝑖𝜃 ·�𝑑−1 +𝑖=0 |𝑖𝑖⟩⟨𝑖𝑖 | +(7) +These are both well-established operations in quantum computing +hardware with well-characterized noise characteristics. Moreover, +they operate in a distinct way (the light-shift acts on phases, the +MS acts on populations) with different entangling power [25] when +applied to qudit systems. +Example 4.2. The compiler solves an ansatz like the one in Fig- +ure 2.— Consider a two-qudit entangling unitary 𝑈 . Figure 1 illus- +trates two different compilation results: (1) The top result uses mul- +tiple pre-selected entangling gates with low entangling power but +potentially less noise. (2) The bottom result uses two pre-selected +entangling gates with high entangling power but more noise per +gate.— The selection of the best result heavily depends on the per- +formance of the gates, i.e., specifics of the underlying hardware. +Both cases will always produce a correct result, as more entan- +glement can be built up with additional gates, and entanglement +generation can also be reduced through judicious use of single qudit +gates between entangling layers. +4.2.3 +Solution of the Ansatz. The optimization of the ansatz works +similar to a variational quantum algorithm. In each step, the full +circuit is simulated by multiplying the gate matrices for the chosen +parameters. This is computationally feasible for two-qudit gates in +dimensions of practical relevance given the capabilities of current +and future quantum hardware. For multi-qudit gates, the applicabil- +ity of this approach is expected to be limited. The resulting unitary +matrix in each optimization step is then compared to the target +matrix according to an objective function. Here, the choice was the +fidelity between the unitaries [40], for two-𝑑-dimensional systems: +Fidelity(𝐴, 𝐵) = 1 +𝑑2 Tr ⟨𝐴† , 𝐵 ⟩ +For the purpose of this work, the fidelity has several desirable prop- +erties that make it desirable for physical applications, compared to +other commonly used objective functions for matrix optimization. +The optimization loop is performed by iteratively simulating the +ansatz and verifying the fidelity achieved by the current solution +of optimization algorithm. +4.2.4 +Optimizer. The used optimization method is the dual anneal- +ing method, a combination of classical and fast simulated anneal- +ing, coupled with the L-BFGS [41] algorithm on boxed constraints, +which is a local search method applied on the neighborhood of the +solutions found by the global method in the high dimensional land- +scape of the problem to optimize. We refer the reader to Ref. [42] for +details on this algorithm. Alternatively, the use of gradient-based +methods is left for future work, due to the unclear trainability of +qudit PQCs. + +ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan +Kevin Mato, Martin Ringbauer, Stefan Hillmich, and Robert Wille +4.2.5 +Binary Search. Although the cost of a circuit depends on its +depth and gates structures, the final resulting circuit should have the +least number of layers that can compile with the desired fidelity the +entangling gate. The maximum number of layers for the search is +heuristically chosen as 2·𝑑2. This number was found to be sufficient +for generating maximally entangled qudit states from the least +powerful operation, acting only on two fixed physical levels. The +compiler uses a binary search and the number of layers is decreased +every time the gate can be decomposed for a certain number of +layers under a time period heuristically chosen, otherwise if the +target fidelity is not reached or the optimizer does not converge +before the end of a timer, the number of layers is increased. +5 +CASE STUDIES +The considerations made so far in the paper already showed that +the problem of entanglement compilation in high-dimensional +systems is complex. The solution proposed above is supposed to +provide a step forward in the field of compilation for qudit sys- +tems. The implementation is available freely under the MIT licence +at github.com/cda-tum/qudit-entanglement-compilation as part of +the Munich Quantum Toolkit (MQT). It is completely written in +Python 3.8, with exception of the external dependencies Numpy [43] +and Scipy [42]. This section provides corresponding case studies, +in order to demonstrate the feasibility of the workflow and its use- +fulness in compiling any multi-level entangling operation. The use +cases focus on the compilation of the controlled-sum gate CSUM, a +characteristic operation in qudit systems as discussed in the previ- +ous sections. +Although constituted by two compilation steps, the computa- +tional complexity of the workflow is dictated on a high level by +three routines. The first routine is the QR decomposition applied +on the unitary to compile; the algorithm has quadratic complexity +in the number of dimension of the two-qudit interaction with size +𝐷, therefore O(𝐷2). The result is a sequence of, in the worst case, +𝐷2 operations. The second routine is the translation of every output +operation in term of local operations and CEX; this algorithm has +complexity O(𝐷2), because linear in the number of output oper- +ations of the QR. Finally, the last routine of the workflow is the +solution by optimization of the ansatz, which has complexity of +O(log 𝐷). In fact, the last compilation step is a binary search, with +every step of duration linear in the number of dimensions of the +original unitary. +The evaluations were performed on a server running GNU/Linux +using an AMD Ryzen 9 3950X and 128 GiB main memory. The +layered compilation step is performed under a time limit in hours +heuristically chosen as 𝑑/4 (e.g., 4 h for a two-ququarts unitary), at +each step in the binary search. The compiler has minimum target +infidelity (1 − Fidelity) of 10−3. Several runs of the optimization +algorithm could lead to better results and, beyond dimension 16, +the computational time and the fidelity are affected. +The considered example follows the default strategies encoded +in the compiler. Every controlled rotation and partial swap is trans- +formed by local rotations into 𝑐𝑅𝑜𝑡1;0,1 and for the partial swaps +𝑝𝑆𝑤𝑎𝑝0,1, while the automatic choice for generating entanglement +is the controlled exchange gates CEX. More precisely, CEX1;0,1 has +control on the first level of the first qudit and the targets on the +first two levels of the second qudit. +The designer can decide to follow a different path and impose a +decomposition for which they have a specific implementation in +terms of the entangling gate involved, or compile it directly for the +machine. +In the studies, we considered the implementation of the CSUM +gate for qutrits and ququarts. Table 1 shows the results. Between +the near-term usable physical qudit dimensions [7], the solution +Table 1: Results of the workflow on CSUM for dimension 9 +and dimension 16 +System +Dim. +cRot +pSwap +CEX𝑡𝑜𝑡 +MS𝑡𝑜𝑡 +LS𝑡𝑜𝑡 +(1 − 𝐹) +two qutrits +9 +44 +24 +184 +1472 +368 +< 10−4 +two ququarts +16 +92 +36 +328 +9184 +10168 +10−3 ∼ 10−4 +for ququarts proves an already difficult and useful design case. The +columns denote the dimension (“Dim.”), followed by the number of +controlled rotations, and partial swaps, as well as the total number +of CEX, MS, and LS gates required for the decomposition, respec- +tively. The last column gives a bound on the achieved infidelity +(i.e., 1 − Fidelity). In a first phase the synthesis of CSUM for 9 di- +mensions (two qutrits) requires 20 controlled rotations and 6 partial +swaps, while the synthesis on two ququarts (dimension 16) outputs +42 controlled rotations and 6 partial swaps. These decompositions +are, by construction, not necessarily optimal. The controlled rota- +tions are further decomposed into 2 CEXs and the partial swap into +4 CEXs, with the resulting sequence being general for all dimen- +sions. +In order to show to the adaptability of the workflow, in the +smaller case we compile the entangling gate in terms of two native +gates used in the latest ion trap qudit processors [7] [25] and in- +troduced before, the MS and LS gate. The manual decomposition +of the CEX gate dedicated for the two-qutrit case by itself is non- +trivial, even for an experienced quantum information specialists. +The inherent difficulty arises not only from the amount of intuition +needed, but also from the inability to easily generalize the single +result found to higher dimensions. The pen-and-paper optimized +sequence for CEX in dimension 9 is made of two LS gates with +angle 𝜋, while two MS gates are also required for performing the +same CEX at any dimension, if the sequence is allowed to exploit +auxiliary levels [7]. +The layered compiler performs a similar decomposition with +the same number of 𝜋-LS gates autonomously. Without auxiliary +levels, the compiler outputs a sequence of 8 MS gates. For both +the compilation results, the infidelity reached is below 10−4. In the +case of two-ququarts, optimized sequences could not be achieved, +while the layered compiler autonomously decomposes the CEX +with infidelity between 10−3 and 10−4, at the cost of 28 MS gates +and in alternative of 31 LS gates. The number of gates required for +a two-ququarts circuit gives a feeling of the increased complexity +of the problem compared to two-qutrits. +The results are in line with the expected automation overhead in +a first automatic solution. In fact, it was predictable that the used +ansatz would ask for an over-parametrization [44], or using more +layers than necessarily, which allows for a great simplification of the +landscapes by eliminating spurious local minima. Although the re- +sults are not expected be efficient to the point of practical applicabil- +ity on a quantum system, this is only one component of a complete +compiler for high-dimensional and mixed-dimensional systems; +future developments will comprise optimization routines[45]–[49]. +In summary, the results show the feasibility of the proposed +workflow to deliver decompositions efficiently for any two-qudit +unitary. Further, the results provide evidence compilation of entan- +gling operations for multi-level systems is not easy, especially com- +pared to binary systems. This work investigates the first key step +towards full automation for qudits. The implementation is available +under the MIT license at github.com/cda-tum/qudit-entanglement- +compilation as part of the Munich Quantum Toolkit (MQT). +6 +CONCLUSION +The challenges for efficient and reliable application of entangling +gates inside qudit circuits arise because of the complexity in under- +standing the amount of entanglement generated by an operation + +Compilation of Entangling Gates for High-Dimensional Quantum Systems +ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan +as well as the corresponding affected levels—commonly referred +to as structure. Consequently, the operations are mostly consid- +ered as black-boxes with the question of whether it is possible to +reach an implementation for a certain qudit platform and with a +given gate-set. So far, the study of feasibility and the implementa- +tion of entangling operations for specific qudit technologies was +performed by quantum information specialists manually, without +the promise of re-utilizing a particular decomposition for a system +certain number of dimension, to those of greater dimensionality. +In this paper, we introduced a complete workflow for compiling +any two-qudit unitary into any target gate set. While the resulting +gate sequences are typically not optimal in terms of gate count, the +method is computationally efficient, since the only step involving +numerical optimization can be pre-computed. The case studies +confirm the feasibility of the workflow as well, for the first time, the +automated results of compilation of generic entangling operations +to any dimensionality. The implementation is available under the +MIT license at github.com/cda-tum/qudit-entanglement-compilation +under the ensemble of the Munich Quantum Toolkit (MQT). In the +future, the proposed approach may be improved up on by utilizing +auxiliary qudit levels, alternative ansatz designs, compression of +synthesis results, and circuit optimization in post-processing. +ACKNOWLEDGMENTS +This work has received funding from the European Union’s Horizon +2020 research and innovation programme under the ERC Consolida- +tor Grant (agreement No 101001318), the Marie Skłodowska-Curie +Grant (agreement No 840450), and the NeQST Grant (agreement +No 101080086). It is part of the Munich Quantum Valley, which is +supported by the Bavarian state government with funds from the +Hightech Agenda Bayern Plus and was partially supported by the +BMK, BMDW, and the State of Upper Austria in the frame of the +COMET program (managed by the FFG). +REFERENCES +[1] +J. Preskill, “Quantum computing in the nisq era and beyond,” Quantum, vol. 2, +p. 79, 2018. +[2] +F. Arute, K. 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Wille, “Adaptive compilation of +multi-level quantum operations,” arXiv preprint arXiv:2206.03842, 2022. + diff --git a/tdE2T4oBgHgl3EQf1wjo/content/tmp_files/load_file.txt b/tdE2T4oBgHgl3EQf1wjo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9c97e47095812d9b7d8303cc5e9bf2dc56dbe5a --- /dev/null +++ b/tdE2T4oBgHgl3EQf1wjo/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf,len=773 +page_content='Compilation of Entangling Gates for High-Dimensional Quantum Systems Kevin Mato∗ Martin Ringbauer+ Stefan Hillmich† Robert Wille∗‡ ∗ Chair for Design Automation, Technical University of Munich, Germany +Institute for Experimental Physics, University of Innsbruck, Austria †Institute for Integrated Circuits, Johannes Kepler University Linz, Austria ‡ Competence Center Hagenberg (SCCH) GmbH, Austria kevin.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='wille@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='de https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='cda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='de/research/quantum/ ABSTRACT Most quantum computing architectures to date natively support multi-valued logic, albeit being typically operated in a binary fash- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Multi-valued, or qudit, quantum processors have access to much richer forms of quantum entanglement, which promise to significantly boost the performance and usefulness of quantum de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' However, much of the theory as well as corresponding design methods required for exploiting such hardware remain insufficient and generalizations from qubits are not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' A partic- ular challenge is the compilation of quantum circuits into sets of native qudit gates supported by state-of-the-art quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In this work, we address this challenge by introducing a complete workflow for compiling any two-qudit unitary into an arbitrary native gate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Case studies demonstrate the feasibility of both, the proposed approach as well as the corresponding implementation (which is freely available at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='com/cda-tum/qudit-entanglement- compilation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' CCS CONCEPTS Hardware → Quantum computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Electronic design au- tomation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Emerging languages and compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' ACM Reference Format: Kevin Mato, Martin Ringbauer, Stefan Hillmich, and Robert Wille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Compilation of Entangling Gates for High-Dimensional Quantum Sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In Proceedings of Asia and South Pacific Design Automation Conference (ASP-DAC ’23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' ACM, New York, NY, USA, 7 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1145/ 3566097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='3567930 1 INTRODUCTION In the future, quantum computers are expected to solve industrial and scientific problems with a reduced consumption of resources and greater algorithmic efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Current quantum devices can host up to hundreds of noisy quantum bits (qubits) and support a limited number of logical operations on these qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For this reason, they are referred to as Noisy Intermediate-Scale Quantum (NISQ) de- vices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' A variety of technology platforms have now implemented such NISQ devices, including superconducting circuits [2], trapped ions [3], and single photons [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Notably, while these devices thus far almost exclusively work with two-level qubits, the underlying hardware almost always natively supports encoding multi-valued logic in high-dimensional quantum digits (qudits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The research on qudit design and computation has a long history, with efforts primarily focusing on conceptual studies of algorithms for idealized qudits and their comparison to qubits [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Fundamen- tally, a qudit can not only store and process more information per quantum particle, but also features a richer set of logical opera- tions [6] that make processing more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Proof-of-principle ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1145/3566097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='3567930 demonstrations [7]–[10] have shown that qudits enable improve- ments in circuit complexity and algorithmic efficiency for a wide class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' These results inspired proposals for and demon- stration of qudit basic control in numerous physical platforms such as trapped ions [7], [11], photonic systems [8], [12]–[14], super- conducting circuits [15], [16], Rydberg atoms [17], [18], nuclear spins [19], cold atoms [20], [21], and nuclear magnetic resonance systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' More recently, efforts have intensified with the demon- stration of universal qudit quantum processors with error rates that are competitive to qubit systems [7], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' However, a key task for using qudit quantum processors remains the efficient compilation of circuits for this hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Compared to qubits, where every entangling gate is equivalent to the CNOT [23] gate, qudits offer much richer possibilities in the form of many inequivalent entangling gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The flip side of these opportunities, however, are challenges in finding suitable gate sets that are native to the hardware and, then, compiling algorithms to these gate sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Thus far, much of the theory required to address these challenges is insufficient and, accordingly, no corresponding design methods are available for this purpose yet—making the compilation of entan- gling gates a mostly manual task and, thus, preventing the further exploitation of high-dimensional quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In this work, we introduce a workflow for the compilation of ar- bitrary two-qudit entangling gates in any high-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The core idea is to map the target unitary operating on two qu- dits to a single-qudit unitary in an appropriately larger space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The latter can then be decomposed into two-level couplings using estab- lished techniques [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The resulting decomposition will feature at most 𝑑2 two-level entangling gates of two standardized types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For decomposing these two standard gates into any hardware-native gate set, we developed an offline optimization routine, which we demonstrate on a recently developed gate set for trapped ion qudit quantum processors [7], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In typical experimental scenarios, where the native gates act only on a subspace of the full Hilbert space, the cost of this pre-computation is independent of the size of the target unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The resulting circuit will express the target unitary using only native gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Overall, this results in a method- ology that, for the first time, enables compiling entangling gates for high-dimensional quantum systems in a fully automatic fash- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The proposed method can be applied to any two-qudit unitary and is computationally efficient, since the numerical optimization step is done offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Case studies demonstrate the feasibility of the proposed method and a corresponding implementation is avail- able in open-source via github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='com/cda-tum/qudit-entanglement- compilation, along with other components of the Munich Quantum Toolkit (MQT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The proposed tool is only a component of a com- plete compiler for high-dimensional and mixed-dimensional sys- tems, that realizes a fully automated and computationally scalable workflow for the compilation of entangling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Therefore the results proposed are not necessarily efficient to the point of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='04155v1 [quant-ph] 10 Jan 2023 ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan Kevin Mato, Martin Ringbauer, Stefan Hillmich, and Robert Wille practical applicability on a quantum system, both in terms of gate count and possible requirement of additional circuit optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The remainder of this paper is structured as follows: Section 2 briefly reviews the basics of quantum information processing (QIP) and entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Section 3 motivates the problem of entanglement compilation for qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Section 4 describes the proposed approach to tackle entanglement compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Section 5 provides case studies on the feasibility of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Finally, Section 6 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 2 BACKGROUND In this work, we provide the basis for efficient compilation of en- tangling operations for high-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' To this end, this section first briefly reviews the fundamentals of QIP (with a focus on high-dimensional quantum logic) and entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1 Quantum Information Processing The fundamental unit of classical information is the bit (binary digit), which can exclusively take the values 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Generalizing this concept to quantum computers gives rise to the qubit as the corresponding unit of quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The crucial difference to the classical case, however, is that qubits can be in any linear combination of the basis states |0⟩ and |1⟩ (using Dirac’s bra-ket notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Since there are no ideal two-level systems in nature, qubits are usually constructed by restricting the natural multi-level struc- ture of the underlying physical carriers of quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' These systems, therefore, natively support multi-level logic with the fundamental unit of information termed a qudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' A qudit is the quantum equivalent of a 𝑑-ary digit with 𝑑 ≥ 2, whose state can be described as a vector in a 𝑑-dimensional complex Hilbert space H𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The quantum state |𝜓⟩ of a qudit can thus be written as a linear combination |𝜓⟩ = 𝛼0 · |0⟩ +𝛼1 · |1⟩ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='+𝛼𝑑−1 · |𝑑 −1⟩, or simplified as vector |𝜓⟩ = [ 𝛼0 𝛼1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 𝛼𝑑−1 ]T, where 𝛼𝑖 ∈ C are the amplitudes relative to the computational basis of the Hilbert space—given by the vectors |0⟩, |1⟩, |2⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' , |𝑑 − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The squared magnitude of an amplitude |𝛼𝑖 |2 gives the probability with which the corresponding basis state 𝑖 would be observed when measuring the qudit in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Normalization of probabilities further requires that �𝑑−1 𝑖=0 |𝛼𝑖 |2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Consider a single qudit with three levels (also re- ferred to as qutrit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The quantum state |𝜓⟩ = √︁ 1/3 · |0⟩ + √︁ 1/3 · |1⟩ + √︁ 1/3 · |2⟩ is a state with equal probability of measuring each basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' A different notation for the same quantum state may be √︁ 1/3 · [ 1 1 1 ]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Two key properties that distinguish quantum computing from classical computing are superposition and entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' A qudit is said to be in a superposition of states in a given basis when at least two amplitudes are non-zero relative to this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Entangle- ment, instead, describes a powerful form of non-classical correla- tion born from interactions in multi-qudit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Quantum com- puter operations are represented by unitary matrices 𝑈 , satisfying 𝑈 †𝑈 = 𝑈𝑈 † = 𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2 Entanglement Structures Classically, the state of a bipartite system can always be written as a product of the state of the individual systems (or a mixture of such products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Entangled quantum states encode information in a non-local way, such that it can only be extracted from the full system, but not from the constituent qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' More precisely, two or more quantum systems are in an entangled state, whenever their state cannot be written as a product of states of the individual subsystems (or a convex mixture of such products) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While systems of two qubits are quite well understood, entanglement in multi-partite or higher dimensional systems still present a range of open questions [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The central scope of the work are quantum logic operations gen- erating quantum entanglement, with a special focus on bipartite entangling gates, which enable the core aspect of error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Generalizations to multipartite entangling operations will be tack- led where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The first notable observation is the much richer entanglement structure of qudits compared to qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Specifically, for qubits, all entangling operations are related to the controlled-NOT (CNOT) gate via local operations on the sub- systems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Hence, for qubit systems, it is sufficient to compile the CNOT gate to the native operations of the quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For qudit systems, instead, this is no longer true and they can be entangled in many inequivalent ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Consequently, while any single entangling gate is sufficient for universal quantum computa- tion [27], not all entangling gates are equally useful for any given application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Consider, the controlled-exchange gate CEX [7] defined by CEX𝑐,𝑡1,𝑡2 : �|𝑐,𝑡1⟩ ↔ |𝑐,𝑡2⟩ |𝑗,𝑘⟩ → |𝑗,𝑘⟩ for 𝑗 ≠ 𝑐,𝑘 ≠ 𝑡1,𝑡2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' (1) This qudit-embedded version of the CNOT gate generates qubit- level entanglement in a high-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' However, there are gates that directly generate genuine qudit entanglement, such as the controlled-SUM gate defined by CSUM : |𝑖, 𝑗⟩ ↦→ |𝑖,𝑖 ⊕ 𝑗⟩, (2) where ⊕ denotes addition modulo 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' These are just two examples of a more general theme in qudit systems, where entangling gates differ in their entangling power or structure, that corresponds to the amount of entanglement generated by an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The reasons why genuine qudit gates produce more entanglement than the first one are more intuitively explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While the CEX gate has a low entangling power, since it only generates two-level entan- glement, it is very flexible and can be used to intuitively build up any entanglement structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In turn, however, it is quite inefficient for constructing highly entangling unitaries such as the CSUM gate out of two-level entangling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Too much entangling power, however, is not always helpful either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' It may happen that a qudit circuit requires the generation of entanglement only within a small subspace of the Hilbert space, in that case applying CSUM becomes disadvantageous and the CEX is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In practice, of course, neither of the gates from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2 might be natively supported by the quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' However, potent qudit quantum devices are expected to support several primitive entangling gates, with a range of entangling powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Hence, it becomes crucial to be able to compile arbitrary qudit unitaries into the native set of operations, while making the best use of the available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 3 MOTIVATION Based on the general setting introduced above, the compilation of entangling operations into native gate sets is of central importance for efficient QIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Since the general problem is NP-hard already for qubits [28], quantum computers critically depend on efficient, if not optimal, compilation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Given the more complicated entanglement structure of qudits, compared to their binary coun- terpart, the compilation problem becomes much more challenging in high-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Compilation of Entangling Gates for High-Dimensional Quantum Systems ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1 Problem Formulation Qubit circuits may be written using several entangling gates, all equivalent to the CNOT gate [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For qudit systems, the situation is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' With the structure of entanglement becoming much richer, there is not just more freedom in designing quantum circuits, but also more opportunities for optimizing their efficiency with the right set of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The challenge for qudit compilation is then to understand and to harness this entanglement structure for efficient decompositions while at the same time, keeping the problem computationally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In order to make the problem more intuitive for the qubit expert, the use of a metaphor can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Using color coding as an analogy: Information encoded in one qubit is like a single grayscale pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Information encoded in a qutrit (𝑑 = 3) is like an RGB pixel that can combine 3 different colors and all their possi- ble shades inbetween.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Now, adding a second pixel, the qubits will only have two parameters, where qutrits have 6 parameters for leveraging the full potential of the two pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This highlights the drastically higher information density in qutrits (and subsequently qudits of even higher dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' However, the qutrit, just like the color pixel, requires a vastly more complex control system, to efficiently use the capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The problem of entanglement compilation can now be formu- lated as follows: Given a unitary 𝑈 representing an interaction be- tween two qudits of dimension 𝑑, find a decomposition of 𝑈 into arbitrary local unitaries and a pre-defined set of entangling gates, in a way that is as close to the optimum as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In an application setting, the native gate set will be defined by the used quantum hardware, which will also set the cost of each component of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While it is in general not known how complex the optimal solution is, the general figure of merit will be the gate fidelity given the experimental noise sources for the various gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Here, we will focus on two-qudit gates, although the methods we present generally also apply to the multi-qudit setting, at the cost of decomposing the tensor product in two-qudit unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2 State of the Art In this section, we briefly review existing approaches to compile entangling operations in high-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While some results exist for special cases [24], [30], these are of limited use in actual quantum hardware, which typically does not natively support the types of interactions we might want to work with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' More precisely, in pioneering work, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [24] introduced a syn- thesis algorithm for qudit entangling gates that produces decompo- sitions in terms of controlled-householder rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' They prove a lower and a constructive upper bound on the number of two-qudit controlled-rotation gates required for an arbitrary two-qudit uni- tary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' These results, however, only apply to the specific controlled rotations used, without considering the constraints or indeed op- portunities of physical qudit quantum processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [31], a compilation algorithm for generic qudit unitaries is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The final goal of the mathematical procedure is to reduce the number of gates that use magic state injection protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The entanglement complexity of qudit systems is ignored, and although an elegant solution is found, the restricted scalability of the algorithm and the final output, too far from being applied to a physical quantum pro- cessor, make the work still theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Other previous works [32] try to lay out routines for the synthesis of qudit circuits, in this particular case for qutrits only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Related to this, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [30] presents a method to construct circuits for generating different forms of qudit entanglement using a specific gate set including the controlled-exchange gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While the goal is the generation of specific quantum states from a fixed input, it is unclear to what extent these methods can be transferred to the more complicated compilation of entangling unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [33], the challenge of multi-qubit compilation is tack- led.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The authors use parametrized quantum circuits and numerical optimization of the fidelity function, for solving an incremental structure of alternating local and entangling gate layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' More pre- cisely, the work focuses on the compilation of global entangling gates based on an ansatz made of alternating global entangling gates and specific equatorial rotations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' the approach cannot be directly applied to qudit entanglement compilation, due to the inefficient search strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Moreover, the solution provided in [33] is unable to express the richer variety of local and partial entangling oper- ations available in this new setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Finally, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [34] arbitrary controlled unitaries are proved to be physically implemented using auxiliary levels in the qudit Hilbert space, regardless of their form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4 EFFICIENT COMPILATION OF ENTANGLEMENT Motivated by the challenges described in Section 3, we now propose a workflow that enables the compilation of any two-qudit unitary into any target gate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The proposed solution consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' First, a synthesis method, which takes the target two-qudit unitary and returns a high-level circuit comprised of only two types of two-level entangling gates, regardless of the initial unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This step is crucial for making the technique scalable to arbitrary dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Second, a compilation step that uses parametrized circuits and numerical optimization to decompose the two standardized two-level entangling gates into any target gate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Although this step can be computationally intensive, it can be pre-computed, since the structure from the first step is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Hence, this step does not affect the scalability of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1 Step 1: QR Decomposition of Entangling Operations Qudit quantum hardware typically exploits two-level couplings between the various qudit levels, despite having access to a full high-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' It has been shown that this is sufficient for implementing arbitrary single-qudit operations with a cost that is at most quadratic in the size of the Hilbert space [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In the simplest instance, the resulting sequence of operations is composed of two-level Givens-rotations between adjacent sites 𝑉𝑖 and a diagonal phase matrix Θ in 𝑈 = 𝑉𝑘 · 𝑉𝑘−1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' · 𝑉1 · Θ [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This algorithm is universally valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' However, it scales quadratically in the Hilbert space dimension and, due to the rigid structure, it can introduce redundant operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Hence, for efficiently compiling local qudit unitaries, more advanced algorithms, that take into account the structure of the qudits and experimental constraints, can be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For the purpose of the present work, however, it is precisely the standardized structure of the QR decomposition that is beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In more detail, we start by interpreting the target two-qudit uni- tary 𝑈 as a single qudit unitary in dimension 𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This is achieved by mapping the two-qudit states to a single qudit as |𝑖, 𝑗⟩ ↦→ |𝑑 · 𝑖 + 𝑗⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' We then apply the QR decomposition algorithm to that higher-dimensional local unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Without loss of generality, we thereby assume a ladder-type coupling, where each of the virtual single-qudit states is coupled to its neighboring states |𝑗⟩ ↔ |𝑗 ±1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The QR decomposition then applies successive rotations between adjacent state in the virtual single qudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The result of the decom- position is a set of two-level rotations of the form 𝑅(𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='𝜙) = � cos 𝜃 2 sin 𝜃 2 (−𝑖 cos𝜙 − sin𝜙) sin 𝜃 2 (−𝑖 cos𝜙 + sin𝜙) cos 𝜃 2 � ASP-DAC ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' January 16–19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Japan Kevin Mato,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Martin Ringbauer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Stefan Hillmich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' and Robert Wille followed by a phase matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' which can be represented as a se- quence of phase rotations on neighboring states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' which again can be decomposed into two-level rotations using the identity Z(𝜃) = 𝑅( 𝜋 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 0) · 𝑅(𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 𝜋 2 ) · 𝑅(− 𝜋 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' As a consequence of our choice of ladder-type coupling, the two-level rotations all occur between adjacent states and are thus embedded into the 𝑑2 dimensional Hilbert space as ˆ𝑅𝑖 (𝜃,𝜙) = �1 · · 0 0 · · [𝑅(𝜃,𝜙)]𝑖,𝑖+1 · · · 0 0 · · 1 � , where the subscript 𝑖,𝑖 + 1 denotes the affected states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The resulting matrix is a two-level rotation ˆR embedded in a higher-dimensional space that translates to an entangling operation in the original two-qudit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The next challenge is now decomposing these entangling gates into the native gate set of the target hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' A straightforward approach would be to simply try and compile each of the O(𝑑2) entangling gates in the sequence into the target gate set, although computationally very costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Instead, one can notice that only two different gates need to be compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This is most easily seen at a simple example with two qutrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' By design, each of the two-level rotation 𝑅𝑖 is represented as 2 × 2 block matrices in a 𝑑2 × 𝑑2 identity matrix as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 𝑐𝑅 𝑐𝑅 𝑐𝑅 𝑐𝑅 𝑝𝑆 𝑝𝑆 𝑝𝑆 𝑝𝑆 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (3) Here, the horizontal and vertical lines indicate the tensor product structure of the original two-qutrit unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' We note that when- ever the rotation 𝑅𝑖 acts on levels that do not cross these lines, it physically corresponds to a controlled-rotation gate cRot in the two-qudit system (indicated above as cR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' When 𝑅𝑖 does cross a boundary, it corresponds to a partial Swap operation pSWAP in the bipartite system (indicated above as pS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While mathematically very similar, these operations are fundamentally different in their complexity, so they must be treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Fortunately, it is sufficient to compile just one example of each, since the other cases (for different qudit indices) can be obtained simply by permuting the states of one operation of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For simplicity and generality, we focus in the following on gates that act in the 0-1 subspace of the qudit Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' After the generation of the abstract sequence of rotations, the compiler simplifies the complexity of the problem by applying permutation gates to every 𝑐𝑅𝑜𝑡 and 𝑝𝑆𝑤𝑎𝑝 gate in order to express the sequence in terms of only two entangling operations, 𝑐𝑅𝑜𝑡1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1, pSwap0,1, and permutation gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This way, every operation can be compiled at the expense of compiling efficiently only one 𝑐𝑅𝑜𝑡 and one 𝑝𝑆𝑤𝑎𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The two operations cannot, in general, be implemented directly on the hardware, therefore one further step is needed before concluding the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Since 𝑐𝑅𝑜𝑡1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1, pSwap0,1 are parametrized in 𝜃 and 𝜙, it would require to compile these default gates for every different value of the two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The default gates chosen will be further decomposed into a sequence of local operations, that will encode the variables 𝜃 and 𝜙 for the equatorial rotations, plus a static entangling gate that will provide the necessary entangling strength and that will act on a two-level subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In this regard, the compiler will follow the decomposition of the controlled rotation 𝑐𝑅𝑜𝑡1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1 (control on the level 1 of the first qudit, target the subspace 0-1 of the second qudit) as, 𝑐𝑅𝑜𝑡1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1(𝜃,𝜙) = [𝐼 ⊗ 𝑅(𝜋/2, −𝜙 − 𝜋/2)]· (4) CEX · [𝐼 ⊗ 𝑍 (𝜃/2)]· CEX · [𝐼 ⊗ (𝑍 (−𝜃/2) · 𝑅(−𝜋/2, −𝜙 − 𝜋/2)), where the decomposition uses the CEX1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 0, 1 that will act on the 0,1 subspace of the second qudit, while the partial swap: 𝑝𝑆𝑤𝑎𝑝0,1(𝜃,𝜙) = [𝐻 ⊗ 𝐻] · CEX · [𝐻 ⊗ 𝐻] (5) [𝐼 ⊗ 𝑅(𝜋/2, −𝜙 − 𝜋/2)] · CEX· [𝐼 ⊗ 𝑍 (𝜃/2)] · CEX · [𝐼 ⊗ 𝑍 (−𝜃/2)]· [𝐼 ⊗ 𝑅(−𝜋/2, −𝜙 − 𝜋/2)]· [𝐻 ⊗ 𝐻] · CEX · [𝐻 ⊗ 𝐻] the CEX1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1 is used once again in the pSwap operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In order to better understand the intermediary com- pilation step, the operations involved, and the new encoding, let us consider the case where two qutrits are interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The controlled rotation on levels |7⟩ and |8⟩ appears in the decomposition, which correspond to |21⟩ and |22⟩ in the two-qutrit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Since the method requires the controlled rotation to be applied to the 0-1 subspace, we have to permute the levels of the two-qutrit system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' We add local rotations with angle 𝜋 and phase 𝜋/2 that will act as permutations and route |21⟩ to |10⟩ and |22⟩ to |11⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The decomposed sequence will be: 𝑐𝑅𝑜𝑡2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1,2(𝜃,𝜙) =(𝑃0,2 · 𝑃1,2 ⊗ 𝑃0,1 · 𝑃1,2)†· 𝑐𝑅𝑜𝑡1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1(𝜃,𝜙)· (𝑃0,2 · 𝑃1,2 ⊗ 𝑃0,1 · 𝑃1,2), where 𝑃𝑖,𝑗 denotes a permutation of levels 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Note that after executing the controlled-rotation, the sequence finishes with the uncomputation of the permutation sequence, in order to restore the original encoding for the subsequent operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' At this point the synthesis returns a sequence of local operations that will encode the angles of rotation and the permutation of the states, interleaved with the chosen CEX gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This is the default choice made in the project, but the user could provide a different one that could act on a different subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The compiler will now proceed with the last step: the compilation of the CEX gate with an entangling gate input by the user and compatible with the target machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2 Step 2: Layered Compilation This section describes the second step of the compiler and the lowest abstraction level before getting to a physical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The software component takes as input the target unitary to compile and the target entangling gate for the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The entangling gates could be a primitive entangling gate of the target quantum hardware, or it could be a higher-level abstract gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The software outputs a compiled sequence made of arbitrary local two-level rotations and the chosen gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' At the core of the second component are parametrized quantum circuits, that solved for different gates, lead to different decomposi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' More precisely, as Figure 1 shows, results present different depths and noise characteristics depending on the noise and power of the single entangling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The section will continue with a brief overview of parametrized quantum circuits, followed by a detailed explanation on the con- struction of the problem representation and its resolution by the layered compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Compilation of Entangling Gates for High-Dimensional Quantum Systems ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan 𝑈 = 𝐿 𝐿 · · 𝐿 𝐿 𝐿 · · · · 𝐿 𝐿 𝐿 · · 𝐿 𝐿 𝐿 · · · · 𝐿 𝐸 𝐸 𝐸 𝑈 = 𝐿 𝐿 · · 𝐿 · · 𝐿 𝐿 𝐿 · · 𝐿 · · 𝐿 𝐸′ 𝐸′ Figure 1: Given a two-qudit unitary 𝑈 , compilation results depend on the structure and the noise inherent to the chosen gate 𝐸;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' new local (𝐿) gates will match the gate 𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='1 Parametrized Quantum Circuits (PQCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' PQCs consist of a series of quantum gates dependent on a set of continuous or discrete parameters 𝜃𝑖 that can be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The particular initial choice of the sequence of gates is called an ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This circuit is then optimized, typically by a classical algorithm, to minimize a cost function that encodes the solution of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This approach can be used for a range of computational ques- tions [35], like Hamiltonian ground-state energy estimation [36] or compilation [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In the following we discuss the important design choices during the ansatz construction, its resolution and how the ansatz evolves during the optimization loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2 Construction of Ansatz and Expressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The first step con- sists of choosing the building blocks for our ansatz and then incre- mentally composing them in order to create the final circuit to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' There are two types of operations involved in the quantum circuit: local operations and entangling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The initial objective is to find a representation for single-qudit operations that is parametric and that achieves maximal expressibil- ity [37] with the minimal number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Here, expressibility refers to the circuit’s ability to generate a large fraction of two-qudit unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The proposed solution exploits theoretical results [38] al- ready present in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The single qudit lives in a 𝑑-dimensional (𝑑 ≥ 2) complex Hilbert space H = C𝑑 spanned by the orthonor- mal basis |0⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' , |𝑑 − 1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Local qudit unitaries will be written as in [38], where elements of the special unitary group SU(𝑑) are parametrized as: 𝑈 = � �𝑑−2 𝑚=0 � �𝑑−1 𝑛=𝑚+1 exp(𝑖𝑍𝑚,𝑛𝜆𝑛,𝑚) exp(𝑖𝑌𝑚,𝑛𝜆𝑚,𝑛)�� � �𝑑−1 𝑙=1 exp(𝑖𝑍𝑙,𝑑𝜆𝑙,𝑙) � using 𝑑2 − 1 real parameters � 𝜆𝑚,𝑛 � in the interval between 0 and 𝜋 for 𝑚 > 𝑛, 𝜋/2 for 𝑚 < 𝑛, as well as 2𝜋 for 𝑚 = 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' On this space, 𝑍 is a diagonal trace-less operator, a generalized form of the Pauli 𝜎𝑧 applied to the subspace spanned by |𝑚⟩ and |𝑛⟩, while 𝑌 is the generalized form of the Pauli 𝜎𝑦 applied to the sub- space on the subspace spanned by |𝑚⟩ and |𝑛⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 𝑍𝑚,𝑛 = |𝑚⟩⟨𝑚| − |𝑛⟩⟨𝑛| for 0 ≤ 𝑚 < 𝑛 ≤ 𝑑 − 1 𝑌𝑚,𝑛 = −𝑖|𝑚⟩⟨𝑛| + 𝑖|𝑛⟩⟨𝑚| for 0 ≤ 𝑚 < 𝑛 ≤ 𝑑 − 1 This formulation comes with the promise of being expressible for the single qudits and it has the minimum number of parameters required for representing the special unitary group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The second objective is an efficient use of user-specified entan- gling gates in the ansatz compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The usage of multiple species of qudit entangling gates inside the same ansatz is future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Although the user can specify the preferred choice for the entan- gling primitive, here, we focus on using a single entangling gate, choosing between two gates commonly used in the trapped-ion platform: the Molmer-Sorenson (MS) gate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' (6) [7] and the generalized light-shift (LS) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' (7), a genuine qudit entangling 𝑈1 · · 𝑈𝑓1 𝑈2 · · 𝑈𝑓2 𝐸 𝑄0 𝑄1 Figure 2: Example of an ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Each layer contains an en- tangling gate, preceded by a generic unitary 𝑈 on each one of the qudits and followed by two more generic unitaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' in this way, every layer is universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' gate [25], [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 𝑀𝑆(𝜃) = 𝑒−𝑖 𝜃 4 ·(𝐼 ⊗𝐼+𝜎𝑥01 ⊗𝜎𝑥01) (6) 𝐿𝑆(𝜃) = 𝑒−𝑖𝜃 ·�𝑑−1 𝑖=0 |𝑖𝑖⟩⟨𝑖𝑖 | (7) These are both well-established operations in quantum computing hardware with well-characterized noise characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Moreover, they operate in a distinct way (the light-shift acts on phases, the MS acts on populations) with different entangling power [25] when applied to qudit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The compiler solves an ansatz like the one in Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='— Consider a two-qudit entangling unitary 𝑈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Figure 1 illus- trates two different compilation results: (1) The top result uses mul- tiple pre-selected entangling gates with low entangling power but potentially less noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' (2) The bottom result uses two pre-selected entangling gates with high entangling power but more noise per gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='— The selection of the best result heavily depends on the per- formance of the gates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=', specifics of the underlying hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Both cases will always produce a correct result, as more entan- glement can be built up with additional gates, and entanglement generation can also be reduced through judicious use of single qudit gates between entangling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='3 Solution of the Ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The optimization of the ansatz works similar to a variational quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In each step, the full circuit is simulated by multiplying the gate matrices for the chosen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This is computationally feasible for two-qudit gates in dimensions of practical relevance given the capabilities of current and future quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For multi-qudit gates, the applicabil- ity of this approach is expected to be limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The resulting unitary matrix in each optimization step is then compared to the target matrix according to an objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Here, the choice was the fidelity between the unitaries [40], for two-𝑑-dimensional systems: Fidelity(𝐴, 𝐵) = 1 𝑑2 Tr ⟨𝐴† , 𝐵 ⟩ For the purpose of this work, the fidelity has several desirable prop- erties that make it desirable for physical applications, compared to other commonly used objective functions for matrix optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The optimization loop is performed by iteratively simulating the ansatz and verifying the fidelity achieved by the current solution of optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='4 Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The used optimization method is the dual anneal- ing method, a combination of classical and fast simulated anneal- ing, coupled with the L-BFGS [41] algorithm on boxed constraints, which is a local search method applied on the neighborhood of the solutions found by the global method in the high dimensional land- scape of the problem to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' We refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' [42] for details on this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Alternatively, the use of gradient-based methods is left for future work, due to the unclear trainability of qudit PQCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan Kevin Mato, Martin Ringbauer, Stefan Hillmich, and Robert Wille 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='5 Binary Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Although the cost of a circuit depends on its depth and gates structures, the final resulting circuit should have the least number of layers that can compile with the desired fidelity the entangling gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The maximum number of layers for the search is heuristically chosen as 2·𝑑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This number was found to be sufficient for generating maximally entangled qudit states from the least powerful operation, acting only on two fixed physical levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The compiler uses a binary search and the number of layers is decreased every time the gate can be decomposed for a certain number of layers under a time period heuristically chosen, otherwise if the target fidelity is not reached or the optimizer does not converge before the end of a timer, the number of layers is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 5 CASE STUDIES The considerations made so far in the paper already showed that the problem of entanglement compilation in high-dimensional systems is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The solution proposed above is supposed to provide a step forward in the field of compilation for qudit sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The implementation is available freely under the MIT licence at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='com/cda-tum/qudit-entanglement-compilation as part of the Munich Quantum Toolkit (MQT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' It is completely written in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='8, with exception of the external dependencies Numpy [43] and Scipy [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This section provides corresponding case studies, in order to demonstrate the feasibility of the workflow and its use- fulness in compiling any multi-level entangling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The use cases focus on the compilation of the controlled-sum gate CSUM, a characteristic operation in qudit systems as discussed in the previ- ous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Although constituted by two compilation steps, the computa- tional complexity of the workflow is dictated on a high level by three routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The first routine is the QR decomposition applied on the unitary to compile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' the algorithm has quadratic complexity in the number of dimension of the two-qudit interaction with size 𝐷, therefore O(𝐷2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The result is a sequence of, in the worst case, 𝐷2 operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The second routine is the translation of every output operation in term of local operations and CEX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' this algorithm has complexity O(𝐷2), because linear in the number of output oper- ations of the QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Finally, the last routine of the workflow is the solution by optimization of the ansatz, which has complexity of O(log 𝐷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In fact, the last compilation step is a binary search, with every step of duration linear in the number of dimensions of the original unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The evaluations were performed on a server running GNU/Linux using an AMD Ryzen 9 3950X and 128 GiB main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The layered compilation step is performed under a time limit in hours heuristically chosen as 𝑑/4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=', 4 h for a two-ququarts unitary), at each step in the binary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The compiler has minimum target infidelity (1 − Fidelity) of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Several runs of the optimization algorithm could lead to better results and, beyond dimension 16, the computational time and the fidelity are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The considered example follows the default strategies encoded in the compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Every controlled rotation and partial swap is trans- formed by local rotations into 𝑐𝑅𝑜𝑡1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1 and for the partial swaps 𝑝𝑆𝑤𝑎𝑝0,1, while the automatic choice for generating entanglement is the controlled exchange gates CEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' More precisely, CEX1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='0,1 has control on the first level of the first qudit and the targets on the first two levels of the second qudit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The designer can decide to follow a different path and impose a decomposition for which they have a specific implementation in terms of the entangling gate involved, or compile it directly for the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In the studies, we considered the implementation of the CSUM gate for qutrits and ququarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Table 1 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Between the near-term usable physical qudit dimensions [7], the solution Table 1: Results of the workflow on CSUM for dimension 9 and dimension 16 System Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' cRot pSwap CEX𝑡𝑜𝑡 MS𝑡𝑜𝑡 LS𝑡𝑜𝑡 (1 − 𝐹) two qutrits 9 44 24 184 1472 368 < 10−4 two ququarts 16 92 36 328 9184 10168 10−3 ∼ 10−4 for ququarts proves an already difficult and useful design case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The columns denote the dimension (“Dim.”), followed by the number of controlled rotations, and partial swaps, as well as the total number of CEX, MS, and LS gates required for the decomposition, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The last column gives a bound on the achieved infidelity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=', 1 − Fidelity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In a first phase the synthesis of CSUM for 9 di- mensions (two qutrits) requires 20 controlled rotations and 6 partial swaps, while the synthesis on two ququarts (dimension 16) outputs 42 controlled rotations and 6 partial swaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' These decompositions are, by construction, not necessarily optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The controlled rota- tions are further decomposed into 2 CEXs and the partial swap into 4 CEXs, with the resulting sequence being general for all dimen- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In order to show to the adaptability of the workflow, in the smaller case we compile the entangling gate in terms of two native gates used in the latest ion trap qudit processors [7] [25] and in- troduced before, the MS and LS gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The manual decomposition of the CEX gate dedicated for the two-qutrit case by itself is non- trivial, even for an experienced quantum information specialists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The inherent difficulty arises not only from the amount of intuition needed, but also from the inability to easily generalize the single result found to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The pen-and-paper optimized sequence for CEX in dimension 9 is made of two LS gates with angle 𝜋, while two MS gates are also required for performing the same CEX at any dimension, if the sequence is allowed to exploit auxiliary levels [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The layered compiler performs a similar decomposition with the same number of 𝜋-LS gates autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Without auxiliary levels, the compiler outputs a sequence of 8 MS gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' For both the compilation results, the infidelity reached is below 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In the case of two-ququarts, optimized sequences could not be achieved, while the layered compiler autonomously decomposes the CEX with infidelity between 10−3 and 10−4, at the cost of 28 MS gates and in alternative of 31 LS gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The number of gates required for a two-ququarts circuit gives a feeling of the increased complexity of the problem compared to two-qutrits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The results are in line with the expected automation overhead in a first automatic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In fact, it was predictable that the used ansatz would ask for an over-parametrization [44], or using more layers than necessarily, which allows for a great simplification of the landscapes by eliminating spurious local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Although the re- sults are not expected be efficient to the point of practical applicabil- ity on a quantum system, this is only one component of a complete compiler for high-dimensional and mixed-dimensional systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' future developments will comprise optimization routines[45]–[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In summary, the results show the feasibility of the proposed workflow to deliver decompositions efficiently for any two-qudit unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Further, the results provide evidence compilation of entan- gling operations for multi-level systems is not easy, especially com- pared to binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' This work investigates the first key step towards full automation for qudits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The implementation is available under the MIT license at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='com/cda-tum/qudit-entanglement- compilation as part of the Munich Quantum Toolkit (MQT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' 6 CONCLUSION The challenges for efficient and reliable application of entangling gates inside qudit circuits arise because of the complexity in under- standing the amount of entanglement generated by an operation Compilation of Entangling Gates for High-Dimensional Quantum Systems ASP-DAC ’23, January 16–19, 2023, Tokyo, Japan as well as the corresponding affected levels—commonly referred to as structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' Consequently, the operations are mostly consid- ered as black-boxes with the question of whether it is possible to reach an implementation for a certain qudit platform and with a given gate-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' So far, the study of feasibility and the implementa- tion of entangling operations for specific qudit technologies was performed by quantum information specialists manually, without the promise of re-utilizing a particular decomposition for a system certain number of dimension, to those of greater dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In this paper, we introduced a complete workflow for compiling any two-qudit unitary into any target gate set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' While the resulting gate sequences are typically not optimal in terms of gate count, the method is computationally efficient, since the only step involving numerical optimization can be pre-computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The case studies confirm the feasibility of the workflow as well, for the first time, the automated results of compilation of generic entangling operations to any dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' The implementation is available under the MIT license at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content='com/cda-tum/qudit-entanglement-compilation under the ensemble of the Munich Quantum Toolkit (MQT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' In the future, the proposed approach may be improved up on by utilizing auxiliary qudit levels, alternative ansatz designs, compression of synthesis results, and circuit optimization in post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the ERC Consolida- tor Grant (agreement No 101001318), the Marie Skłodowska-Curie Grant (agreement No 840450), and the NeQST Grant (agreement No 101080086).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE2T4oBgHgl3EQf1wjo/content/2301.04155v1.pdf'} +page_content=' It is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus and was partially supported by the BMK, BMDW, and the State of Upper Austria in the frame of the COMET program (managed by 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0000000000000000000000000000000000000000..92e86b52d90bdde1cc0d6c3a5c3abd9edbc90c06 --- /dev/null +++ b/vNE3T4oBgHgl3EQfkgr7/content/tmp_files/2301.04599v1.pdf.txt @@ -0,0 +1,16770 @@ +arXiv:2301.04599v1 [math.AP] 11 Jan 2023 +UNIFORM IN GRAVITY ESTIMATES FOR 2D WATER WAVES +SIDDHANT AGRAWAL +Abstract. We consider the 2D gravity water waves equation on an infinite domain. We +prove a local wellposedness result which allows interfaces with corners and cusps as initial +data and which is such that the time of existence of solutions is uniform even as the gravity +parameter g → 0. For g > 0, we prove an improved blow up criterion for these singular +solutions and we also prove an existence result for g = 0. Moreover the energy estimate +used to prove this result is scaling invariant. +As an application of this energy estimate, we then consider the water wave equation +with no gravity where the fluid domain is homeomorphic to the disc. We prove a local +wellposedness result which allows for interfaces with angled crests and cusps as initial data +and then by a rigidity argument, we show that there exists initial interfaces with angled +crests for which the energy blows up in finite time, thereby proving the optimality of this +local wellposedness result. For smooth initial data, this local wellposedness result gives a +longer time of existence as compared to previous results when the initial velocity is small +and we also improve upon the blow up criterion. +Contents +1 +Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +2 +Notation, preliminaries and equations of motion . . . . . . . . . . . . . +5 +2.1 +Notation for the unbounded domain case . . . . . . . . . . . . . . . . . . +5 +2.2 +The system of equations on the boundary for the unbounded domain case +8 +2.3 +Notation for the bounded domain case +. . . . . . . . . . . . . . . . . . . +9 +2.4 +The system of equations on the boundary for the bounded domain case . +13 +3 +Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3.1 +The unbounded domain case . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3.2 +The bounded domain case +. . . . . . . . . . . . . . . . . . . . . . . . . . +19 +4 +Proof of main results for unbounded domain case . . . . . . . . . . . . +24 +4.1 +Some useful identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +4.2 +The main equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.3 +Quantities controlled by the energy Ea(t) . . . . . . . . . . . . . . . . . . +29 +4.4 +Quantities controlled by the energy E(t) +. . . . . . . . . . . . . . . . . . +50 +4.5 +Closing the energy estimate +. . . . . . . . . . . . . . . . . . . . . . . . . +57 +4.6 +Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +5 +Proof of the main results for bounded domain case . . . . . . . . . . . +85 +5.1 +Derivation of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +5.2 +The a priori estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +93 +5.3 +Proof of Theorem 3.6 and Theorem 3.8 . . . . . . . . . . . . . . . . . . . +99 +1 + +2 +SIDDHANT AGRAWAL +6 +Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 +References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 +1. Introduction +We are concerned with the motion of a fluid in dimension two with a free boundary. In +this work we will identify 2D vectors with complex numbers. The fluid is assumed to be +inviscid, incompressible and irrotational. The fluid domain Ω(t) ⊂ C and the air region +are separated by an interface ∂Ω(t). The air and the fluid are assumed to have constant +densities of 0 and 1 respectively. We will consider two different models: +In the first model we assume that the interface ∂Ω(t) is homeomorphic to R and tends +to real line at infinity. The fluid is below the air region and there is no bottom. The fluid is +subject to a uniform gravitational field −gi acting in the downward direction (here g ≥ 0). +The motion of the fluid is then governed by the Euler equation +vt + (v.∇)v = −gi − ∇P +on Ω(t) +div v = 0, +curl v = 0 +on Ω(t) +P = 0 +on ∂Ω(t) +(1, v) is tangent to the free surface (t, ∂Ω(t)) +(1) +along with the decay conditions v → 0, ∇P → −gi as |(x, y)| → ∞. We will consider this +problem for all values of the gravity parameter g ≥ 0. +For the second model we will consider the problem when the domain Ω(t) is bounded +and is homeomorphic to the unit disc D. In this case we assume that there is no gravity. +With these assumptions, the equations become +vt + (v.∇)v = −∇P +on Ω(t) +div v = 0, +curl v = 0 +on Ω(t) +P = 0 +on ∂Ω(t) +(1, v) is tangent to the free surface (t, ∂Ω(t)) +(2) +The earliest results on local well-posedness for the Cauchy problem are for small data in +2D and were obtained by Nalimov [26], Yoshihara [36, 37] and Craig [16]. In the case of zero +surface tension and gravity g = 1, Wu [31, 32] obtained the proof of local well-posedness +for arbitrary data in Sobolev spaces. This result has been extended in various directions, +see the works in [12, 24, 23, 15, 38, 11, 6, 22, 20, 19, 7, 17, 3, 5, 4, 8]. +An important quantity related to the well-posedness of the problem in the zero surface +tension case is the Taylor sign condition proposed in [29]. This says that there should exist +a constant c > 0 such that +−∂P +∂n ≥ c > 0 +on ∂Ω(t) +where n is the outward unit normal. In [18], Ebin gave an example of initial data with non- +zero vorticity not satisfying the Taylor sign condition, for which the problem is ill posed. + +2D WATER WAVES +3 +In [9] the authors proved that the linearized problem around a solution is wellposed if the +Taylor sign condition is satisfied. In [31] for gravity g = 1, Wu proved that the Taylor sign +condition is satisfied for the infinite bottom case if the interface is C1,α for α > 0. This was +later shown to be true for flat bottoms and with perturbations to flat bottom by Lannes +[23]. See also [20, 28]. +In [11] the authors proved the existence of splash and splat singularities for the water +wave equation. In order to study solutions with non C1 interfaces, Kinsey and Wu [21] +proved an a priori estimate for angled crested water waves in the case of zero surface +tension. Using this, Wu [35] proved a local wellposedness result which allows the initial +data to have interfaces with corners and cusps. In [1] the author proved that the singular +solutions constructed in [35] are rigid and in particular the angle of the corner does not +change with time. In a recent work [14] for a zero gravity model, the authors construct +solutions with interfaces which have corners and cusps, with the property that the angle +of the corner changes with time. Note that at the corners and cusps in [35, 14] the Taylor +sign condition is not satisfied and one has − ∂P +∂n = 0 at those singularities. +In this paper we explore the question of local wellposedness when the gravity parameter +g → 0. For example consider the following question: For some fixed initial data, does one +have a uniform time of existence for the solutions as g → 0? All the above results give a +time of existence T → 0 as g → 0. We now give a heuristic argument to illustrate one of +the main difficulties of this problem. Consider the following simplified model of the water +wave equation +D2 +t f + +� +−∂P +∂n +� +|∂α′|f = l.o.t +Here Dt is the material derivate, l.o.t are lower order terms and f is some variable such as +the velocity on the interface. To prove energy estimates for this equation, one can naturally +take the following energy +E(t) = +� +|Dtf|2 + +� � +−∂P +∂n +����|∂α′| +1 +2 f +��� +2 +and take the time derivative to get +d +dtE(t) ≈ +� +Dt +� +−∂P +∂n +����|∂α′| +1 +2 f +��� +2 ++ errors +(3) +Now if g > 0 and fixed, and the interface is C1,α, then as shown by Wu [31], we have a +positive lower bound on − ∂P +∂n and hence the energy E(t) controls the ˙H +1 +2 norm of f. Then +one shows that we can control +��Dt +� +− ∂P +∂n +��� +∞ from the energy and hence the first term on +the right hand side of (3) is controlled. +Now as we show in §3.1, − ∂P +∂n → g at infinity. So as g → 0, the energy E(t) no longer +controls the +˙H +1 +2 norm of f uniformly. +In the extreme case of g = 0, we show that if +the interface is C1,α and the velocity is in H1 and is not identically zero, then − ∂P +∂n > 0 +everywhere but − ∂P +∂n → 0 at infinity. Hence to control the first term of (3), it is no longer + +4 +SIDDHANT AGRAWAL +enough to control +��Dt +� +− ∂P +∂n +��� +∞. To solve this issue, we prove the following new estimate +����� +Dt +� +− ∂P +∂n +� +� +− ∂P +∂n +� +����� +∞ +≲ +����� +Zt,α′ +Z,α′ +����� +L∞∩ ˙H +1 +2 ++ +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +Here Z and Zt are the interface and the velocity on the boundary in conformal coordinates. +Note that the term +Zt,α′ +Z,α′ is nothing but the boundary value of ∂zv. This estimate shows +that if one has control of the right hand side of the above inequality, then Dt +� +− ∂P +∂n +� +decays +at least as fast as +� +− ∂P +∂n +� +at infinity. This estimate is in fact also scaling invariant with +respect to the 2 parameter family of scaling transformations (28) and the only term on the +right hand side which controls the interface namely +���∂α′ +1 +Z,α′ +��� +2, allows corners of angle less +than 90◦ and cusps. +Using this and several other similar estimates, we prove an energy estimate which works +uniformly in gravity. Let us now summarize the main results of §3.1 which are the results +for model (1): +(1) We construct an energy E(t) which is uniform in gravity and prove an a priori +estimate for it in Theorem 3.1. The estimate is scaling invariant with respect to the +2 parameter family of scaling transformations (28) and allows both smooth initial +data and initial interfaces with angled crests and cusps. +(2) In Theorem 3.3, we prove a local wellposedness result which allows for both smooth +initial data and initial data with interfaces having angled crests and cusps. The +time of existence of solutions is shown to be uniform in gravity g. The result also +establishes an existence result for g = 0. We also prove a blow up criterion which +is an improvement from previous one in [35] for these singular solutions. +By modifying the energy estimate for the energy E(t), we then also prove an energy estimate +for the model (2). Let us now summarize the main results of §3.1 which are the results for +model (2): +(1) Similar to Theorem 3.3, we prove a local wellposedness result Theorem 3.6 which +allows for both smooth initial data and initial data with interfaces having angled +crests and cusps. For smooth initial data, this result gives a longer time of existence +as compared to previous results in the case where the initial velocity is small. We +also prove a blow up criterion which improves upon previous blow up criterions for +both smooth and singular solutions. +(2) Using a rigidity argument Theorem 3.7, we prove that there exists initial data with +singular interfaces for which the energy blows up in finite time. This shows the +optimality of the local wellposedness result Theorem 3.6. +This also shows that +the time of existence obtained in Theorem 3.6 is optimal in a suitable sense. This +blow up result does not say anything about blow up for smooth solutions. We now +remark that if one were to prove a local wellposedness result in an exactly analogous +manner to [35] without using our new energy estimate, then by the same rigidity +argument one can indeed show that there is a finite maximal time of existence for +these singular solutions. However what is not clear is whether the energy blows up + +2D WATER WAVES +5 +or not at the maximal time. The blow up criterion of Theorem 3.6, which is proved +using the new energy estimate, ensures that the energy indeed blows up. See §3.2 +for more details. +In this paper we follow the framework of Wu [35]. We also use several identities and +estimates proved in [2]. +The paper is organized as follows: In §2 we introduce the notation and write down the +system of equations in conformal coordinates. In §3 we explain our main results in detail. +In §4 we prove the results for the unbounded domain case stated in §3.1 and in §5 we prove +the results for the bounded domain case stated in §3.2. Finally in §6 we collect some of the +identities and estimates used throughout the paper. +Acknowledgment: The author was supported by the National Science Foundation un- +der Grant No. DMS-1928930 while participating in a program hosted by MSRI during the +Spring 2021 semester. The author also received funding from the European Research Coun- +cil (ERC) under the European Union’s Horizon 2020 research and innovation programme +through the grant agreement 862342. +2. Notation, preliminaries and equations of motion +In this paper we write a ≲ b if there exists a universal constant C > 0 so that a ≤ Cb. +We write a ≲M b if there exits a constant C(M) depending only on M so that a ≤ C(M)b. +Similar definitions for a ≲M1,M2 b etc. For singular integrals, all integrals will be understood +in the principle value sense and we will suppress writing p.v. in front of the integrals. +2.1. Notation for the unbounded domain case +In this section we recall the notation used in [2]. The Fourier transform is defined as +ˆf(ξ) = +1 +√ +2π +� +R +e−ixξf(x) dx +We will denote by S(R) the Schwartz space of rapidly decreasing functions and S′(R) is the +space of tempered distributions. A Fourier multiplier with symbol a(ξ) is the operator Ta +defined formally by the relation � +Taf = a(ξ) ˆf(ξ). For s ≥ 0 the operators |∂α′|s and ⟨∂α′⟩s +are defined as the Fourier multipliers with symbols |ξ|s and (1 + |ξ|2) +s +2 respectively. The +Sobolev space Hs(R) for s ≥ 0 is the space of functions with ∥f∥Hs = ∥⟨∂α′⟩sf∥L2(dx) < ∞. +The homogenous Sobolev space +˙H +1 +2(R) is the space of functions modulo constants with +∥f∥ ˙H +1 +2 = ∥|∂α′| +1 +2 f∥L2(dx) < ∞. +From now on compositions of functions will always be in the spatial variables. +We +write f = f(·, t), g = g(·, t), f ◦ g(·, t) := f(g(·, t), t). Define the operator Ug as given by +Ugf = f ◦ g. Observe that UfUg = Ug◦f. Let [A, B] := AB − BA be the commutator of +the operators A and B. If A is an operator and f is a function, then (A + f) will represent +the addition of the operators A and the multiplication operator Tf where Tf(g) = fg. We +denote the convolution of f and g by f ∗ g. We will denote the spacial coordinates in Ω(t) +with z = x + iy, whereas z′ = x′ + iy′ will denote the coordinates in the lower half plane + +6 +SIDDHANT AGRAWAL +P− = +� +(x, y) ∈ R2 �� y < 0 +� +. As we will frequently work with holomorphic functions, we will +use the holomorphic derivatives ∂z = 1 +2(∂x − i∂y) and ∂z′ = 1 +2(∂x′ − i∂y′). In this paper +all norms will be taken in the spacial coordinates unless otherwise specified. For example +for a function f : R × [0, T] → C we write ∥f∥2 = ∥f(·, t)∥2 = ∥f(·, t)∥L2(R,dα′). Also for a +function f : P− → C we write supy′<0∥f∥L2(R,dx′) = supy′<0∥f(·, y′)∥L2(R,dx′). The Poisson +kernel is given by +Kǫ(x) = +ǫ +π(ǫ2 + x2) +for ǫ > 0 +(4) +Let the interface be parametrized in Lagrangian coordinates by z(·, t) : R → Σ(t) satis- +fying zα(α, t) ̸= 0 for all α ∈ R. Hence zt(α, t) = v(z(α, t), t) is the velocity of the fluid on +the interface and ztt(α, t) = (vt + (v.∇)v)(z(α, t), t) is the acceleration. +Let Ψ(·, t) : P− → Ω(t) be conformal maps satisfying limz→∞ Ψz(z, t) = 1 and that +limz→∞ Ψt(z, t) = 0. With this, the only ambiguity left in the definition of Ψ is that of the +choice of translation of the conformal map at t = 0, which does not play any role in the +analysis. Let Φ(·, t) : Ω(t) → P− be the inverse of the map Ψ(·, t) and define h(·, t) : R → R +as +h(α, t) = Φ(z(α, t), t) +(5) +hence h(·, t) is a homeomorphism. As we use both Lagrangian and conformal parameter- +izations, we will denote the Lagrangian parameter by α and the conformal parameter by +α′. Let h−1(·, t) be its spacial inverse i.e. +h(h−1(α′, t), t) = α′ +From now on, we will fix our Lagrangian parametrization at t = 0 by imposing +h(α, 0) = α +for all α ∈ R +Hence the Lagrangian parametrization is the same as conformal parametrization at t = 0. +Define the variables +Z(α′, t) = z ◦ h−1(α′, t) +Z,α′(α′, t) = ∂α′Z(α′, t) +Hence +( zα +hα +) ◦ h−1 = Z,α′ +Zt(α′, t) = zt ◦ h−1(α′, t) +Zt,α′(α′, t) = ∂α′Zt(α′, t) +Hence +(ztα +hα +) ◦ h−1 = Zt,α′ +Ztt(α′, t) = ztt ◦ h−1(α′, t) +Ztt,α′(α′, t) = ∂α′Ztt(α′, t) +Hence +(zttα +hα +) ◦ h−1 = Ztt,α′ +Hence Z(α′, t), Zt(α′, t) and Ztt(α′, t) are the parameterizations of the boundary, the veloc- +ity and the acceleration in conformal coordinates and in particular Z(·, t) is the boundary +value of the conformal map Ψ(·, t). Note that as Z(α′, t) = z(h−1(α′, t), t) we see that +∂tZ ̸= Zt. Similarly ∂tZt ̸= Ztt. The substitute for the time derivative is the material + +2D WATER WAVES +7 +derivative. Define: +Dt = material derivative = ∂t + b∂α′ +where b = ht ◦ h−1 +Dα′ = +1 +Z,α′ ∂α′ +Dα′ = +1 +Z,α′ ∂α′ +|Dα′| = +1 +��Z,α′ +��∂α′ +H = Hilbert transform = Fourier multiplier with symbol − sgn(ξ) +Hf(α′) = 1 +iπ p.v. +� +1 +α′ − β′ f(β′) dβ′ +PH = Holomorphic projection = I + H +2 +PA = Antiholomorphic projection = I − H +2 +|∂α′| = iH∂α′ = +√ +−∆ = Fourier multiplier with symbol |ξ| +|∂α′|1/2 = Fourier multiplier with symbol |ξ|1/2 +ω = Z,α′ +��Z,α′ +�� +(6) +Now we have DtZ = Zt and DtZt = Ztt and more generally Dt(f(·, t)◦h−1) = (∂tf(·, t))◦h−1 +or equivalently ∂t(F(·, t) ◦ h) = (DtF(·, t)) ◦ h. This means that Dt = U −1 +h ∂tUh i.e. Dt is +the material derivative in conformal coordinates. +Define U : P − × [0, T) → C and P : P − × [0, T) → R as +U = v ◦ Ψ +P = P ◦ Ψ +(7) +and observe that U(·, t) is a holomorphic function on P−. Also note that its boundary value +is given by Zt(α′, t) = U(α′, t) for all α′ ∈ R. Hence we can write the Euler equations (1) +as equations on P− +Ut − Ψt +Uz′ +Ψz′ + U Uz′ +Ψz′ = − 1 +Ψz′ (∂x′ − i∂y′)P + ig +on P− +U(·, t) is holomorphic +on P− +P = 0 +on ∂P− +Trace of +1 +Ψz′ (U − Ψt) is real valued +on ∂P− +(8) +along with the condition that Ψ(·, t) is conformal and the decay conditions U → 0, (∂x′ − +i∂y′)P → ig, Ψz′ → 1 and Ψt → 0 as z′ → ∞. The condition that the particles on the +boundary stay on the boundary is equivalent to saying that the trace of +1 +Ψz′ (U − Ψt) is +real valued and in particular +� +1 +Ψz′ (U − Ψt) +���� +∂P− = b where Dt = ∂t + b∂α′ is the material +derivative on the boundary ∂P−. Also it should be noted that the process of obtaining +(8) from (1) is reversible so long as the interface ∂Ω(t) = {Z(α′, t) | α′ ∈ R} is non-self +intersecting. See [1] for more details. +The Hilbert transform defined above in (6) satisfies the following property. + +8 +SIDDHANT AGRAWAL +Lemma 2.1 ([30]). Let 1 < p < ∞ and let F : P− → C be a holomorphic function in the +lower half plane with F(z) → 0 as z → ∞. Then the following are equivalent +(1) sup +y<0 +∥F(· + iy)∥p < ∞ +(2) F(z) has a boundary value f, non-tangentially almost everywhere with f ∈ Lp and +H(f) = f. +In particular this says if U decays appropriately at infinity, then the boundary value +of U namely Zt will satisfy HZt = Zt. Similarly as Ψz → 1 as z → ∞, we have that +H +� +1 +Z,α′ − 1 +� += +1 +Z,α′ −1. Also note that as product of holomorphic functions is holomorphic, +we can use the above lemma to conclude that several other functions on the boundary also +satisfy similar identities e.g. H( Zt +Z,α′ ) = +Zt +Z,α′ , H(Dα′Zt) = Dα′Zt etc. +For n ≥ 1 and for functions f1, · · · , fn, g : R → C we define the function [f1, · · · , fn; g] : +R → C as +[f1, · · · , fn; g](α′) = 1 +iπ +� �f1(α′) − f1(β′) +α′ − β′ +� +· · · +�fn(α′) − fn(β′) +α′ − β′ +� +g(β′) dβ′ +(9) +Hence with this notation we see that +[f, H]g = 1 +iπ +� f(α′) − f(β′) +α′ − β′ +g(β′) dβ′ = [f; g] +2.2. The system of equations on the boundary for the unbounded domain case +To solve the system (1), in [2] we obtained a system for the variables (Z,α′, Zt) which we +then solve. The system is as follows: +b = Re(I − H) +� Zt +Z,α′ +� +Ag = g − Im[Zt, H]Zt,α′ +(∂t + b∂α′)Z,α′ = Zt,α′ − bα′Z,α′ +(∂t + b∂α′)Zt = ig − i Ag +Z,α′ +(10) +along with the condition that their harmonic extensions, namely Ψz′(· + iy) = K−y ∗ Z,α′ +and U(· + iy) = K−y ∗ Zt for all y < 0, 1 are holomorphic functions on P− and satisfy 2 +lim +c→∞ sup +|z′|≥c +���Ψz′(z′) − 1 +�� + +��U(z′) +��� += 0 +and +Ψz′(z′) ̸= 0 +for all z′ ∈ P− +After solving the above system one can obtain Z(·, t) by the formula +Z(α′, t) = Z(α′, 0) + +� t +0 +� +Zt(α′, s) − b(α′, s)Z,α′(α′, s) +� +ds +1here K−y is the Poisson kernel (4) +2We observe that for such a Ψz′ we can uniquely define log(Ψz′) : P− → C such that log(Ψz′) is a +continuous function with Ψz′ = exp{log(Ψz′)} and (log(Ψz′))(z′) → 0 as z′ → ∞. + +2D WATER WAVES +9 +and hence (∂t + b∂α′)Z = DtZ = Zt. Hence one can view the system being in variables +(Z, Zt) instead of the variables (Z,α′, Zt). Note that once one has a solution to the above +system, we can recover a solution to the system (8) by letting Ψ and U be defined as above +and defining P as the unique solution to the equation +∆P = −2|Uz′|2 +on P−, +P = 0 +on ∂P− +(11) +along with the condition (∂x′ − i∂y′)P → ig as z′ → ∞. See [35] for the details. +We observe that the above system allows self intersecting interfaces. +However if the +interface is self-intersecting then it becomes nonphysical and so its relation to the Euler +equation (1) is lost. See [2] for more details. +Now to get the function h(α, t) (recall the definition (5)), we solve the ODE +dh +dt = b(h, t) +h(α, 0) = α +(12) +Observe that as long as sup[0,T]∥bα′∥∞(t) < ∞ we can solve this ODE uniquely and for any +t ∈ [0, T] we have that h(·, t) is a homeomorphism. Hence it makes sense to talk about the +functions z = Z ◦h, zt = Zt ◦h which are Lagrangian parameterizations of the interface and +the velocity on the boundary. We also note that the last equation in (10) can be written as +Ztt − ig = −i Ag +Z,α′ +(13) +We note here that from the calculations in [31], we see that the gradient of the pressure on +the boundary in conformal coordinates is (here ˆn is the outward unit normal) +−∂P +∂ˆn ◦ h−1 = +Ag +��Z,α′ +�� +(14) +2.3. Notation for the bounded domain case +Let the unit disc be D = +� +(x, y) ∈ R2 �� x2 + y2 < 1 +� +and let S1 = ∂D. In the following we +will identify functions f : S1 → C with their pullbacks ˜f : R → C, where ˜f(α′) = f(eiα′). +We will frequently abuse notation and for a function f : S1 → C we will usually write f(α′) +instead of f(eiα′). In particular if there exists a function F : D → C whose boundary value +is f : S1 → C, then by abuse of notation we will also say that the boundary value of F is ˜f. +We will denote the boundary value of F by Tr(F). If g : R → C is a 2π periodic function, +then in this paper whenever we use the Lp or Sobolev norms of g, what we mean is that we +are computing the norms by looking at g as a function on S1 and not as a function on R. +We define the Fourier transform for a function f : S1 → C as +ˆf(n) = 1 +2π +� 2π +0 +f(α′)e−inα′ dα′ + +10 +SIDDHANT AGRAWAL +and the inverse Fourier transform as +f(α′) = +∞ +� +n=−∞ +ˆf(n)einα′ +The Lp norm of f is defined as +∥f∥p = +�� 2π +0 +|f(α′)|p dα′ +� 1 +p +and the Sobolev norms of f are defined in the same way as in §2.1. Hence in particular we +observe that +∥f∥2 +˙H +1 +2 = +��|∂α′| +1 +2f +��2 +2 = +� 2π +0 +f|∂α′|f dα′ +Similar to §2.1, compositions of functions are again only taken in spacial coordinates and +we maintain the notation for the operator Ug and convolution of functions. We keep the +notation z = x + iy for z ∈ Ω(t) and z′ = x′ + iy′ for z′ ∈ D. We have the same definitions +for ∂z, ∂z′ and we also suppress the time variables when writing norms. The Poisson kernel +for the disc is given by +Pr(θ) = +1 − r2 +1 − 2r cos(θ) + r2 +for 0 ≤ r < 1 +Let the interface be parametrized in Lagrangian coordinates by a 2π periodic counter- +clockwise parametrization z(·, t) : R → ∂Ω(t) satisfying zα(α, t) ̸= 0 for all α ∈ R. Hence +we see that zt(α, t) = v(z(α, t), t) is the velocity of the fluid on the interface and ztt(α, t) = +(vt + (v.∇)v)(z(α, t), t) is the acceleration. +We fix a point x0 ∈ Ω(0) and let X(x0, t) be the Lagrangian trajectory of this particle. +Hence ∂tX(x0, t) = v(X(x0, t), t). Let Ψ(·, t) : D → Ω(t) be conformal maps satisfying +Ψ(0, t) = X(x0, t) and Ψz′(0, t) > 0. Let Φ(·, t) : Ω(t) → D be the inverse of the map Ψ(·, t) +and observe that Φ(z(α, t), t) ∈ ∂D. Define h(·, t) : R → R as +h(α, t) = −i log(Φ(z(α, t), t)). +(15) +where the ambiguity in the definition of log(z) is removed by ensuring that h is a continuous +function on R × [0, T) and that h(0, 0) ∈ [0, 2π). From this it is easy to see that h(·, t) is an +increasing function and is a homeomorphism on R satisfying h(α+2π, t) = h(α, t)+2π. As +we use both Lagrangian and conformal parameterizations, we will denote the Lagrangian +parameter by α and the conformal parameter by α′. Let h−1(·, t) be its spacial inverse i.e. +h(h−1(α′, t), t) = α′ +From now on, we will fix our Lagrangian parametrization at t = 0 by imposing +h(α, 0) = α +for all α ∈ R +Hence the Lagrangian parametrization is the same as conformal parametrization at t = 0. +The functions Z, Zt, Ztt are defined as in §2.1. Hence the functions Z(·, t) : R → ∂Ω(t) +and Zt(·, t), Ztt(·, t) : R → C are 2π periodic functions and are the parameterizations of +the boundary, the velocity and the acceleration in conformal coordinates. In particular we + +2D WATER WAVES +11 +have Z(α′, t) = Ψ(eiα′, t). We define the material derivative Dt and the weighted derivatives +Dα′, Dα′, |Dα′| in the same way as in §2.1. The variable ω is also defined in the same way. +Define +Av(f) = 1 +2π +� 2π +0 +f(β′) dβ′ +(�Hf)(α′) = +1 +2πi +� 2π +0 +f(β′) cot +�β′ − α′ +2 +� +dβ′ +Observe that we have +Av(f) = ˆf(0) +� +(�Hf)(n) = sgn(n) ˆf(n) +|∂α′| = −i�H∂α′ +where +sgn(n) = + + + + + +1 +if n ≥ 1 +0 +if n = 0 +−1 +if n ≤ −1 +We define the Hilbert transform as +H = �H + Av +The projection operators PH and PA are defined in the same way as in §2.1. Using the +identity +eiα′ − eiβ′ = 2iei +� +α′+β′ +2 +� +sin +�α′ − β′ +2 +� +(16) +it is easily seen that +(Hf)(α′) = +1 +2πi +� 2π +0 +f(β′) cot +�β′ − α′ +2 +� +dβ′ + 1 +2π +� 2π +0 +f(β′) dβ′ += e−i α′ +2 +2πi +� 2π +0 +f(β′) +sin +� +β′−α′ +2 +�ei β′ +2 dβ′ += 1 +iπ +� 2π +0 +f(β′) +eiβ′ − eiα′ ieiβ′ dβ′ +The Hilbert transform H defined above has the following property similar to Lemma 2.1 +and is proved in a similar manner. +Lemma 2.2. Let 1 < p < ∞ and let F : D → C be a holomorphic function. Then the +following are equivalent +(1) sup0 0. +The con- +dition that the particles on the boundary stay on the boundary is equivalent to saying +that the trace of +−i +z′Ψz′ (U − Ψt) is real valued and in particular from §5.1 we see that +� +−i +z′Ψz′ (U − Ψt) +���� +∂D = b where Dt = ∂t + b∂α′ is the material derivative on the boundary +∂D. Again note that the process of obtaining (17) from (2) is reversible so long as the +interface ∂Ω(t) is non-self intersecting. +For f1, f2, f3, g : S1 → C, we define the following functions +[f1; g](α′) = ([f1, �H]g)(α′) = +1 +2πi +� 2π +0 +(f1(α′) − f1(β′)) cot +�β′ − α′ +2 +� +g(β′) dβ′ +(18) +[f1, f2; g](α′) = − 1 +4πi +� 2π +0 +� +f1(α′) − f1(β′) +sin +�β′−α′ +2 +� +�� +f2(α′) − f2(β′) +sin +� β′−α′ +2 +� +� +g(β′) dβ′ +(19) +and +[f1, f2, f3; g](α′) += +1 +8πi +� 2π +0 +� +f1(α′) − f1(β′) +sin +� β′−α′ +2 +� +�� +f2(α′) − f2(β′) +sin +�β′−α′ +2 +� +�� +f3(α′) − f3(β′) +sin +� β′−α′ +2 +� +� +cos +�β′ − α′ +2 +� +g(β′) dβ′ +(20) + +2D WATER WAVES +13 +2.4. The system of equations on the boundary for the bounded domain case +Similar to §2.2, to solve (17) we obtain a system of equations on the boundary of the +domain in the variables (Z,α′, Zt). We derive these equations in §5.1. The equations are: +b = Re(I − �H) +�Zt − Av(Zt) +Z,α′ +� +A0 = Im +� +Zt, �H +� +Zt,α′ +(∂t + b∂α′)Z,α′ = Zt,α′ − bα′Z,α′ +(∂t + b∂α′)Zt = i A0 +Z,α′ +(21) +along with the condition that the functions +Ψz′(reiθ, t) = 1 +2π +� 2π +0 +Pr(θ − β′) +� +−ie−iβ′Z,α′(β′, t) +� +dβ′ +U(reiθ, t) = 1 +2π +� 2π +0 +Pr(θ − β′)Zt(β′, t) dβ′ +are holomorphic functions on D and satisfy for t ≥ 0 +Ψz′(0, t) > 0 +and +Ψz′(z′, t) ̸= 0 +for all z′ ∈ D +We note that one can obtain Z(·, t) by the formula +Z(α′, t) = Z(α′, 0) + +� t +0 +� +Zt(α′, s) − b(α′, s)Z,α′(α′, s) +� +ds +In particular instead of the variables (Z,α′, Zt), one can view the system being in the +variables (Z, Zt). Similar to the unbounded case we note that once one has a solution to +the above system, we can recover a solution to the system (17) by letting Ψ and U be +defined as above and defining P as the unique solution to the equation +∆P = −2|Uz′|2 +on D, +P = 0 +on ∂D +(22) +Similar to the unbounded case, we again observe that the above system allows for the +interface to self intersect. Also similarly, we can recover the function h(α, t) as the solution +to the ODE +dh +dt = b(h, t) +h(α, 0) = α +where b is given by (21). From this we easily see that as long as sup[0,T]∥bα′∥∞(t) < ∞ we +can solve this ODE and for any t ∈ [0, T] we have that h(·, t) is a homeomorphism. Also +using the fact that b(·, t) is a 2π periodic function, it is also easy to see that h satisfies +h(α + 2π, t) = h(α, t) + 2π. Hence the functions z = Z ◦ h, zt = Zt ◦ h are 2π periodic and + +14 +SIDDHANT AGRAWAL +are the Lagrangian parameterizations of the interface and the velocity on the boundary. +We also note that the last equation in (21) can be written as +Ztt = i A0 +Z,α′ +(23) +From the calculations from §5.1, we see that the gradient of the pressure on the boundary +in conformal coordinates is (here ˆn is the outward unit normal) +−∂P +∂ˆn ◦ h−1 = +A0 +��Z,α′ +�� +(24) +3. Main results +3.1. The unbounded domain case +In this subsection we collect our main results for the model (8) which is the same as (1) +but written in conformal coordinates. We define the energies +E1(t) = +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +2 +E2(t) = +���Zt,α′ +��2 +2 + g +� + + + +����Dt∂α′ +1 +Z,α′ +���� +2 +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + +E3(t) = +���Zt,α′ +��2 +2 + g +� + + + +��DtD2 +α′Zt +��2 +2 + +����� +�Ag +Z,α′ D2 +α′Zt +����� +2 +˙H +1 +2 + + + +(25) +and also define +Ea(t) = +� +E1(t)2 + E2(t) +� 1 +2 +(26) +E(t) = +� +E1(t)3 + E2(t) +3 +2 + E3(t) +� 1 +3 +(27) +Note that Ea(t) ≲ E(t). To understand the scaling properties of the energy, recall that if +(Z(α′, t), Zt(α′, t)) is a solution to the gravity water wave equations (10) in the time interval +[0, T] with gravity parameter g ≥ 0 (we assume surface tension σ = 0), then for any λ > 0 +and s ∈ R, the functions ((Zλ)(α′, t), (Zt)λ(α′, t)) given by +(Zλ)(α′, t) = λ−1Z(λα′, λst), +(Zt)λ(α′, t) = λs−1Zt(λα′, λst) +(28) +is a solution to the gravity water wave equation in the time interval [0, λ−sT] with gravity +gλ = λ2s−1g 3. Note that if g > 0, then s = 1 +2 keeps gλ = g and if g = 0, then for all s ∈ R +we have gλ = g = 0. In this paper when we talk about scaling transformations, we will +consider all transformations of the type (28) and not restrict ourselves to the case where +gλ = g, as we are interested in the entire family of equations (10) with all possible values +of g ≥ 0. +3More generally if one had σ ̸= 0, then σλ = λ2s−3σ. + +2D WATER WAVES +15 +If (Ea)λ(t) and Eλ(t) are the energies defined by (26) and (27) respectively for the solution +((Zλ)(α′, t), (Zλ)t(α′, t)) with gravity gλ, then it is easy to see that +(Ea)λ(t) = λ2sEa(λst) +and +Eλ(t) = λ2sE(λst) +(29) +Hence both Ea(t) and E(t) scale like ∥∇u∥2 +∞(t) under all the scaling transformations (28). +Similarly the energies E1(t), E2(t) and E3(t) scale like ∥∇u∥2 +∞(t), ∥∇u∥4 +∞(t) and ∥∇u∥6 +∞(t) +respectively under these scalings. +We now state our first main result which is an a priori estimate for the energies Ea(t) +and E(t). +Theorem 3.1. Let T > 0 and let (Z, Zt)(t) be a solution to the gravity water wave equation +(10) with gravity parameter g ≥ 0 in the time interval [0, T) with (Z,α′ − 1, +1 +Z,α′ − 1, Zt) ∈ +L∞([0, T], Hs(R) × Hs(R) × Hs+ 1 +2 (R)) for some s ≥ 4. Then E(t) < ∞ for all t ∈ [0, T) +and there exists a universal constant c > 0 so that for all t ∈ [0, T) we have +dEa(t) +dt +≤ c +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Ea(t) ≤ c2Ea(t)3/2 +(30) +and also +dE(t) +dt +≤ cEa(t) +1 +2E(t) ≤ c2E(t)3/2 +(31) +We prove this theorem in §4, from §4.1 to §4.5. Let us now highlight some salient features +about the energies Ea(t) and E(t) and the above result: +(1) The energies Ea(t) and E(t) are uniform in gravity. This is because the only place +where the gravity parameter g shows up in these energies is in the term ( +��Zt,α′ +��2 +2+g) +and in the variable Ag. The variable Ag = g + A0 where A0 ≥ 0 and is independent +of g (see (45) and (46)). The above a priori estimate works for all g ≥ 0, in particular +even for g = 0, and the estimates are uniform in gravity. See Corollary 3.2 below. +(2) The energies Ea(t) and E(t) allow both smooth solutions and solutions with angled +crests/cusps similar to [35]. More precisely they allow interfaces with corners of +angle νπ for 0 < ν < 1 +2 and for cusps (which corresponds to ν = 0). To see this, +it is easier to look at the energy E(t) defined below and observe from Lemma 4.5 +that the energy E(t) dominates the energy E(t). Then from [1] we easily see that +for interfaces with angled crests/cusps, we have that the energy E(t) is finite, which +implies that the energy E(t) is also finite. +(3) From (14) we see that the gradient of the pressure on the boundary is given by +−∂P +∂ˆn ◦ h−1 = +Ag +��Z,α′ +�� +(32) +Now from Lemma 6.10 and the formula (46), we see that if Zt,α′(·, t) ∈ L2 then +A0(α′, t) → 0 as |α′| → ∞. Also as Z,α′(·, t) → 1 at infinity, we see that − ∂P +∂ˆn ◦ +h−1(α′, t) → g as |α′| → ∞. If g = 0, the velocity Zt(·, t) is not a constant function +and satisfies Zt,α′(·, t) ∈ L2 and if the interface is C1,α (to make sure that +��Z,α′ +�� is + +16 +SIDDHANT AGRAWAL +bounded), then − ∂P +∂ˆn ◦ h−1(α′, t) > 0 for all α′ ∈ R but − ∂P +∂ˆn ◦ h−1(α′, t) → 0 as +|α′| → ∞. +Hence if g → 0, one does not have a uniform positive lower bound on the Taylor +sign condition term (32), even for smooth decaying initial data. On the other hand +if the interface has a corner or cusp of angle νπ for 0 ≤ ν < +1 +2, then one has +1 +|Z,α′| = 0 at that point and as Ag ∈ L∞ (from the assumption Zt,α′ ∈ L2), we see +that − ∂P +∂ˆn ◦h−1 = 0 at the singularity, for any value of g. Hence in this case as well, +we also don’t have the Taylor sign condition satisfied. One of the main features +of the above a priori estimate is that it makes no assumptions on the Taylor sign +condition term and works in both cases mentioned above. +(4) The energy estimates (30) and (31) are invariant under all the scaling transforma- +tions (28). Scaling invariant estimates for g > 0 and s = 1 +2 were first obtained in +[20] (see also [5]). The estimates (30) and (31) are invariant not only for s = 1 +2 but +for all s ∈ R. +An important consequence of the above a priori estimate is the following: +Corollary 3.2. Assume that the initial data (Z, Zt)(0) satisfies (Z,α′ − 1, +1 +Z,α′ − 1, Zt)(0) ∈ +Hs(R) × Hs(R) × Hs+ 1 +2(R) for some s ≥ 4 and fix g0 > 0. Then for 0 < g ≤ g0, there +exists a time T > 0 independent of g, such that on [0, T] the initial value problem for +(10) with gravity g has a unique solution (Z, Zt)(t) satisfying (Z,α′ − 1, +1 +Z,α′ − 1, Zt) ∈ +Cl([0, T], Hs−l(R) × Hs−l(R) × Hs+ 1 +2−l(R)) for l = 0, 1. +This result shows that the time of existence of the solutions is uniform even as g → 0. +In all previous results, for the type of initial data taken above, one would get that the time +of existence T → 0 as g → 0. Note that for periodic boundary conditions, this result was +already known as in that case, one does have a positive lower bound on − ∂P +∂ˆn ◦ h−1 for +g = 0 (assuming the initial velocity is not identically zero). This lower bound in the case +of periodic solutions is due to compactness. +More generally, using the a priori estimates of Theorem 3.1, one can prove a local well- +posedness result for initial data which allows for interfaces with angled crests and cusps +(and also allows smooth initial data). In this case we use the following energy: +E1(t) = +� +sup +y′<0 +��∂zU(· + iy′, t) +��2 +L2(R,dα′) + g +� +sup +y′<0 +����∂z +1 +Ψz′ (· + iy′, t) +���� +2 +L2(R,dα′) +E2(t) = +� +sup +y′<0 +��∂zU(· + iy′, t) +��2 +L2(R,dα′) + g +� +sup +y′<0 +���� +1 +Ψz′ ∂z +� 1 +Ψz′ ∂zU +� +(· + iy′, t) +���� +2 +L2(R,dα′) ++ +� +sup +y′<0 +��∂zU(· + iy′, t) +��2 +L2(R,dα′) + g +�2 +sup +y′<0 +���� +1 +Ψz′ ∂z +1 +Ψz′ (· + iy′, t) +���� +2 +˙H +1 +2 (R,dα′) + +2D WATER WAVES +17 +E3(t) = +� +sup +y′<0 +��∂zU(· + iy′, t) +��2 +L2(R,dα′) + g +�3 +sup +y′<0 +���� +1 +Ψz′ ∂z +� 1 +Ψz′ ∂z +1 +Ψz′ +� +(· + iy′, t) +���� +2 +L2(R,dα′) ++ +� +sup +y′<0 +��∂zU(· + iy′, t) +��2 +L2(R,dα′) + g +�2 +sup +y′<0 +���� +1 +Ψ2 +z′ +∂z +� 1 +Ψz′ ∂zU +� +(· + iy′, t) +���� +2 +˙H +1 +2 (R,dα′) +E(t) = +� +E1(t)3 + E2(t) +3 +2 + E3(t) +� 1 +3 +If Z and Zt are smooth enough then this energy is equivalent to +E1(t) = +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +2 +E2(t) = +���Zt,α′ +��2 +2 + g +���D2 +α′Zt +��2 +2 + +���Zt,α′ +��2 +2 + g +�2����Dα′ +1 +Z,α′ +���� +2 +˙H +1 +2 +E3(t) = +���Zt,α′ +��2 +2 + g +�3����D2 +α′ +1 +Z,α′ +���� +2 +2 ++ +���Zt,α′ +��2 +2 + g +�2���� +1 +Z,α′ D2 +α′Zt +���� +2 +˙H +1 +2 +We now write down the precise conditions on the initial data and the main existence +result. In the result below, we will construct solutions in a class called smoothly approx- +imable solutions denoted by SA which are defined in Definition 4.3. For g = 0, we also use +an energy called Eb(t) which is a slight modification of E(t), and is defined in (79). +Initial data: Let g ≥ 0 and let (U, Ψ)(0) : P− → C, P(0) : P− → R be such that U(0) and +Ψ(0) are holomorphic functions with Ψz′ ̸= 0 for z′ ∈ P−. We assume that P solves (11) +and that limz′→∞(U, Ψz′, (∂x′ − i∂y′)P)(z′, 0) → (0, 1, ig). We also assume that +c0 := sup +y′<0 +��U(· + iy′, 0) +�� +H1(R,dα′) + sup +y′<0 +���� +1 +Ψz′ (· + iy′, 0) − 1 +���� +H1(R,dα′) +< ∞ +(33) +and that E(0) < ∞. +Theorem 3.3. Let g ≥ 0 and let the initial data (U, Ψ, P)(0) be as given above with +E(0) < ∞. Then there exists a universal constant c > 0 such that there exists T ≥ +c +√ +E(0) so +that there exists a solution (U, Ψ, P)(t) to the water wave equation (8) in the time interval +[0, T) in the class SA with the given initial data satisfying supt∈[0,T) E(t) < ∞ for g > 0 +and supt∈[0,T) Eb(t) < ∞ for g = 0. Moreover if g > 0, then the solution is unique in the +class SA and if T ∗ > 0 is the maximal time of existence, then either T ∗ = ∞ or T ∗ < ∞ +and we have +lim sup +t→T ∗ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 (t) + +���Zt,α′ +�� +2(t) + √g +�����∂α′ +1 +Z,α′ +���� +2 +(t) +� += ∞ +We prove this theorem in §4.6. Let us now explain some important features of this result: +(1) The energy E(t) is clearly uniform in the gravity parameter g and hence the time +of existence in the above result is also uniform in gravity. + +18 +SIDDHANT AGRAWAL +(2) As mentioned before, the above result allows for initial data which are smooth and +for initial interfaces which have angled crests and cusps, similar to [35]. We refer +the reader to section 5 of [1] for further details. +(3) In the above result, the Taylor sign condition is not assumed on the initial data and +indeed there does not exist a c > 0 such that − ∂P +∂ˆn ◦ h−1 ≥ c > 0 on R, if g = 0 or +if the interface has a corner or a cusp. +(4) The energy E(t) scales in the same manner as E(t) and hence the estimate T ≥ +c +√ +E(0), which is a lower bound on the time of existence, is scaling invariant with +respect to all the scaling transformations (28). +(5) Under the assumptions on the initial data, if g > 0 then E(0) = 0 if and only if the +interface is flat i.e. Z,α′(·, 0) = 1 on R and the velocity is zero Zt(·, 0) = 0 on R. If +g = 0, then E(0) = 0 if Zt(·, 0) = 0 on R and no other assumptions on the interface +are needed. Note that these are the trivial solutions and from the above result we +have their time of existence as T = ∞. +(6) For g = 1, Wu in [35] obtained a local wellposedness result for singular solutions +and the blow up criterion there, although not explicitely stated, is of the form +lim supt→T ∗ E(t) = ∞ (here we have replaced the energy used in [35] with the +energy E(t)). In the above result, from the lower bound T ≥ +c +√ +E(0) we indeed get +the blow up criterion lim supt→T ∗ E(t) = ∞, though the blow up criterion above is a +significant improvement over this. We also observe that the quantity which appears +as part of the blow up criterion in the above theorem scales in the same way as +∥∇u∥∞(t) under the transformations (28). +We note that for smooth solutions, the blow up criterion in [20] does not require +control on the Lipschitz norm of the velocity, whereas we do require such a control +in the above result. On the other hand, the above blow up criterion or the one +in [35] does not require a control on the C1,α norm of the interface, whereas it is +required in [20]. +To the best of our knowledge the above existence result for g = 0 is new even for small +smooth initial data. However there are some drawbacks to the above existence result for g = +0. First we do not prove uniqueness of solutions when g = 0. This is because of the nature +of the weight A0 which goes to zero at infinity, which causes issues in the known uniqueness +proofs. Another issue is the fact that for g = 0, we have assumed at time t = 0 that +E(0) < ∞ however this is not propagated in time i.e. we do not know whether E(t) < ∞ for +t > 0, we only know that Eb(t) < ∞. Another natural and related question is what happens +if the initial data (Z, Zt)(0) satisfies (Z,α′ − 1, +1 +Z,α′ − 1, Zt)(0) ∈ Hs(R) × Hs(R) × Hs+ 1 +2 (R) +for some s ≥ 2. We do not know if the solution remains in the same Sobolev space for +t > 0, as the natural energy for g = 0 contains terms involving A0 which decays to zero at +infinity, which makes the energy non-equivalent to the Sobolev norms. We believe that all +these issues can be resolved however we do not address them in this paper. +Remark 3.4. The singular solutions established in Theorem 3.3 also have a rigidity prop- +erty namely that the singularities (where singularities are defined as the set of all points + +2D WATER WAVES +19 +where +1 +Ψz′ (α′, t) = 0) propagate via the Lagrangian flow, angled crests/cusps remain angled +crested/cusped, the acceleration at the tip is −ig, and the angle of the crest does not change +nor does it tilt. We refer the reader to [1] for the precise statement of the results (see also +Theorem 3.7 in the next section). The results of [1] hold in exactly the same manner here +as well, with the energy used in [1] replaced by E(t). The proof of the rigidity results for +g > 0 follow exactly in the same way and for g = 0, the proof goes through with some +minor modifications. +Remark 3.5. One can construct higher order energies to the energies E1(t), E2(t) and E3(t) +defined in (25). To do this, look at the terms in the main brackets of the energies E2(t) +and E3(t). From (43) we see that Dt∂α′ +1 +Z,α′ ≈ ω2D2 +α′Zt and hence we have +��DtD2 +α′Zt +�� +2 ≈ +����D2 +t ∂α′ +1 +Z,α′ +���� +2 +and +����� +�Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 +≈ +�����Dt +��Ag +Z,α′ ∂α′ +1 +Z,α′ +������ ˙H +1 +2 +Therefore for n ≥ 3 we can write an approximate energy for En(t) which is +En(t) ≈ +���Zt,α′ +��2 +2 + g +� + + + +����D(n−1) +t +∂α′ +1 +Z,α′ +���� +2 +2 ++ +�����D(n−2) +t +�� +Ag +Z,α′ ∂α′ +1 +Z,α′ +������ +2 +˙H +1 +2 + + + +and the main energy E(t) as +E(t) = +� +E1(t)n + E2(t) +n +2 + E3(t) +n +3 + · · · + En(t) +� 1 +n +One can then prove an estimate similar to (31). +3.2. The bounded domain case +In this subsection we collect our main results for the model (17) which is the same as +(2) but written in conformal coordinates. The results of this subsection mimic the results +established in the previous section §3.1. Similar to the a priori estimate Theorem 3.1 we +also prove an a priori estimate Theorem 5.2. Let us now define the analogue of the energy +E(t) defined in the previous section. Define the energy +�E1(t) = sup +0 0 and assume that P +solves (22). We also assume that +c0 := sup +0 0 such that there exists T ≥ +c +√ �E(0) so that there exists +a unique solution (U, Ψ, P)(t) to the water wave equation (17) in the time interval [0, T) +in the class SA and we have supt∈[0,T) �E(t) < ∞. Moreover if T ∗ is the maximal time of +existence then either T ∗ = ∞ or T ∗ < ∞ along with +lim sup +t→T ∗ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 (t) + +��Zt,α′ +�� +2(t) +�����∂α′ +1 +Z,α′ +���� +2 +(t) + +���� +1 +Z,α′ +���� +2 +(t) +�� += ∞ +We prove this result in §5.3. Let us now explain some important features of this result: +(1) Similar to the unbounded case, the above result allows for initial data which are +smooth and for initial interfaces which have angled crests and cusps, similar to [35]. +(2) Under the assumptions on the initial data, the energy �E(0) = 0 if and only if the +initial velocity is a constant function. In this case, the solution is the trivial solution +and from the above result we see that the time of existence T = ∞. + +2D WATER WAVES +21 +(3) In the above result, the Taylor sign condition is not assumed on the initial data and +indeed − ∂P +∂ˆn ◦ h−1 = 0 at all points where the interface has a corner or a cusp. +For smooth initial data, if the initial velocity is constant then as mentioned above +the solution is trivial and in this case − ∂P +∂ˆn ◦ h−1 is identically zero. If the initial +data is smooth and the velocity is non-constant, then from (119) and (24) we see +that there exists a constant c > 0 such that − ∂P +∂ˆn ◦ h−1 ≥ c > 0 on [0, 2π] and so +the Taylor sign condition is satisfied. Note that we get this lower bound because of +the compactness of the interval [0, 2π], and this contrasts from the fact that there +is no positive lower bound for the case of zero gravity for unbounded domains we +looked in §3.1. +However note that in the above result, the time of existence does not depend +quantitatively on this lower bound. This has important consequences regarding the +time of existence of solutions. For example, consider a smooth initial domain and +initial velocity ǫv0 with v0 fixed. If ǫ > 0 is small, then we see that − ∂P +∂ˆn ◦ h−1 ≥ +c > 0, where c is of the size ǫ2. Note that the size of the interface is of order 1. +So if we follow a classical argument for local wellposedness, this would result in the +energy being of size 1, hence a bound on +��Dt +� ∂P +∂ˆn ◦ h−1��� +∞ is of size 1, and so we +have an upper bound on the time of existence of ǫ2 to ensure that the Taylor sign +condition − ∂P +∂ˆn ◦ h−1 ≥ c/2 is satisfied. So the time of existence T → 0 as ǫ → 0. +To the best of our knowledge, all previous existence results have the property that +T → 0 as ǫ → 0. +On the other hand, we can clearly see that for ǫ = 0, the initial velocity is +identically zero and so the solution is the trivial solution, which is global. Hence +one expects the time of existence T → ∞ as ǫ → 0. This is indeed what we see if +we use the above result where we get T ≳ 1 +ǫ. Note that not only does the energy +�E(0) → 0 as ǫ → 0, it is also crucial to note that we do not use the lower bound +on the Taylor sign condition quantitatively at any point in the energy estimate (see +Theorem 5.2). +(4) For smooth solutions, one of the blow up criterions is the one given by Theorem 5.5. +Even though previously there were no local wellposedness result for singular solu- +tions and a corresponding blow up criterion for the model (17) (the model used in +[14] is slightly different), if one were to follow the approach of [35] directly, then +one would get a blow up criterion of the form lim supt→T ∗ +� +�E(t) + +��� 1 +A0 +��� +∞(t) +� += ∞. +Note that in both situations we have the term +��� 1 +A0 +��� +∞(t) in the blow up criterion. +This is because in general for bounded domain, one needs to control the lower bound +on − ∂P +∂ˆn ◦ h−1 to be able to continue the solution. For smooth solutions, controlling +the C1,α norm of the interface and controlling +��� 1 +A0 +��� +∞(t) gives a lower bound on +− ∂P +∂ˆn ◦ h−1 from (24), and in the case of singular solutions having a lower bound +on A0(t) is equivalent to having a weighted Taylor sign condition being satisfied +(which is enough in this case). It is important to note that having an upper bound + +22 +SIDDHANT AGRAWAL +on the Sobolev norms of the solution or controlling �E(t) for any given fixed t does +not give any quantitative lower bound on A0(t). This is because if +��Zt,α′ +�� +2(t) ≤ ǫ, +then +��� 1 +A0 +��� +∞(t) ≳ +1 +ǫ2 from (119) and Proposition 6.3, whereas the Sobolev norms +and �E(t) can remain of size 1. +One of the main features of the blow up criterion of Theorem 3.6 is that there is +no term of the form +��� 1 +A0 +��� +∞(t). We then further improve the blow up criterion from +lim supt→T ∗ �E(t) = ∞ to the one given by Theorem 3.6. The blow up criterion of +Theorem 3.6 thus improves upon the blow up criterion for both smooth and singular +solutions, as there are no terms related to the Taylor sign condition in the blow up +criterion. +The singular solutions from Theorem 3.6 satisfy a rigidity property similar to the un- +bounded case and [1]. As we will need this rigidity result for our later blow up result, we +now state the rigidity result precisely. +We define the singular set of the interface at time t as +S(t) = +� +α′ ∈ R +��� Tr +� 1 +Ψz′ +� +(α′, t) = 0 +� +The non-singular set is then defined as NS(t) = R\S(t). Observe that both S(t) and NS(t) +are 2π periodic sets and that corners and cusps are indeed included in the singular set. The +rigidity result is now as follows: +Theorem 3.7. Let (U, Ψ, P)(t) be a solution to the water wave equation (17) in the time +interval [0, T] in the class SA with supt∈[0,T] �E(t) < ∞. Then +(1) S(t) = {h(α, t) ∈ R | α ∈ S(0)} for all t ∈ [0, T]. +Moreover S(t) is a closed set of +measure zero for all t ∈ [0, T]. +(2) For every fixed t ∈ [0, T], the functions +� +1 +Ψz′ ∂z′U +� +(·, t) and +� +1 +Ψz′ ∂z′U +� +(·, t) extend to +continuous functions on D with +Tr +� 1 +Ψz′ ∂z′U +� +(α′, t) = Tr +� 1 +Ψz′ ∂z′U +� +(α′, t) = 0 +∀α′ ∈ S(t) +(3) If α0 ∈ S(0), and if {αn}n≥1 is any sequence such that αn ∈ NS(0) for all n ≥ 1 with +αn → α0, then for any t ∈ [0, T] +lim +αn→α0 +Z,α′ +|Z,α′|(h(αn, t), t)) +Z,α′ +|Z,α′|(αn, 0) += 1 +(35) +(4) For all α′ ∈ S(t), we have Ztt(α′, t) = 0. +The first statement of the result says that the singularities are propagated by the La- +grangian flow. The second statement says that the gradient of the velocity extends con- +tinuous to the boundary and vanishes at the singularities. The third statements says that +the nature of the singularity is preserved for later time i.e. a corner remains a corner and + +2D WATER WAVES +23 +a cusp remains a cusp with the angle of the corner remaining the same with no tilting. +The last statement says that the acceleration at the singularity is zero. The proof of the +above rigidity result follows in exactly the same manner as the proof in [1] with virtually +no changes and hence we skip the proof here. +We can now state our result on blow up. +Figure 1. An example of blow up of the energy �E(t) +Theorem 3.8. There exist initial data (U, Ψ, P)(0) satisfying the assumptions of Theo- +rem 3.6 with 0 < �E(0) < ∞ and in addition satisfying the following: +(1) The initial data is symmetric with respect to the y-axis, namely +Z(α′, 0) = Z(−α′, 0) and Zt(α′, 0) = Zt(−α′, 0) for α′ ∈ [0, π] +(2) We have Tr +� +1 +Ψz′ +� +(0, 0) = Tr +� +1 +Ψz′ +� +(π, 0) = 0. Let d = |Z(0, 0) − Z(π, 0)|. +(3) We have Zt(0, 0) < 0 and Zt(π, 0) > 0. Let v = |Zt(0, 0) − Zt(π, 0)|. +For all such initial data, there exists T ∗ > 0 satisfying 0 < +c +√�E(0) ≤ T ∗ ≤ d +v < ∞, where c is +a universal constant, so that the water wave equation (17) has a unique solution (U, Ψ, P)(t) +in the class SA in the time interval [0, T ∗) with supt∈[0,T] �E(t) < ∞ for all T ∈ [0, T ∗) and +we have +lim sup +t→T ∗ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 (t) + +��Zt,α′ +�� +2(t) +�����∂α′ +1 +Z,α′ +���� +2 +(t) + +���� +1 +Z,α′ +���� +2 +(t) +�� += ∞ +In particular lim supt→T ∗ �E(t) = ∞. +The above result follows essentially directly from Theorem 3.6 and Theorem 3.7, and +we give the details of the proof in §5.3. This result shows that the local in time solutions +of Theorem 3.6 cannot in general be extended to global solutions. This also shows that +the time of existence obtained in Theorem 3.6 is optimal in the following sense: Consider +an initial data (U, Ψ, P)(0) satisfying the conditions of the above theorem and replace the +initial velocity by Uǫ(0) = ǫU(0) for some ǫ > 0 small. It is clear that this initial data also +satisfies the conditions of the theorem and we have �Eǫ(0) = ǫ2 �E(0) and vǫ = ǫv (here v is +the variable used in the above theorem). Hence from the result above we see that as ǫ → 0, +we have that T ∗ +ǫ ∼ 1 +ǫ. This shows that the lower bound on the time of existence T ≥ +c +√ �E(0) +from Theorem 3.6 cannot be replaced with T ≥ +c +�E(0) +1 +2 +δ for any δ > 0 even for initial data +with small �E(0). + +24 +SIDDHANT AGRAWAL +We would now like to remark that if one were to prove a local wellposedness result of +(17) in an exactly analogous manner to [35] and take the above initial data, then by the +same rigidity argument one can indeed show that there is a finite maximal time of existence +for these singular solutions. However what is not clear is whether the energy blows up or +not at the maximal time. As mentioned before, if one were to do this, then one would +essentially get a blow up criterion of the form lim supt→T ∗ +� +�E(t) + +��� 1 +A0 +��� +∞(t) +� += ∞. From +this it is not clear whether the energy �E(t) blows up or not. The benefit of Theorem 3.6 is +that it shows that the energy �E(t) indeed blows up and even improves on this. +The proof of the above result strongly relies on the rigidity result which requires that the +interface be non-smooth, and the result does not say anything about blowup for smooth +initial data. Note that the blow up in the above result is not related in any way to splash or +splat singularities of [11]. As the local wellposedness result is done in conformal coordinates +and there are no assumptions on the chord arc condition of the interface, the solutions +constructed here can be continued even if there is a splash singularity. Finally we note +that the proof of the above result is via a contradiction argument and so we do not know +the exact behavior of the solution near the blow up and we do not know the exact time of +blow up. Note that if T ∗ = d +v, then this would correspond to the situation of a pinch type +singularity. +4. Proof of main results for unbounded domain case +4.1. Some useful identities +Here we first collect the main identities commonly used in this paper. The identity (44) +is from [21] whereas the rest are from Section 4 of [2]. +a) +We have +Z,α′ +��Z,α′ +��∂α′ +1 +Z,α′ = ∂α′ +1 +��Z,α′ +�� + ω|Dα′|ω +Observe that ∂α′ +1 +��Z,α′ +�� is real valued and ω|Dα′|ω is purely imaginary. From this we +obtain +Re +� +Z,α′ +��Z,α′ +��∂α′ +1 +Z,α′ +� += ∂α′ +1 +��Z,α′ +�� +Im +� +Z,α′ +��Z,α′ +��∂α′ +1 +Z,α′ +� += i(ω|Dα′|ω) +(36) +b) For any complex valued function f, we have H(Ref) = iIm(Hf) and H(iImf) = +Re(Hf). Hence we get the following identities +(I + H)(Ref) = f − iIm(I − H)f +(37) +(I + H)(iImf) = f − Re(I − H)f +(38) +c) +We have +Ag = g + iZtZt,α′ − i(I + H) +� +Re(ZtZt,α′) +� +(39) + +2D WATER WAVES +25 +d) We have +b = Zt +Z,α′ − i(I + H) +� +Im +� Zt +Z,α′ +�� +and hence +bα′ = Dα′Zt + Zt∂α′ +1 +Z,α′ − i∂α′(I + H) +� +Im +� Zt +Z,α′ +�� +(40) +e) +We now record some frequently used commutator identities. +[∂α′, Dt] = bα′∂α′ +[|Dα′|, Dt] = Re(Dα′Zt)|Dα′| = Re(Dα′Zt)|Dα′| +[Dα′, Dt] = (Dα′Zt)Dα′ +[Dα′, Dt] = +� +Dα′Zt +� +Dα′ +(41) +Using these we also obtain the following formulae +Dt +��Z,α′ +�� = DteRe log Z,α′ = +��Z,α′ +��{Re(Dα′Zt) − bα′} +(42) +Dt +1 +Z,α′ = −1 +Z,α′ (Dα′Zt − bα′) = +1 +Z,α′ +� +(bα′ − Dα′Zt − Dα′Zt) + Dα′Zt +� +(43) +Observe that (bα′ − Dα′Zt − Dα′Zt) is real valued and this fact will be useful later on. +Using the above commutator relations and (13) we also get +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′, Dα′ +� += −2(Dα′Ztt)Dα′ − 2(Dα′Zt)DtDα′ +(44) +We now obtain some identities related to the function Ag. These will be very important +in proving the a priori estimate. From (10) we see that (here A0 is Ag with g = 0) +Ag = g + A0 +(45) +where from a calculation from [31] and using the definition (9) we have +A0(α′) = −(Im[Zt, H]Zt,α′)(α′) = −Im +� 1 +iπ +� Zt(α′) − Zt(β′) +α′ − β′ +Zt,β′(β′) dβ′ +� += 1 +2π +� ���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +2 +dβ′ += 1 +2π +���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +2 +L2(dβ′) += i +2 +� +Zt, Zt; 1 +� +(α′) +(46) +Hence Ag ≥ g ≥ 0. Moreover if Zt is not a constant function, then A0 > 0 everywhere and +hence Ag > 0 everywhere. On the other hand if Zt,α′ ∈ L2, then A0(α′) → 0 as |α′| → ∞ +from Lemma 6.10. Hence A0 is positive everywhere but does not have a uniform positive +lower bound. +Let us now compute the material derivative of Ag. + +26 +SIDDHANT AGRAWAL +Lemma 4.1. We have +DtAg +Ag += Im +� +−2 +� +Zt, Ztt; 1 +� ++ +� +Zt, Zt; Dα′Zt +�� +Ag ++ 2Re(Dα′Zt) − bα′ += 2Re +�� +Zt, +1 +Z,α′ ; 1 +� +(α′) +� ++ 1 +Ag +Im +� +2i +� +Zt, Ag; +1 +Z,α′ +� +(α′) + +� +Zt, Zt; Dα′Zt +� +(α′) +� ++ 2Re(Dα′Zt) − bα′ +Proof. Following [31] (see also [34]), the variable a is defined such that we have the identity +Ag = +�a|zα|2 +hα +� +◦ h−1 +Hence we see that +DtAg +Ag += +�at +a + 2Re +�ztα +zα +� +− htα +hα +� +◦ h−1 = at +a ◦ h−1 + 2Re(Dα′Zt) − bα′ +(47) +Now in [31] (see also [34]), the following formula was derived +at +a ◦ h−1 = −Im +� +2[Zt, H]Ztt,α′ + 2[Ztt, H]∂α′Zt − +� +Zt, Zt; Dα′Zt +�� +Ag +The above formula was derived for g = 1 but the same proof works for g ≥ 0 as well. Now +we simplify the first two terms of the numerator of at +a ◦ h−1 +Im +� +[Zt, H]Ztt,α′ + [Ztt, H]∂α′Zt +� += Im +� 1 +iπ +� Zt(α′) − Zt(β′) +α′ − β′ +Ztt,β′(β′) dβ′ + 1 +iπ +� Ztt(α′) − Ztt(β′) +α′ − β′ +Zt,β′(β′) dβ′ +� += − 1 +πRe +�� Zt(α′) − Zt(β′) +α′ − β′ +Ztt,β′(β′) dβ′ + +� Ztt(α′) − Ztt(β′) +α′ − β′ +Zt,β′(β′) dβ′ +� += − 1 +πRe +�� +∂β′ +�Zt(α′) − Zt(β′) +α′ − β′ +�� +Ztt(α′) − Ztt(β′) +� +dβ′ + +� Ztt(α′) − Ztt(β′) +α′ − β′ +Zt,β′(β′) dβ′ +� += − 1 +πRe +� �Zt(α′) − Zt(β′) +α′ − β′ +��Ztt(α′) − Ztt(β′) +α′ − β′ +� +dβ′ += −Re +� +i +� +Zt, Ztt; 1 +�� += Im +� +Zt, Ztt; 1 +� +Hence we have +DtAg +Ag += Im +� +−2 +� +Zt, Ztt; 1 +� ++ +� +Zt, Zt; Dα′Zt +�� +Ag ++ 2Re(Dα′Zt) − bα′ + +2D WATER WAVES +27 +Now from (13) we get +Ztt(α′) − Ztt(β′) +α′ − β′ += −i + + +Ag(α′) + + +1 +Z,α′ (α′) − +1 +Z,α′ (β′) +α′ − β′ + + + +�Ag(α′) − Ag(β′) +α′ − β′ +� 1 +Z,α′ (β′) + + + +(48) +Hence +� +Zt, Ztt; 1 +� +(α′) = −iAg(α′) +� +Zt, +1 +Z,α′ ; 1 +� +(α′) − i +� +Zt, Ag; +1 +Z,α′ +� +(α′) +Therefore +DtAg +Ag += 2Re +�� +Zt, +1 +Z,α′ ; 1 +� +(α′) +� ++ 1 +Ag +Im +� +2i +� +Zt, Ag; +1 +Z,α′ +� +(α′) + +� +Zt, Zt; Dα′Zt +� +(α′) +� ++ 2Re(Dα′Zt) − bα′ +□ +4.2. The main equations +We will now derive our main equations used to obtain the energy estimates. Define +J0 = Dt(bα′ − Dα′Zt − Dα′Zt) +(49) +Observe that J0 is real valued. Apply Dt to the formula for Dt +1 +Z,α′ in (43) to get +D2 +t +1 +Z,α′ = +1 +Z,α′ +� +(bα′ − Dα′Zt)2 + J0 + DtDα′Zt +� +Now from (13) we see that +DtDα′Zt = −(Dα′Zt)2 + Dα′Ztt += −(Dα′Zt)2 − +i +��Z,α′ +��2 ∂α′Ag − iAgDα′ +1 +Z,α′ +Define +Q0 = (bα′ − Dα′Zt)2 − (Dα′Zt)2 − +i +��Z,α′ +��2 ∂α′Ag +(50) +Hence we see that +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +1 +Z,α′ = +1 +Z,α′ (J0 + Q0) + +28 +SIDDHANT AGRAWAL +Now we apply ∂α′ and commute. First using [∂α′, Dt] = bα′∂α′ we see that +� +∂α′, D2 +t +� 1 +Z,α′ = [∂α′, Dt]Dt +1 +Z,α′ + Dt[∂α′, Dt] 1 +Z,α′ += bα′ +� +∂α′Dt +1 +Z,α′ +� ++ Dt +� +bα′∂α′ +1 +Z,α′ +� += bα′ +� +∂α′Dt +1 +Z,α′ + Dt∂α′ +1 +Z,α′ +� ++ (Dtbα′) +� +∂α′ +1 +Z,α′ +� +Hence we get our main equation as +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ = Dα′J0 + R0 +(51) +where +R0 = +� +∂α′ +1 +Z,α′ +� +(J0 + Q0) + Dα′Q0 − bα′ +� +∂α′Dt +1 +Z,α′ + Dt∂α′ +1 +Z,α′ +� +− (Dtbα′) +� +∂α′ +1 +Z,α′ +� +− 2iAg +� +|Dα′| +1 +��Z,α′ +�� +�� +∂α′ +1 +Z,α′ +� +− i +� +1 +��Z,α′ +��2 ∂α′Ag +�� +∂α′ +1 +Z,α′ +� +(52) +We will also need another equation for D2 +α′Zt. To obtain it, we first apply Dt to (13) +and then use (43) to get +D2 +t Zt = −iDtAg +Z,α′ − iAgDt +1 +Z,α′ += −i Ag +Z,α′ Dα′Zt − i 1 +Z,α′ +� +DtAg + Ag +� +bα′ − Dα′Zt − Dα′Zt +�� +Define +J1 = DtAg + Ag +� +bα′ − Dα′Zt − Dα′Zt +� +(53) +Observe that J1 is real valued. The above equation can now be written as +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +Zt = −i J1 +Z,α′ + +2D WATER WAVES +29 +Now applying D2 +α′ to both sides and using (44) we get +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +D2 +α′Zt += +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′, D2 +α′ +� +Zt − iD2 +α′ +� J1 +Z,α′ +� += Dα′ +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′, Dα′ +� +Zt + +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′, Dα′ +� +Dα′Zt +− iDα′ +� +J1Dα′ +1 +Z,α′ + +ω2 +��Z,α′ +��2 ∂α′J1 +� += Dα′� +−2(Dα′Ztt)(Dα′Zt) − 2(Dα′Zt)(DtDα′Zt) +� +− 2(Dα′Ztt)(D2 +α′Zt) +− 2(Dα′Zt)(DtD2 +α′Zt) − i(Dα′J1) +� +Dα′ +1 +Z,α′ +� +− iJ1D2 +α′ +1 +Z,α′ +− 2iω(Dα′ω) +� +1 +��Z,α′ +��2 ∂α′J1 +� +− i ω3 +��Z,α′ +��∂α′ +� +1 +��Z,α′ +��2 ∂α′J1 +� +Hence we get the equation +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +D2 +α′Zt = R1 − i ω3 +��Z,α′ +��∂α′ +� +1 +��Z,α′ +��2 ∂α′J1 +� +(54) +where +R1 = −2(D2 +α′Ztt)(Dα′Zt) − 4(Dα′Ztt)(D2 +α′Zt) − 2(D2 +α′Zt)(DtDα′Zt) +− 2(Dα′Zt)(Dα′DtDα′Zt) − 2(Dα′Zt)(DtD2 +α′Zt) − i(Dα′J1) +� +Dα′ +1 +Z,α′ +� +− iJ1D2 +α′ +1 +Z,α′ − 2iω(Dα′ω) +� +1 +��Z,α′ +��2 ∂α′J1 +� +(55) +4.3. Quantities controlled by the energy Ea(t) +To prove the energy estimate for the energy Ea(t) namely (30), we need to prove sev- +eral estimates first before we can start taking the time derivative of the energy. In this +subsection, we collect some of these estimates which are then used in §4.5 to prove (30). +Some of the estimates in this section, especially the earlier estimates from estimate 1 to +estimate 11, have already been proven in [21, 35] or [2]. However we still prove them here +as we need the explicit upper bound on them to prove the more refined estimate (30). Most +of the later estimates (from estimate 12 onwards) are completely new, especially all later +estimates involving Ag in one way or another. + +30 +SIDDHANT AGRAWAL +1) +We have +∥Ag∥ ˙H +1 +2 + ∥A0∥L∞∩ ˙H +1 +2 ≲ +��Zt,α′ +��2 +2 +∥Ag∥∞ ≲ g + +��Zt,α′ +��2 +2 +��� +� +Ag +��� +∞ ≲ √g + +��Zt,α′ +�� +2 +Proof: From (45) and (46) we see that Ag = g + A0 where A0 = −Im[Zt, H]Zt,α′. The +estimates now follow from Proposition 6.4. +2) +����∂α′ +1 +��Z,α′ +�� +���� +2 ++ ∥|Dα′|ω∥2 ≲ +����∂α′ +1 +Z,α′ +���� +2 +Proof: This follows directly from (36). +3) +We have +���∂α′PA +� Zt +Z,α′ +���� +∞ ≲ +��Dα′Zt +�� +∞ + +��Zt,α′ +�� +2 +���∂α′ +1 +Z,α′ +��� +2 +(56) +Proof: We see that +2∂α′PA +� Zt +Z,α′ +� += (I − H)(Dα′Zt) + (I − H) +� +Zt∂α′ +1 +Z,α′ +� += 2Dα′Zt − (I + H)(Dα′Zt) + (I − H) +� +Zt∂α′ +1 +Z,α′ +� += 2Dα′Zt + +� 1 +Z,α′ , H +� +Zt,α′ + [Zt, H]∂α′ +1 +Z,α′ +The estimate now follows from Proposition 6.4. +4) +∥bα′∥L∞ ≲ +��Dα′Zt +�� +L∞ + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +Proof: Applying Re(I − H) to (40) we get +bα′ = Re +� +(I − H) +�Zt,α′ +Z,α′ +� ++ [Zt, H] +� +∂α′ +1 +Z,α′ +�� += Re +�� 1 +Z,α′ , H +� +Zt,α′ + 2Dα′Zt + [Zt, H] +� +∂α′ +1 +Z,α′ +�� +(57) +The estimate now follows from Proposition 6.4. +5) +��|Dα′|(bα′ − Dα′Zt − Dα′Zt) +�� +2 ≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +2D WATER WAVES +31 +Proof: Observe that as (bα′ − Dα′Zt − Dα′Zt) is real valued we have +|Dα′|(bα′ − Dα′Zt − Dα′Zt) = Re +� ω +Z,α′ (I − H)∂α′(bα′ − Dα′Zt − Dα′Zt) +� += Re +� +ω(I − H)Dα′(bα′ − Dα′Zt − Dα′Zt) +� +− Re +� +ω +� 1 +Z,α′ , H +� +∂α′(bα′ − Dα′Zt − Dα′Zt) +� +(58) +From (40) we have bα′ = Dα′Zt + Zt∂α′ +1 +Z,α′ − i∂α′(I + H) +� +Im +� Zt +Z,α′ +�� +. Hence we get +(I − H)Dα′(bα′ − Dα′Zt − Dα′Zt) += (I − H) +� +−Dα′Dα′Zt + (Dα′Zt) +� +∂α′ +1 +Z,α′ +� ++ Zt +Z,α′ ∂α′ +� +∂α′ +1 +Z,α′ +�� += (I − H) +� +−Dα′Dα′Zt + (Dα′Zt) +� +∂α′ +1 +Z,α′ +�� ++ +� +PA +� Zt +Z,α′ +� +, H +� +∂α′ +� +∂α′ +1 +Z,α′ +� +(59) +Now observe that as Dα′Dα′Zt = Dα′(ω2Dα′Zt) we have +(I − H)Dα′Dα′Zt = (I − H) +� +2w(Dα′ω)Dα′Zt +� ++ (I − H) +� +ω2D2 +α′Zt +� += (I − H) +� +2w(Dα′ω)Dα′Zt +� ++ +� ω2 +Z,α′ , H +� +∂α′Dα′Zt +Hence as ∥|Dα′|ω∥2 ≲ +���∂α′ +1 +Z,α′ +��� +2 we have from Proposition 6.4 +��|Dα′|(bα′ − Dα′Zt − Dα′Zt) +�� +2 +≲ +���∂α′ +1 +Z,α′ +��� +2 +���Dα′Zt +�� +∞ + +���∂α′PA +� Zt +Z,α′ +���� +∞ + ∥bα′∥∞ +� +≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ +6) +��Dα′Dα′Zt +�� +2 ≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Proof: Recall from (43) that Dt +1 +Z,α′ = +1 +Z,α′ (bα′ − Dα′Zt) and hence +∂α′Dt +1 +Z,α′ = Dα′(bα′ − Dα′Zt − Dα′Zt) + Dα′Dα′Zt + (bα′ − Dα′Zt) +� +∂α′ +1 +Z,α′ +� +Now using [∂α′, Dt] = bα′∂α′ we see that +Dt∂α′ +1 +Z,α′ = Dα′(bα′ − Dα′Zt − Dα′Zt) + Dα′Dα′Zt − Dα′Zt +� +∂α′ +1 +Z,α′ +� +(60) + +32 +SIDDHANT AGRAWAL +Hence we see that +��Dα′Dα′Zt +�� +2 ≲ +��Dα′(bα′ − Dα′Zt − Dα′Zt) +�� +2 + +����∂α′ +1 +Z,α′ +���� +2 +��Dα′Zt +�� +∞ + +����Dt∂α′ +1 +Z,α′ +���� +2 +≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +����Dt∂α′ +1 +Z,α′ +���� +2 +7) +��Dα′Zt +�� +L∞∩ ˙H +1 +2 ≲ +���∂α′ +1 +Z,α′ +��� +2 +��Zt,α′ +�� +2 + +����Dt +� +∂α′ +1 +Z,α′ +����� +1 +2 +2 +��Zt,α′ +�� +1 +2 +2 ≲ Ea(t) +1 +2 +Proof: Using Proposition 6.7 with f = Dα′Zt and w = +1 +Z,α′ +��Dα′Zt +��2 +L∞∩ ˙H +1 +2 +≲ +��Zt,α′ +�� +2 +��Dα′Dα′Zt +�� +2 + +��Zt,α′ +��2 +2 +����∂α′ +1 +Z,α′ +���� +2 +2 +≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +��2 +2 + +��Zt,α′ +�� +2 +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +����Dt∂α′ +1 +Z,α′ +���� +2 +��Zt,α′ +�� +2 +(61) +Hence using ab ≤ a2 +2ǫ + ǫb2 +2 for ǫ small on the second term we get the required estimate. +8) +We have +��D2 +α′Zt +�� +2 + +���|Dα′|2Zt +��� +2 + +��D2 +α′Zt +�� +2 + +����� +1 +��Z,α′ +��2 ∂α′Zt,α′ +����� +2 +≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +(62) +Proof: Observe that +D2 +α′Zt = Dα′(ω2Dα′Zt) = 2ω(Dα′ω)Dα′Zt + ω2Dα′Dα′Zt +The estimate now follows from the estimate of +��Dα′Dα′Zt +�� +2. The other estimates are +similarly proved. +9) +We have +��Dα′Zt +�� +˙H +1 +2 + +��|Dα′|Zt +�� +˙H +1 +2 + +��Dα′Zt +�� +˙H +1 +2 +≲ +��Dα′Zt +�� +˙H +1 +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Zt,α′ +�� +2 +≲ +���∂α′ +1 +Z,α′ +��� +2 +��Zt,α′ +�� +2 + +����Dt +� +∂α′ +1 +Z,α′ +����� +1 +2 +2 +��Zt,α′ +�� +1 +2 +2 +≲ Ea(t) +1 +2 +(63) + +2D WATER WAVES +33 +Proof: Using Proposition 6.8 with f = Zt,α′, w = +1 +Z,α′ and h = ω we obtain +��|Dα′|Zt +�� +˙H +1 +2 ≲ ∥ω∥∞ +��Dα′Zt +�� +˙H +1 +2 + +�����∂α′ +1 +��Z,α′ +�� +����� +2 +��Zt,α′ +�� +2 + ∥ω∥∞ +����∂α′ +1 +Z,α′ +���� +2 +��Zt,α′ +�� +2 +≲ +��Dα′Zt +�� +˙H +1 +2 + +����∂α′ +1 +Z,α′ +���� +2 +��Zt,α′ +�� +2 +We can control +��Dα′Zt +�� +˙H +1 +2 similarly. Now by following the proof of the estimate for +��Dα′Zt +�� +L∞∩ ˙H +1 +2 we similarly get +��Dα′Zt +�� +L∞∩ ˙H +1 +2 ≲ +���∂α′ +1 +Z,α′ +��� +2 +��Zt,α′ +�� +2 + +����Dt +� +∂α′ +1 +Z,α′ +����� +1 +2 +2 +��Zt,α′ +�� +1 +2 +2 ≲ E(t) +1 +2 +10) ∥bα′∥L∞∩ ˙H +1 +2 ≲ +��Dα′Zt +�� +L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +Proof: From (57) we have +bα′ = Re +�� 1 +Z,α′ , H +� +Zt,α′ + 2Dα′Zt + [Zt, H] +� +∂α′ +1 +Z,α′ +�� +Hence from Proposition 6.4 we have +∥bα′∥L∞∩ ˙H +1 +2 ≲ ∥Dα′Zt∥L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +11) ∥|Dα′|bα′∥2 ≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Proof: Obvious from the previous estimate for +��|Dα′|(bα′ − Dα′Zt − Dα′Zt) +�� +2. +12) We have +����� +1 +� +Ag +|Dα′|Ag +����� +2 +≲ ∥|Dα′|Zt∥ ˙H +1 +2 + +��Zt,α′ +�� +2 +�����∂α′ +1 +��Z,α′ +�� +����� +2 +≲ +��Dα′Zt +�� +˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +Proof: Recall from (45) and (46) that Ag = g + A0 = g + i +2 +� +Zt, Zt; 1 +� +. Hence from +Proposition 6.1 we obtain +|Dα′|Ag += i +2 +� +� +|Dα′|Zt, Zt; 1 +� ++ +� +Zt, |Dα′|Zt; 1 +� ++ +� +Zt, Zt; ∂α′ +1 +��Z,α′ +�� +� +− 2 +� +1 +��Z,α′ +��, Zt, Zt; 1 +�� + +34 +SIDDHANT AGRAWAL +(a) We see that +��� +|Dα′|Zt, Zt; 1 +� +(α′) +�� ≲ +���� +|Dα′|Zt(α′) − |Dα′|Zt(β′) +α′ − β′ +���� +L2(dβ′) +���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +L2(dβ′) +Therefore from (46), using Ag ≥ A0 and Proposition 6.3 we have +����� +1 +� +Ag +� +|Dα′|Zt, Zt; 1 +� +����� +2 +≲ +���� +|Dα′|Zt(α′) − |Dα′|Zt(β′) +α′ − β′ +���� +L2(dβ′ dα′) +≲ ∥|Dα′|Zt∥ ˙H +1 +2 +Similarly +����� +1 +�Ag +� +Zt, |Dα′|Zt; 1 +� +����� +2 +≲ ∥|Dα′|Zt∥ ˙H +1 +2 +(b) Observe that using (46) and Ag ≥ A0 we obtain +����� +� +Zt, Zt; ∂α′ +1 +��Z,α′ +�� +� +(α′) +����� +≲ +���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +L2(dβ′) +����� +�Zt(α′) − Zt(β′) +α′ − β′ +�� +∂α′ +1 +��Z,α′ +�� +� +(β′) +����� +L2(dβ′) +≲ +� +Ag(α′) +����∂α′ +1 +��Z,α′ +�� +���� +2 +���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +L∞(dβ′) +Hence from Proposition 6.3 +����� +1 +� +Ag +� +Zt, Zt; ∂α′ +1 +��Z,α′ +�� +������ +2 +≲ +��Zt,α′ +�� +2 +�����∂α′ +1 +��Z,α′ +�� +����� +2 +(c) We have from (46), Ag ≥ A0 and Proposition 6.3 +����� +� +1 +��Z,α′ +��, Zt, Zt; 1 +� +(α′) +����� +≲ +���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +L2(dβ′) +������ + + +1 +|Z,α′|(α′) − +1 +|Z,α′|(β′) +α′ − β′ + + +�Zt(α′) − Zt(β′) +α′ − β′ +������� +L2(dβ′) +≲ +� +Ag(α′) +�����∂α′ +1 +��Z,α′ +�� +����� +2 +���� +Zt(α′) − Zt(β′) +α′ − β′ +���� +L∞(dβ′) +Hence from Proposition 6.3 we have +����� +1 +� +Ag +� +1 +��Z,α′ +��, Zt, Zt; 1 +������ +2 +≲ +��Zt,α′ +�� +2 +�����∂α′ +1 +��Z,α′ +�� +����� +2 + +2D WATER WAVES +35 +Hence +����� +1 +�Ag +|Dα′|Ag +����� +2 +≲ ∥|Dα′|Zt∥ ˙H +1 +2 + +��Zt,α′ +�� +2 +�����∂α′ +1 +��Z,α′ +�� +����� +2 +The second estimate follows from this from estimates proved earlier for +����∂α′ +1 +|Z,α′| +���� +2 +and ∥|Dα′|Zt∥ ˙H +1 +2 . +13) We have +��Ztt,α′ +�� +2 ≲ +���Zt,α′ +�� +2 + √g +����Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Proof: From (13) we see that +Ztt,α′ = −i 1 +Z,α′ ∂α′Ag − iAg∂α′ +1 +Z,α′ +Hence using previous estimates we see that +��Ztt,α′ +�� +2 ≲ +��� +� +Ag +��� +∞ +����� +1 +� +Ag +|Dα′|Ag +����� +2 ++ ∥Ag∥∞ +����∂α′ +1 +Z,α′ +���� +2 +≲ ( +��Zt,α′ +�� +2 + √g) +���Dα′Zt +�� +˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +� ++ +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +14) For any n ∈ Z we have +�����ωn +� +Ag +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +�����ωn +� +Ag +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 ++ +�����ωn +� +Ag +��Z,α′ +��2 ∂α′ω +����� ˙H +1 +2 +≲n +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Proof: For the first estimate we use the weighted +˙H +1 +2 estimate Proposition 6.8 with +f = ∂α′ +1 +Z,α′ weight w = +� +Ag +Z,α′ and h = ωn+1 to get +�����ωn +�Ag +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 +≲n +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +�����∂α′ +�� +Ag +Z,α′ +������ +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +����� +� +Ag +��Z,α′ +��∂α′ω +����� +2 + +36 +SIDDHANT AGRAWAL +≲n +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +2 +���Zt,α′ +�� +2 + √g +� ++ +����∂α′ +1 +Z,α′ +���� +2 +��Dα′Zt +�� +˙H +1 +2 +Now from (36) we have the relations +Re +� +Z,α′ +��Z,α′ +��∂α′ +1 +Z,α′ +� += ∂α′ +1 +��Z,α′ +�� +Im +� +Z,α′ +��Z,α′ +��∂α′ +1 +Z,α′ +� += i(ω|Dα′|ω) +Hence from the first estimate we see that +����� +� +Ag +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 ++ +�����ω +� +Ag +��Z,α′ +��2 ∂α′ω +����� ˙H +1 +2 +≲ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +2 +���Zt,α′ +�� +2 + √g +� ++ +����∂α′ +1 +Z,α′ +���� +2 +��Dα′Zt +�� +˙H +1 +2 +The other estimates follow similarly from Proposition 6.8. +15) For any n ∈ Z we have +�����ωn Ag +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +�����ωn Ag +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 ++ +�����ωn +Ag +��Z,α′ +��2 ∂α′ω +����� ˙H +1 +2 +≲n +���Zt,α′ +�� +2 + √g +� +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 +Proof: Obvious from weighted ˙H +1 +2 estimate Proposition 6.8 with f = ωn� +Ag∂α′ +1 +Z,α′ , +f = ωn� +Ag∂α′ +1 +��Z,α′ +�� or f = ωn +�Ag +��Z,α′ +��∂α′ω and weight w = +1 +��Z,α′ +�� and h = +� +Ag. +16) We have the following temporary estimate +����|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +∞ + +2D WATER WAVES +37 +Proof: Observe that |Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +� += Re +� ω3ω3 +��Z,α′ +��(I − H)∂α′ +� +1 +��Z,α′ +��2 ∂α′Ag +�� +. +Now +ω3 +��Z,α′ +��(I − H)∂α′ +� +1 +��Z,α′ +��2 ∂α′Ag +� += (I − H) +� +ω2Dα′ +� +1 +��Z,α′ +��2 ∂α′Ag +�� +− +� +ω3 +��Z,α′ +��, H +� +∂α′ +� +1 +��Z,α′ +��2 ∂α′Ag +� += (I − H) +� +Dα′ +� 1 +Z2 +,α′ +∂α′Ag +� +− 2 +� +ω +��Z,α′ +��2 ∂α′Ag +� +(Dα′ω) +� +− +� +ω3 +��Z,α′ +��, H +� +∂α′ +� +1 +��Z,α′ +��2 ∂α′Ag +� +Using the formula of Ag from (39) we see that +(I − H) +� +Dα′ +� 1 +Z2 +,α′ +∂α′Ag +�� += i(I − H) +� +Dα′ +��Zt,α′ +Z2 +,α′ +� +Zt,α′ + Zt +� 1 +Z2 +,α′ +∂α′Zt,α′ +��� += i(I − H) +�� +∂α′ Zt,α′ +Z2 +,α′ +�� +Dα′Zt +� ++ 2(Dα′Zt) +� 1 +Z2 +,α′ +∂α′Zt,α′ +�� ++ i +� +PA +� Zt +Z,α′ +� +, H +� +∂α′ +� 1 +Z2 +,α′ +∂α′Zt,α′ +� +Hence from Proposition 6.4 we have +����|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +����� +2 +≲ +���� +1 +Z2 +,α′ +∂α′Zt,α′ +���� +2 +� +∥Dα′Zt∥∞ + +���∂α′PA +� Zt +Z,α′ +���� +∞ +� ++ +����∂α′ Zt,α′ +Z2 +,α′ +���� +2 +��Dα′Zt +�� +∞ ++ +����∂α′ +1 +Z,α′ +���� +2 +���� +1 +��Z,α′ +��2 ∂α′Ag +���� +∞ +≲ +����� +1 +��Z,α′ +��2 ∂α′Zt,α′ +����� +2 +� +∥Dα′Zt∥∞ + +���∂α′PA +� Zt +Z,α′ +���� +∞ +� ++ +����∂α′ +1 +Z,α′ +���� +2 +��Dα′Zt +��2 +∞ ++ +����∂α′ +1 +Z,α′ +���� +2 +���� +1 +��Z,α′ +��2 ∂α′Ag +���� +∞ +The required estimate now follows from using previously proved estimates. + +38 +SIDDHANT AGRAWAL +17) We have +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +L∞∩ ˙H +1 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Proof: We use Proposition 6.7 with f = +1 +|Z,α′| +2∂α′Ag and w = +√ +Ag +|Z,α′| to get +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +2 +L∞∩ ˙H +1 +2 +≲ +����� +1 +�Ag +|Dα′|Ag +����� +2 +��wf ′�� +2 + +����� +1 +�Ag +|Dα′|Ag +����� +2 +2 +�����∂α′ +� � +Ag +��Z,α′ +�� +������ +2 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +����Zt,α′ +�� +2 + √g +� +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +������ +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�4 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����� +1 +��Z,α′ +��2 ∂α′Ag +����� +L∞∩ ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�4 +Now we can use the estimate ab ≤ a2 +2ǫ + ǫb2 +2 with ǫ small for the +���� +1 +|Z,α′| +2∂α′Ag +���� +L∞∩ ˙H +1 +2 +term, to move it on the left hand side and hence obtain +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +2 +L∞∩ ˙H +1 +2 +≲ +���Zt,α′ +�� +2 + √g +�2 +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +������ +2 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�4 +Hence proved. + +2D WATER WAVES +39 +18) Hence we have the estimate +����|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Proof: This follows directly from the previous two estimates. +19) +���� +DtAg +Ag +���� +∞ +≲ +��Dα′Zt +�� +∞ + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +Proof: From Lemma 4.1 we have that +DtAg +Ag += 2Re +�� +Zt, +1 +Z,α′ ; 1 +� +(α′) +� ++ 1 +Ag +Im +� +2i +� +Zt, Ag; +1 +Z,α′ +� +(α′) + +� +Zt, Zt; Dα′Zt +� +(α′) +� ++ 2Re(Dα′Zt) − bα′ +Observe from (45), (46) that +��� +Zt, Zt; Dα′Zt +� +(α′) +�� ≲ +��Dα′Zt +�� +∞Ag(α′) +and also from Proposition 6.3 we get +���� +� +Zt, +1 +Z,α′ ; 1 +����� +∞ ++ ∥bα′∥∞ ≲ +��Dα′Zt +�� +∞ + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +We now observe from (45), (46) that +���� +� +Zt, Ag; +1 +Z,α′ +� +(α′) +���� ≲ +� +Ag(α′) +���� +Ag(α′) − Ag(β′) +α′ − β′ +1 +Z,α′ (β′) +���� +L2(dβ′) +Hence it is enough to show that for all α′ ∈ R we have +���� +Ag(α′) − Ag(β′) +α′ − β′ +1 +Z,α′ (β′) +���� +L2(dβ′) +≲ +���Dα′Zt +�� +∞ + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +�� +Ag(α′) +(64) + +40 +SIDDHANT AGRAWAL +Now from (45), (46) and the fact that H(Zt,α′) = Zt,α′ we get +Ag(α′) − Ag(β′) +α′ − β′ += − +1 +α′ − β′ Im +� 1 +iπ +� �Zt(α′) − Zt(s) +α′ − s +− Zt(β′) − Zt(s) +β′ − s +� +Zt,β′(s) ds +� += − +1 +α′ − β′ Im +� 1 +iπ +� � +Zt(α′) − Zt(s) +�� +1 +α′ − s − +1 +β′ − s +� +Zt,β′(s) ds +� +− +1 +α′ − β′ Im +� 1 +iπ +� �Zt(α′) − Zt(β′) +β′ − s +� +Zt,β′(s) ds +� += Im +� 1 +iπ +� +1 +β′ − s +Zt(α′) − Zt(s) +α′ − s +Zt,β′(s) ds − Zt(α′) − Zt(β′) +α′ − β′ +Zt,β′(β′) +� +(65) +Define +Zt(α′, β′) = Zt(α′) − Zt(β′) +α′ − β′ +Hence for fixed α′ we have +Ag(α′) − Ag(β′) +α′ − β′ += Im +� +H +� +Zt(α′, β′)Zt,α′(β′) +� +− Zt(α′, β′)Zt,α′(β′) +� += −Im +� +(I − H) +� +Zt(α′, β′)Zt,α′(β′) +�� +where we consider the functions as functions of β′ and α′ is some fixed number. Now +observe that +���� +1 +Z,α′ (β′) +�Ag(α′) − Ag(β′) +α′ − β′ +����� +≲ +���� +� 1 +Z,α′ (β′), H +� +Zt(α′, β′)Zt,α′(β′) +���� + +��(I − H) +� +Zt(α′, β′)Dα′Zt(β′) +��� +Now we have from Proposition 6.4 and (45), (46) +���� +� 1 +Z,α′ (β′), H +� +Zt(α′, β′)Zt,α′(β′) +���� +L2(dβ′) +≲ +����∂α′ +1 +Z,α′ +���� +2 +��Zt(α′, β′)Zt,α′(β′) +�� +L1(dβ′) +≲ +����∂α′ +1 +Z,α′ +���� +2 +��Zt(α′, β′) +�� +L2(dβ′) +��Zt,α′ +�� +2 +≲ +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +� +Ag(α′) + +2D WATER WAVES +41 +and also from (45), (46) +��(I − H) +� +Zt(α′, β′)Dα′Zt(β′) +��� +L2(dβ′) +≲ +��Zt(α′, β′)Dα′Zt(β′) +�� +L2(dβ′) +≲ +��Dα′Zt +�� +∞ +��Zt(α′, β′) +�� +L2(dβ′) +≲ +��Dα′Zt +�� +∞ +� +Ag(α′) +Hence (64) is shown thereby proving the main estimate. +20) We have +����� +|Dα′|DtAg +� +Ag +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +Proof: From Lemma 4.1 we see that +DtAg = Im +� +−2 +� +Zt, Ztt; 1 +� ++ +� +Zt, Zt; Dα′Zt +�� ++ Ag{2Re(Dα′Zt) − bα′} +Hence we have +|Dα′|DtAg +� +Ag += +1 +� +Ag +|Dα′|Im +� +−2 +� +Zt, Ztt; 1 +� ++ +� +Zt, Zt; Dα′Zt +�� ++ +� +1 +� +Ag +|Dα′|Ag +� +{2Re(Dα′Zt) − bα′} + +� +Ag|Dα′|{2Re(Dα′Zt) − bα′} +Using previous estimates it is easy to see that +����� +� +1 +� +Ag +|Dα′|Ag +� +{2Re(Dα′Zt) − bα′} +����� +2 ++ +��� +� +Ag|Dα′|{2Re(Dα′Zt) − bα′} +��� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Now let us control the other terms. +(a) From Proposition 6.1 we see that +|Dα′| +� +Zt, Zt; Dα′Zt +� += 2 +� +|Dα′|Zt, Zt; Dα′Zt +� ++ +� +Zt, Zt; ∂α′ +� +1 +��Z,α′ +��Dα′Zt +�� +− 2 +� +1 +��Z,α′ +��, Zt, Zt; Dα′Zt +� + +42 +SIDDHANT AGRAWAL +Now using (45) and (46) we have +��� +|Dα′|Zt, Zt; Dα′Zt +� +(α′) +�� +≲ +� +Ag(α′) +���� +|Dα′|Zt(α′) − |Dα′|Zt(β′) +α′ − β′ +Dα′Zt(β′) +���� +L2(dβ′) +Hence from Proposition 6.3 +����� +1 +� +Ag +� +|Dα′|Zt, Zt; Dα′Zt +� +����� +2 +≲ +��Dα′Zt +�� +∞∥|Dα′|Zt∥ ˙H +1 +2 +We also see using (45), (46) that +����� +� +Zt, Zt; ∂α′ +� +1 +��Z,α′ +��Dα′Zt +�� +(α′) +����� +≲ +� +Ag(α′) +����� +Zt(α′) − Zt(β′) +α′ − β′ +∂β′ +� +1 +��Z,α′ +��Dα′Zt +� +(β′) +����� +L2(dβ′) +Hence using Proposition 6.3 +����� +1 +� +Ag +� +Zt, Zt; ∂α′ +� +1 +��Z,α′ +��Dα′Zt +������� +2 +≲ +��Zt,α′ +�� +2 +�����∂α′ +� +1 +��Z,α′ +��Dα′Zt +������ +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +��Zt,α′ +�� +2 +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Finally we consider the last term and observe using (45), (46) +����� +� +1 +��Z,α′ +��, Zt, Zt; Dα′Zt +� +(α′) +����� +≲ +� +Ag(α′) +������ +1 +|Z,α′|(α′) − +1 +|Z,α′|(β′) +α′ − β′ +Zt(α′) − Zt(β′) +α′ − β′ +Dα′Zt(β′) +������ +L2(dβ′) +Hence using Proposition 6.3 we obtain +����� +1 +� +Ag +� +1 +��Z,α′ +��, Zt, Zt; Dα′Zt +������ +2 +≲ +��Dα′Zt +�� +∞ +�����∂α′ +1 +��Z,α′ +�� +����� +2 +��Zt,α′ +�� +2 + +2D WATER WAVES +43 +(b) From Proposition 6.1 we see that +|Dα′| +� +Zt, Ztt; 1 +� += +� +|Dα′|Zt, Ztt; 1 +� ++ +� +Zt, |Dα′|Ztt; 1 +� ++ +� +Zt, Ztt; ∂α′ +1 +��Z,α′ +�� +� +− 2 +� +1 +��Z,α′ +��, Zt, Ztt; 1 +� +We have from (48), Proposition 6.3 and (64) +��� +|Dα′|Zt, Ztt; 1 +� +(α′) +�� +≲ +���� +|Dα′|Zt(α′) − |Dα′|Zt(β′) +α′ − β′ +���� +L2(dβ′) +���� +Ztt(α′) − Ztt(β′) +α′ − β′ +���� +L2(dβ′) +≲ +� +Ag(α′) +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +����� +|Dα′|Zt(α′) − |Dα′|Zt(β′) +α′ − β′ +���� +L2(dβ′) +Therefore from Proposition 6.3 +����� +1 +� +Ag +� +|Dα′|Zt, Ztt; 1 +� +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +� +∥|Dα′|Zt∥ ˙H +1 +2 +Now we see from (13) +|Dα′|Ztt = −iAg|Dα′| 1 +Z,α′ − +i +Z,α′ |Dα′|Ag +Hence +� +Zt, |Dα′|Ztt; 1 +� += −i +� +Zt, Ag|Dα′| 1 +Z,α′ ; 1 +� +− i +� +Zt, +1 +Z,α′ |Dα′|Ag; 1 +� +Now from (45), (46) +���� +� +Zt, Ag|Dα′| 1 +Z,α′ ; 1 +� +(α′) +���� ≲ +� +Ag(α′) +������ +Ag|Dα′| +1 +Z,α′ (α′) − Ag|Dα′| +1 +Z,α′ (β′) +α′ − β′ +������ +L2(dβ′) +Therefore from Proposition 6.3 +����� +1 +� +Ag +� +Zt, Ag|Dα′| 1 +Z,α′ ; 1 +������ +2 +≲ +����Ag|Dα′| 1 +Z,α′ +���� ˙H +1 +2 +≲ +���Zt,α′ +�� +2 + √g +� +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 + +44 +SIDDHANT AGRAWAL +Now we also have from (45), (46) +���� +� +Zt, +1 +Z,α′ |Dα′|Ag; 1 +� +(α′) +���� ≲ +� +Ag(α′) +������ +1 +Z,α′ |Dα′|Ag(α′) − +1 +Z,α′ |Dα′|Ag(β′) +α′ − β′ +������ +L2(dβ′) +Hence by Proposition 6.3 +����� +1 +�Ag +� +Zt, +1 +Z,α′ |Dα′|Ag; 1 +������ +2 +≲ +����� +ω +��Z,α′ +��2 ∂α′Ag +����� ˙H +1 +2 +Now using Proposition 6.8 with f = +1 +� +Ag +|Dα′|Ag, w = +� +Ag +��Z,α′ +�� and h = ω, we get +����� +1 +� +Ag +� +Zt, +1 +Z,α′ |Dα′|Ag; 1 +������ +2 +≲ +����� +1 +��Z,α′ +��2 ∂α′Ag +����� ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 +≲ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 +Let us now come back to the third term. We see from (45), (46) that +����� +� +Zt, Ztt; ∂α′ +1 +��Z,α′ +�� +� +(α′) +����� ≲ +� +Ag(α′) +����� +Ztt(α′) − Ztt(β′) +α′ − β′ +� +∂α′ +1 +��Z,α′ +�� +� +(β′) +����� +L2(dβ′) +Hence from Proposition 6.3 +����� +1 +�Ag +� +Zt, Ztt; ∂α′ +1 +��Z,α′ +�� +������ +2 +≲ +�����∂α′ +1 +��Z,α′ +�� +����� +2 +��Ztt,α′ +�� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 +Finally let us control the last term. We see from (45), (46) that +����� +� +1 +��Z,α′ +��, Zt, Ztt; 1 +� +(α′) +����� ≲ +� +Ag(α′) +��Ztt,α′ +�� +2 +������ +1 +|Z,α′|(α′) − +1 +|Z,α′|(β′) +α′ − β′ +������ +L∞(dβ′) +Hence from Proposition 6.3 +����� +1 +�Ag +� +1 +��Z,α′ +��, Zt, Ztt; 1 +������ +2 +≲ +��Ztt,α′ +�� +2 +�����∂α′ +1 +��Z,α′ +�� +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 + +2D WATER WAVES +45 +Hence proved. +21) We have +∥J0∥L∞∩ ˙H +1 +2 += +��Dt(bα′ − Dα′Zt − Dα′Zt) +�� +L∞∩ ˙H +1 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +and +∥Dtbα′∥ ˙H +1 +2 + ∥∂α′Dtb∥ ˙H +1 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +Proof: Recall from (40) that +bα′ = Dα′Zt + Zt∂α′ +1 +Z,α′ − i∂α′(I + H) +� +Im +� Zt +Z,α′ +�� +Observe that (bα′ − Dα′Zt − Dα′Zt) is real valued and hence by applying Re(I − H) we +get +bα′ − Dα′Zt − Dα′Zt = Re +� +[Zt, H] +� +∂α′ +1 +Z,α′ +� +− +� +1 +Z,α′ , H +� +Zt,α′ +� +Applying Dt we obtain from Proposition 6.1 +Dt(bα′ − Dα′Zt − Dα′Zt) += Re +� +[Ztt, H] +� +∂α′ +1 +Z,α′ +� ++ [Zt, H] +� +∂α′Dt +1 +Z,α′ +� +− +� +b, Zt; ∂α′ +1 +Z,α′ +� +− +� +Dt +1 +Z,α′ , H +� +Zt,α′ +− +� +1 +Z,α′ , H +� +Ztt,α′ + +� +b, +1 +Z,α′ ; Zt,α′ +�� + +46 +SIDDHANT AGRAWAL +Hence by Proposition 6.4 and Proposition 6.6 +��Dt(bα′ − Dα′Zt − Dα′Zt) +�� +L∞∩ ˙H +1 +2 +≲ +��Ztt,α′ +�� +2 +���∂α′ +1 +Z,α′ +��� +2 + +��Zt,α′ +�� +2 +����∂α′Dt +1 +Z,α′ +���� +2 ++ ∥bα′∥∞ +��Zt,α′ +�� +2 +���∂α′ +1 +Z,α′ +��� +2 +≲ +��Ztt,α′ +�� +2 +���∂α′ +1 +Z,α′ +��� +2 + +��Zt,α′ +�� +2 +����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ ∥bα′∥∞ +��Zt,α′ +�� +2 +���∂α′ +1 +Z,α′ +��� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +This proves the first estimate. Now using Proposition 6.5 we have +∥Dtbα′∥ ˙H +1 +2 + ∥∂α′Dtb∥ ˙H +1 +2 +≲ ∥Dtbα′∥ ˙H +1 +2 + ∥bα′∥2 +L∞∩ ˙H +1 +2 +≲ +��Dt(bα′ − Dα′Zt − Dα′Zt) +�� +˙H +1 +2 + +��DtDα′Zt +�� +˙H +1 +2 + ∥bα′∥2 +L∞∩ ˙H +1 +2 +≲ +��Dt(bα′ − Dα′Zt − Dα′Zt) +�� +˙H +1 +2 + +��Dα′Zt +��2 +L∞∩ ˙H +1 +2 + +��Dα′Ztt +�� +˙H +1 +2 + ∥bα′∥2 +L∞∩ ˙H +1 +2 +Hence we only need to control +��Dα′Ztt +�� +˙H +1 +2 . Applying Dα′ to (13) we get +��Dα′Ztt +�� +˙H +1 +2 ≲ +����� +1 +��Z,α′ +��2 ∂α′Ag +����� ˙H +1 +2 ++ +�����ω Ag +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 +The required estimate now follows from previously proved estimates. +22) We have +∥Q0∥L∞∩ ˙H +1 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +and +∥|Dα′|Q0∥2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +Proof: Recall from (50) that +Q0 = (bα′ − Dα′Zt)2 − (Dα′Zt)2 − +i +��Z,α′ +��2 ∂α′Ag + +2D WATER WAVES +47 +Hence we see that +∥Q0∥L∞∩ ˙H +1 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +and we also see that +∥|Dα′|Q0∥2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 +23) We have +∥R0∥2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +Proof: Recall from (52) that +R0 = +� +∂α′ +1 +Z,α′ +� +(J0 + Q0) + Dα′Q0 − bα′ +� +∂α′Dt +1 +Z,α′ + Dt∂α′ +1 +Z,α′ +� +− (Dtbα′) +� +∂α′ +1 +Z,α′ +� +− 2iAg +� +|Dα′| +1 +��Z,α′ +�� +�� +∂α′ +1 +Z,α′ +� +− i +� +1 +��Z,α′ +��2 ∂α′Ag +�� +∂α′ +1 +Z,α′ +� +(a) We have +����∂α′ +1 +Z,α′ +���� +2 +(∥J0∥∞ + ∥Q0∥∞) + ∥|Dα′|Q0∥2 + ∥bα′∥∞ +�����∂α′Dt +1 +Z,α′ +���� +2 ++ +����Dt∂α′ +1 +Z,α′ +���� +2 +� +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 + +48 +SIDDHANT AGRAWAL +(b) We see from (49), (13) and the fact that Ag is real valued +Dtbα′ = J0 + DtDα′Zt + DtDα′Zt += J0 − (Dα′Zt)2 − (Dα′Zt)2 + 2Re +� +Dα′Ztt +� += J0 − (Dα′Zt)2 − (Dα′Zt)2 + 2Re +� +−i +��Z,α′ +��2 ∂α′Ag − i ωAg +��Z,α′ +��∂α′ +1 +Z,α′ +� += J0 − (Dα′Zt)2 − (Dα′Zt)2 + 2Re +� +−i ωAg +��Z,α′ +��∂α′ +1 +Z,α′ +� +Now using the second estimate of Proposition 6.8 with f = g = ∂α′ +1 +Z,α′ and +w = +� +Ag +��Z,α′ +�� we get +����� +�Ag +��Z,α′ +�� +� +∂α′ +1 +Z,α′ +�� +∂α′ +1 +Z,α′ +������ +2 +≲ +����� +� +Ag +��Z,α′ +�� +� +∂α′ +1 +Z,α′ +������ ˙H +1 +2 +����∂α′ +1 +Z,α′ +���� +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +2 +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Hence +����(Dtbα′) +� +∂α′ +1 +Z,α′ +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +The last two terms of R0 are controlled similarly and hence the estimate follows. +24) We have +����(I − H)D2 +t +� +∂α′ +1 +Z,α′ +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� + +2D WATER WAVES +49 +Proof: For a function f satisfying PAf = 0 we have from Proposition 6.1 +(I − H)D2 +t f = [Dt, H]Dtf + Dt[Dt, H]f += [b, H]∂α′Dtf + Dt[b, H]∂α′f += 2[b, H]∂α′Dtf + [Dtb, H]∂α′f − [b, b; ∂α′f] +(66) +Hence we have from Proposition 6.4 and Proposition 6.6 +����(I − H)D2 +t +� +∂α′ +1 +Z,α′ +����� +2 +≲ ∥bα′∥ ˙H +1 +2 +����Dt∂α′ +1 +Z,α′ +���� +2 ++ ∥∂α′Dtb∥ ˙H +1 +2 +����∂α′ +1 +Z,α′ +���� +2 ++ ∥bα′∥2 +∞ +����∂α′ +1 +Z,α′ +���� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +25) We have +�����(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ +������� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +Proof: Observe that +(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ +�� += i +� +Ag +��Z,α′ +��2 , H +� +∂α′ +� +∂α′ +1 +Z,α′ +� +Hence we have from Proposition 6.4 +�����(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ +������� +2 +≲ +������ +1 +��Z,α′ +��2 ∂α′Ag +����� ˙H +1 +2 ++ +����� +Ag +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 +�����∂α′ +1 +Z,α′ +���� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� + +50 +SIDDHANT AGRAWAL +26) We have +∥|Dα′|J0∥2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +Proof: As J0 given by (49) is real valued, we see that +|Dα′|J0 = Re +� ω +Z,α′ (I − H)∂α′J0 +� += Re{ω(I − H)Dα′J0} − Re +� +ω +� 1 +Z,α′ , H +� +∂α′J0 +� +From equation (51) we have +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ = Dα′J0 + R0 +Applying (I−H) to the above equation and using Proposition 6.4 we obtain the estimate +∥|Dα′|J0∥2 +≲ ∥(I − H)Dα′J0∥2 + +����∂α′ +1 +Z,α′ +���� +2 +∥J0∥∞ +≲ +����(I − H) +� +D2 +t +� +∂α′ +1 +Z,α′ +������ +2 ++ +�����(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ +������� +2 ++ ∥R0∥2 + +����∂α′ +1 +Z,α′ +���� +2 +∥J0∥∞ +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +4.4. Quantities controlled by the energy E(t) +This subsection is similar to §4.3 but for the energy E(t) instead of Ea(t). Here we collect +some of the estimates required to prove the energy estimate for E(t) namely (31). These +estimates are then used in §4.5 to complete the proof of (31). + +2D WATER WAVES +51 +1) +We have +��Dα′DtDα′Zt +�� +2 + +��D2 +α′Ztt +�� +2 + +����AgD2 +α′ +1 +Z,α′ +���� +2 ++ +��D2 +α′Ztt +�� +2 +≲ +��DtD2 +α′Zt +�� +2 + +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +≲ +��DtD2 +α′Zt +�� +2 + Ea(t) +1 +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +����∂α′ +1 +Z,α′ +���� +2 +Proof: We see that +Dα′DtDα′Zt = (Dα′Zt)D2 +α′Zt + DtD2 +α′Zt +(67) +and also +D2 +α′Ztt = Dα′� +(Dα′Zt)Dα′Zt + DtDα′Zt +� += (D2 +α′Zt)Dα′Zt + (Dα′Zt)D2 +α′Zt + Dα′DtDα′Zt +(68) +From these identities we see that +��Dα′DtDα′Zt +�� +2 + +��D2 +α′Ztt +�� +2 ≲ +��DtD2 +α′Zt +�� +2 + +��Dα′Zt +�� +∞ +���D2 +α′Zt +�� +2 + +��D2 +α′Zt +�� +2 +� +and hence the required estimates for these two quantities follow from previously proved +estimates. Now we observe from (13) +D2 +α′Ztt = −iD2 +α′ +� Ag +Z,α′ +� += −iDα′ +� +AgDα′ +1 +Z,α′ + +ω2 +��Z,α′ +��2 ∂α′Ag +� += −i +� +1 +Z2 +,α′ +∂α′Ag +� +∂α′ +1 +Z,α′ − iAgD2 +α′ +1 +Z,α′ − 2i(ωDα′ω) +� +1 +��Z,α′ +��2 ∂α′Ag +� +− iω3|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +� +(69) +Therefore we get +����AgD2 +α′ +1 +Z,α′ +���� +2 +≲ +��D2 +α′Ztt +�� +2 + +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 ++ +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′Ag +������ +2 + +52 +SIDDHANT AGRAWAL +The required estimate now follows from previously proved estimates. Finally using (13) +we get that +D2 +α′Ztt = ω2Dα′(ω2Dα′Ztt) += 2ω3(Dα′ω)(Dα′Ztt) + ω4D2 +α′Ztt += −2iω3(Dα′ω) +� +1 +Z2 +,α′ +∂α′Ag + AgDα′ +1 +Z,α′ +� ++ ω4D2 +α′Ztt += −2iω3(Dα′ω) +� +1 +Z2 +,α′ +∂α′Ag +� +− 2iω(|Dα′|ω) +� +Ag +��Z,α′ +��∂α′ +1 +Z,α′ +� ++ ω4D2 +α′Ztt +Hence we get +��D2 +α′Ztt +�� +2 ≲ +����∂α′ +1 +Z,α′ +���� +2 +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +∞ ++ +�����(|Dα′|ω) +� +Ag +��Z,α′ +��∂α′ +1 +Z,α′ +������ +2 ++ +��D2 +α′Ztt +�� +2 +All terms other than the middle term have already been controlled. For that term we +use Proposition 6.8 with f = |Dα′|ω, g = ∂α′ +1 +Z,α′ and w = +Ag +|Z,α′| to get +�����(|Dα′|ω) +� +Ag +��Z,α′ +��∂α′ +1 +Z,α′ +������ +2 +≲ +����� +Ag +��Z,α′ +��2 ∂α′ω +����� ˙H +1 +2 +����∂α′ +1 +Z,α′ +���� +2 ++ +����� +Ag +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 +����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 +Hence by combining these estimates, we get the required estimate for +��D2 +α′Ztt +�� +2. +2) +We have the estimate +��Dα′Ztt +�� +∞ + +����AgDα′ +1 +Z,α′ +���� +∞ ++ +��DtDα′Zt +�� +∞ ≲ Ea(t) +1 +4E3(t) +1 +4 + Ea(t) ≲ E(t) +Proof: Letting f = Dα′Ztt and w = +1 +Z,α′ in Proposition 6.7 and by using previously +proved estimates we get +��Dα′Ztt +��2 +L∞∩ ˙H +1 +2 ≲ +��Ztt,α′ +�� +2 +��D2 +α′Ztt +�� +2 + +��Ztt,α′ +��2 +2 +����∂α′ +1 +Z,α′ +���� +2 +2 +≲ Ea(t) +1 +2 E3(t) +1 +2 + Ea(t)2 +This proves the estimate for +��Dα′Ztt +�� +∞. Using (13) we see that +Dα′Ztt = −iAgDα′ +1 +Z,α′ − i 1 +Z2 +,α′ +∂α′Ag + +2D WATER WAVES +53 +Hence we get +����AgDα′ +1 +Z,α′ +���� +∞ +≲ +��Dα′Ztt +�� +∞ + +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +∞ +≲ Ea(t) +1 +4 E3(t) +1 +4 + Ea(t) ≲ E(t) +Now as Dα′Ztt = (Dα′Zt)Dα′Zt + DtDα′Zt we see that +��DtDα′Zt +�� +∞ ≲ +��Dα′Ztt +�� +∞ + +��Dα′Zt +��2 +∞ ≲ Ea(t) +1 +4 E3(t) +1 +4 + Ea(t) +3) +We have the estimates +���� +J1 +Ag +���� +∞ +≲ +��Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +≲ Ea(t) +1 +2 +and +����� +1 +� +Ag +|Dα′|J1 +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2 ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +≲ Ea(t) +Proof: From (53) we see that +���� +J1 +Ag +���� +∞ +≲ +���� +DtAg +Ag +���� +∞ ++ ∥bα′∥∞ + +��Dα′Zt +�� +∞ +≲ +��Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +Now again from (53) we obtain +1 +� +Ag +|Dα′|J1 = +1 +� +Ag +|Dα′|DtAg + +� +1 +� +Ag +|Dα′|Ag +� +(bα′ − Dα′Zt − Dα′Zt) ++ +� +Ag|Dα′|(bα′ − Dα′Zt − Dα′Zt) +Hence we get +����� +1 +�Ag +|Dα′|J1 +����� +2 +≲ +����� +1 +�Ag +|Dα′|DtAg +����� +2 ++ +����� +1 +�Ag +|Dα′|Ag +����� +2 +� +∥bα′∥∞ + +��Dα′Zt +�� +∞ +� ++ ∥Ag∥ +1 +2∞ +��|Dα′|(bα′ − Dα′Zt − Dα′Zt) +�� +2 +The required estimate now follows. +4) +We have the estimate +��(I − H)D2 +t D2 +α′Zt +�� +2 ≲ Ea(t) +1 +2��DtD2 +α′Zt +�� +2 + Ea(t) +����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ Ea(t) +3 +2 +����∂α′ +1 +Z,α′ +���� +2 + +54 +SIDDHANT AGRAWAL +Proof: As PAD2 +α′Zt = 0, we see from (66) that +(I − H)D2 +t D2 +α′Zt = 2[b, H]∂α′DtD2 +α′Zt + [Dtb, H]∂α′D2 +α′Zt − +� +b, b; ∂α′D2 +α′Zt +� +Hence we have from Proposition 6.4 and Proposition 6.6 +��(I − H)D2 +t D2 +α′Zt +�� +2 +≲ ∥bα′∥ ˙H +1 +2 +��DtD2 +α′Zt +�� +2 + ∥∂α′Dtb∥ ˙H +1 +2 +��D2 +α′Zt +�� +2 + ∥bα′∥2 +∞ +��D2 +α′Zt +�� +2 +The required estimate now follows from previously proved estimates. +5) +We have the estimate +�����(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′D2 +α′Zt +������ +2 +≲ Ea(t) +����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ Ea(t) +3 +2 +����∂α′ +1 +Z,α′ +���� +2 +Proof: Observe that +(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′D2 +α′Zt +� += i +� +Ag +��Z,α′ +��2 , H +� +∂α′D2 +α′Zt +Hence we have from Proposition 6.4 +�����(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′D2 +α′Zt +������ +2 +≲ +������ +1 +��Z,α′ +��2 ∂α′Ag +����� ˙H +1 +2 ++ +����� +Ag +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 +� +��D2 +α′Zt +�� +2 +The required estimate now follows. +6) +We have the temporary estimate +∥R1∥2 ≲ Ea(t) +1 +2 ��DtD2 +α′Zt +�� +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +1 +2 E(t) +����∂α′ +1 +Z,α′ +���� +2 ++ +����� +1 +��Z,α′ +��2 ∂α′J1 +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 +Proof: Recall from (55) that +R1 = −2(D2 +α′Ztt)(Dα′Zt) − 4(Dα′Ztt)(D2 +α′Zt) − 2(D2 +α′Zt)(DtDα′Zt) +− 2(Dα′Zt)(Dα′DtDα′Zt) − 2(Dα′Zt)(DtD2 +α′Zt) − i(Dα′J1) +� +Dα′ +1 +Z,α′ +� +− iJ1D2 +α′ +1 +Z,α′ − 2iω(Dα′ω) +� +1 +��Z,α′ +��2 ∂α′J1 +� + +2D WATER WAVES +55 +Hence we see that +∥R1∥2 ≲ +��Dα′Zt +�� +∞ +���D2 +α′Ztt +�� +2 + +��Dα′DtDα′Zt +�� +2 + +��DtD2 +α′Zt +�� +2 +� ++ +� +∥Dα′Ztt∥∞ + +��DtDα′Zt +�� +∞ +����D2 +α′Zt +�� +2 + +��D2 +α′Zt +�� +2 +� ++ +����� +1 +��Z,α′ +��2 ∂α′J1 +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 ++ +���� +J1 +Ag +���� +∞ +����AgD2 +α′ +1 +Z,α′ +���� +2 +The estimate now follows using previous estimates. +7) +We have the estimate +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +≲ Ea(t) +1 +2 ��DtD2 +α′Zt +�� +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +1 +2E(t) +����∂α′ +1 +Z,α′ +���� +2 +Proof: Using the fact that J1 is real valued, we observe that +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +≲ +����� +ω3 +��Z,α′ +��(I − H)∂α′ +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +≲ +����� +� +ω3 +��Z,α′ +��, H +� +∂α′ +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 ++ +�����(I − H) +� +ω3 +��Z,α′ +��∂α′ +� +1 +��Z,α′ +��2 ∂α′J1 +������� +2 +Now using Proposition 6.4 and (54) we see that +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +≲ +����∂α′ +1 +Z,α′ +���� +2 +����� +1 +��Z,α′ +��2 ∂α′J1 +����� +∞ ++ ∥R1∥2 + +��(I − H)D2 +t D2 +α′Zt +�� +2 ++ +�����(I − H) +� +i +Ag +��Z,α′ +��2 ∂α′D2 +α′Zt +������ +2 +≲ Ea(t) +1 +2 ��DtD2 +α′Zt +�� +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +1 +2E(t) +����∂α′ +1 +Z,α′ +���� +2 ++ +����� +1 +��Z,α′ +��2 ∂α′J1 +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 + +56 +SIDDHANT AGRAWAL +Now in Proposition 6.7 we put f = +1 +|Z,α′| +2 ∂α′J1 and w = +1 +|Z,α′| to get +����� +1 +��Z,α′ +��2 ∂α′J1 +����� +2 +∞ +≲ ∥|Dα′|J1∥2 +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 ++ ∥|Dα′|J1∥2 +2 +�����∂α′ +1 +��Z,α′ +�� +����� +2 +2 +≲ ∥Ag∥ +1 +2∞ +����� +1 +� +Ag +|Dα′|J1 +����� +2 +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 ++ ∥Ag∥∞ +����� +1 +� +Ag +|Dα′|J1 +����� +2 +2 +����∂α′ +1 +Z,α′ +���� +2 +2 +≲ Ea(t) +���Zt,α′ +�� +2 + √g +� +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 ++ Ea(t)3 +(70) +Using this estimate in the previous estimate we get +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +≲ Ea(t) +1 +2 ��DtD2 +α′Zt +�� +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +1 +2 E(t) +����∂α′ +1 +Z,α′ +���� +2 ++ Ea(t) +1 +2���Zt,α′ +�� +2 + √g +� 1 +2 +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +1 +2 +2 +����∂α′ +1 +Z,α′ +���� +2 +Now using the estimate ab ≤ a2 +2ǫ + ǫb2 +2 +with ǫ small on the last term and absorbing it +on the left hand side we get +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +≲ Ea(t) +1 +2 ��DtD2 +α′Zt +�� +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +1 +2E(t) +����∂α′ +1 +Z,α′ +���� +2 +8) +We have the estimate +����� +1 +��Z,α′ +��2 ∂α′J1 +����� +∞ +≲ Ea(t) +1 +2 E(t) + +2D WATER WAVES +57 +This in particular implies that we have +∥R1∥2 ≲ Ea(t) +1 +2 ��DtD2 +α′Zt +�� +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +1 +2 E(t) +����∂α′ +1 +Z,α′ +���� +2 +Proof: The first estimate follows directly from the estimate for +����|Dα′| +� +1 +|Z,α′| +2∂α′J1 +����� +2 +and (70). The second estimate then follows immediately from the temporary estimate +proved for ∥R1∥2. +4.5. Closing the energy estimate +We first note the following lemma from Sec 5.2 of [2] +Lemma 4.2. Let T > 0 and let f, b ∈ C2([0, T), H2(R)) with b being real valued. +Let +Dt = ∂t + b∂α′. Then +(1) d +dt +� +f dα′ = +� +Dtf dα′ + +� +bα′f dα′ +(2) +��� d +dt +� +|f|2 dα′ − 2Re +� +¯f(Dtf) dα′��� ≲ ∥f∥2 +2∥bα′∥∞ +(3) +���� +d +dt +� +(|∂α′| ¯f)f dα′ − 2Re +�� +(|∂α′| ¯f)Dtf dα′ +����� ≲ ∥f∥2 +˙H +1 +2 ∥bα′∥∞ +Now using the above lemma we see that +d +dtE1(t) += d +dt +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +2 +≲ ∥bα′∥∞ +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +2 ++ +��Zt,α′ +�� +2 +��Ztt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +2 ++ +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +����Dt∂α′ +1 +Z,α′ +���� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Ea(t) +(71) +Now let us control E2(t). We first need to control +∂t + + + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + + +58 +SIDDHANT AGRAWAL +Let f = ∂α′ +1 +Z,α′ . Note that here Hf = f. Hence we need to control the time derivative of +∥Dtf∥2 +2 + +����� +� +Ag +Z,α′ f +����� +2 +˙H +1 +2 +for a function f satisfying Hf = f. This will also help us in controlling E3. Such a type +of estimate was done in [2] but the estimate proved there strongly used the fact that g = 1 +and does not work here. We will crucially use the new estimates proved in §4.3 to obtain +the estimate (74). The estimates from §4.3 and §4.4 will then be further used to close the +energy estimates. +We see from Lemma 4.2 that +���� +d +dt +� +|Dtf|2 dα′ − 2Re +� +(D2 +t f)(Dt ¯f) dα′ +���� +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� +2 +� +∥Dtf∥2 +2 +(72) +We also see from Lemma 4.2 that +����� +d +dt +� ����|∂α′| +1 +2 +��Ag +Z,α′ f +����� +2 +dα′ − 2Re +� � +|∂α′| +��Ag +Z,α′ f +�� +Dt +��Ag +Z,α′ +¯f +� +dα′ +����� +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +������ +� +Ag +Z,α′ f +����� +2 +˙H +1 +2 +Now observe from (43) +Dt +�� +Ag +Z,α′ +¯f +� += +�DtAg +2Ag ++ bα′ − Dα′Zt +�� +Ag +Z,α′ +¯f + +� +Ag +Z,α′ Dt ¯f +Hence +�����2Re +� � +|∂α′| +�� +Ag +Z,α′ f +�� +Dt +�� +Ag +Z,α′ +¯f +� +dα′ − 2Re +� � +|∂α′| +�� +Ag +Z,α′ f +���� +Ag +Z,α′ Dt ¯f +� +dα′ +����� +≲ +����� +� +Ag +Z,α′ f +����� ˙H +1 +2 +����� +�DtAg +2Ag ++ bα′ − Dα′Zt +�� +Ag +Z,α′ +¯f +����� ˙H +1 +2 +From Proposition 6.8 with f = ¯f, w = +� +Ag +Z,α′ and h = +�DtAg +2Ag ++ bα′ − Dα′Zt +� +we get +����� +�DtAg +2Ag ++ bα′ − Dα′Zt +�� +Ag +Z,α′ +¯f +����� ˙H +1 +2 + +2D WATER WAVES +59 +≲ +���� +DtAg +2Ag ++ bα′ − Dα′Zt +���� +∞ +����� +�Ag +Z,α′ +¯f +����� ˙H +1 +2 ++ +���� +DtAg +2Ag ++ bα′ − Dα′Zt +���� +∞ +�����∂α′ +��Ag +Z,α′ +������ +2 +�� ¯f +�� +2 ++ +�� ¯f +�� +2 +����� +� +Ag +Z,α′ ∂α′ +�DtAg +2Ag ++ bα′ − Dα′Zt +������ +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +������ +� +Ag +Z,α′ +¯f +����� ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +�2�� ¯f +�� +2 ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +�� ¯f +�� +2 +Therefore +����� +d +dt +� ����|∂α′| +1 +2 +�� +Ag +Z,α′ f +����� +2 +dα′ − 2Re +� � +|∂α′| +�� +Ag +Z,α′ f +���� +Ag +Z,α′ Dt ¯f +� +dα′ +����� +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +������ +� +Ag +Z,α′ f +����� +2 +˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +�2�� ¯f +�� +2 +����� +� +Ag +Z,α′ f +����� ˙H +1 +2 ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +�� ¯f +�� +2 +����� +� +Ag +Z,α′ f +����� ˙H +1 +2 +(73) +We simplify further using |∂α′| = iH∂α′ and Hf = f +� +Ag +Z,α′ |∂α′| +�� +Ag +Z,α′ f +� += i +�� +Ag +Z,α′ , H +� +∂α′ +�� +Ag +Z,α′ f +� ++ iH +�� +Ag +Z,α′ ∂α′ +�� +Ag +Z,α′ +� +f + +Ag +��Z,α′ +��2 ∂α′f +� += i +�� +Ag +Z,α′ , H +� +∂α′ +�� +Ag +Z,α′ f +� ++ iH +� +1 +2 +� +1 +��Z,α′ +��2 ∂α′Ag +� +f + Ag +� +Dα′ +1 +Z,α′ +� +f +� +− i +� +Ag +��Z,α′ +��2 , H +� +∂α′f + i +Ag +��Z,α′ +��2 ∂α′f + +60 +SIDDHANT AGRAWAL +Observe that in the second part of Proposition 6.8 by letting f = f, g = �Agω∂α′ +1 +Z,α′ and +w = +√ +Ag +|Z,α′| we get +����Ag +� +Dα′ +1 +Z,α′ +� +f +���� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +������ +� +Ag +��Z,α′ +��f +����� ˙H +1 +2 ++ +����� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +∥f∥2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +�2 +∥f∥2 +Hence we have the following estimate by using Proposition 6.4 +���� +� +Ag +Z,α′ |∂α′| +�� +Ag +Z,α′ f +� +− i +Ag +��Z,α′ +��2 ∂α′f +���� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +������ +� +Ag +Z,α′ f +����� ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +�2 +∥f∥2 ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +∥f∥2 +Therefore combining the above estimate with (73) we get +����� +d +dt +� ����|∂α′| +1 +2 +�� +Ag +Z,α′ f +����� +2 +dα′ − 2Re +� � +i +Ag +��Z,α′ +��2 ∂α′f +� +(Dt ¯f) dα′ +����� +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +������ +� +Ag +Z,α′ f +����� ˙H +1 +2 +������ +� +Ag +Z,α′ f +����� ˙H +1 +2 ++ +��Dt ¯f +�� +2 +� ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +�2�� ¯f +�� +2 +������ +�Ag +Z,α′ f +����� ˙H +1 +2 ++ +��Dt ¯f +�� +2 +� ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +�� ¯f +�� +2 +������ +� +Ag +Z,α′ f +����� ˙H +1 +2 ++ +��Dt ¯f +�� +2 +� + +2D WATER WAVES +61 +Finally combining the above estimate with (72) we obtain +������ +d +dt + + +∥Dtf∥2 +2 + +����� +� +Ag +Z,α′ f +����� +2 +˙H +1 +2 + + + − 2Re +� � +D2 +t f + i +Ag +��Z,α′ +��2 ∂α′f +� +(Dt ¯f) dα′ +������ +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +� + +����� +�Ag +Z,α′ f +����� +2 +˙H +1 +2 ++ +��Dt ¯f +��2 +2 + + ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� ˙H +1 +2 +�2�� ¯f +�� +2 +������ +� +Ag +Z,α′ f +����� ˙H +1 +2 ++ +��Dt ¯f +�� +2 +� ++ +���Zt,α′ +�� +2 + √g +� +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +�� ¯f +�� +2 +������ +� +Ag +Z,α′ f +����� ˙H +1 +2 ++ +��Dt ¯f +�� +2 +� +(74) +We are now ready to control E2(t). Recall from (51) that +� +D2 +t + i +Ag +��Z,α′ +��2 ∂α′ +� +∂α′ +1 +Z,α′ = Dα′J0 + R0 +Hence for f = ∂α′ +1 +Z,α′ we have +�����2Re +� � +D2 +t f + i +Ag +��Z,α′ +��2 ∂α′f +� +(Dt ¯f) dα′ +����� +≲ (∥|Dα′|J0∥2 + ∥R0∥2) +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 +����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +�Ag +Z,α′ +� +∂α′ +1 +Z,α′ +������ +2 +˙H +1 +2 +� + +62 +SIDDHANT AGRAWAL +Therefore by using the above estimate and putting f = ∂α′ +1 +Z,α′ in (74) we get +������ +d +dt + + + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + +������ +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ +� +∂α′ +1 +Z,α′ +������ ˙H +1 +2 +� ++ +� +��Dα′Zt +�� +L∞∩ ˙H +1 +2 ++ +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +������Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +� +Ag +Z,α′ +� +∂α′ +1 +Z,α′ +������ +2 +˙H +1 +2 +� +Therefore we get from Lemma 4.2 the estimate +d +dtE2(t) += d +dt + + + +���Zt,α′ +��2 +2 + g +� + + + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + + + + +≲ +� +∥bα′∥∞ +��Zt,α′ +��2 +2 + +��Ztt,α′ +�� +2 +��Zt,α′ +�� +2 +� + + + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + ++ +���Zt,α′ +��2 +2 + g +� d +dt + + + +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +E2(t) +1 +2 Ea(t) +(75) +Combining this estimate with (71) we see that +d +dtEa(t) += 1 +2 +2E1(t)∂tE1(t) + ∂tE2(t) +(E1(t)2 + E2(t)) +1 +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Ea(t) +≲ Ea(t) +3 +2 +thereby proving (30). + +2D WATER WAVES +63 +Let us now control E3(t). Let f = D2 +α′Zt and observe that Hf = f. We see from (54) +that�����2Re +� � +D2 +t f + i +Ag +��Z,α′ +��2 ∂α′f +� +(Dt ¯f) dα′ +����� +≲ +� +∥R1∥2 + +�����|Dα′| +� +1 +��Z,α′ +��2 ∂α′J1 +������ +2 +� +��DtD2 +α′Zt +�� +2 +≲ Ea(t) +1 +2��DtD2 +α′Zt +��2 +2 + E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� +��DtD2 +α′Zt +�� +2 ++ Ea(t) +1 +2E(t) +����∂α′ +1 +Z,α′ +���� +2 +��DtD2 +α′Zt +�� +2 +Hence from (74) we see that +������ +d +dt + + + +��DtD2 +α′Zt +��2 +2 + +����� +� +Ag +Z,α′ D2 +α′Zt +����� +2 +˙H +1 +2 + + + +������ +≲ Ea(t) +1 +2 + +��DtD2 +α′Zt +��2 +2 + +����� +�Ag +Z,α′ D2 +α′Zt +����� +2 +˙H +1 +2 + + ++ E(t) +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +�� +��DtD2 +α′Zt +�� +2 + +����� +� +Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 +� ++ Ea(t) +1 +2 E(t) +����∂α′ +1 +Z,α′ +���� +2 +� +��DtD2 +α′Zt +�� +2 + +����� +� +Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 +� +Therefore we get from Lemma 4.2 the estimate +d +dtE3(t) += d +dt + + + +���Zt,α′ +��2 +2 + g +� + + + +��DtD2 +α′Zt +��2 +2 + +����� +� +Ag +Z,α′ D2 +α′Zt +����� +2 +˙H +1 +2 + + + + + + +≲ +� +∥bα′∥∞ +��Zt,α′ +��2 +2 + +��Ztt,α′ +�� +2 +��Zt,α′ +�� +2 +� + + + +��DtD2 +α′Zt +��2 +2 + +����� +� +Ag +Z,α′ D2 +α′Zt +����� +2 +˙H +1 +2 + + + ++ +���Zt,α′ +��2 +2 + g +� d +dt + + + +��DtD2 +α′Zt +��2 +2 + +����� +� +Ag +Z,α′ D2 +α′Zt +����� +2 +˙H +1 +2 + + + +≲ Ea(t) +1 +2E3(t) + Ea(t)E(t)E3(t) +1 +2 +≲ Ea(t) +1 +2E(t)3 + +64 +SIDDHANT AGRAWAL +Hence by combining this estimate with (71) and (75) we get +d +dtE(t) += 1 +3 +3E1(t)2∂tE1(t) + 3 +2E2(t) +1 +2 ∂tE2(t) + ∂tE3(t) +(E1(t)3 + E2(t) +3 +2 + E3(t)) +2 +3 +≲ Ea(t) +1 +2E(t) +≲ E(t) +3 +2 +This completes the proof of Theorem 3.1. +4.6. Proof of Theorem 3.3 +In this section we complete the proof of Theorem 3.3. Let us first define the class SA +(smoothly approximable solutions) which is the class of solutions in which the solutions lie. +To do this, we first need to define the appropriate norm to take the difference. +Let (Z, Zt)a and (Z, Zt)b be two solutions of the water wave equation (10). Let ha, hb be +the homeomorphisms from (12) for the respective solutions and define +�h = hb ◦ h−1 +a +and +�U = U�h = U −1 +ha Uhb +(76) +While taking the difference of the two solutions, we will subtract in Lagrangian coordinates +and then bring it to the conformal coordinate system of solution A. The operator �U takes +a function in the conformal coordinate system of B to the conformal coordinate system of +A. We define +∆(f) = fa − �U(fb) +(77) +For example we have ∆(Z) = Za − �U(Zb). +We will usually write �U(fb) as �U(f)b for +convenience. Define +F∆((Z, Zt)a, (Z, Zt)b)(t) += ∥∆(Zt)∥ ˙H +1 +2 + ∥∆(Ztt)∥ ˙H +1 +2 + +����∆ +� 1 +Z,α′ +����� ˙H +1 +2 ++ +����hα′ − 1 +��� +2 ++ ∥∆(Dα′Zt)∥2 + ∥∆(Ag)∥2 + ∥∆(bα′)∥2 +(78) +This is the norm which is used to compute the difference between two solutions and was +introduced in [35]. We also need to define the energy Eb(t) which appeared in Theorem 3.3. + +2D WATER WAVES +65 +It is defined as follows: +Eb,1(t) = +���Zt,α′ +��2 +2 + g +�����∂α′ +1 +Z,α′ +���� +2 +2 +Eb,2(t) = +���Zt,α′ +��2 +2 + g +� + + + +�����∂α′ +� +Zt,α′ +Z2 +,α′ +������ +2 +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� +2 +˙H +1 +2 + + + +Eb,3(t) = +���Zt,α′ +��2 +2 + g +� + + + +�����∂α′ +� +Ag +Z2 +,α′ +∂α′ +1 +Z,α′ +������ +2 +2 ++ +����� +�Ag +Z,α′ ∂α′ +� +Zt,α′ +Z2 +,α′ +������ +2 +˙H +1 +2 + + + +Eb(t) = +� +Eb,1(t)3 + Eb,2(t) +3 +2 + Eb,3(t) +� 1 +3 +(79) +We are now ready to define the class SA. +Definition 4.3. Let g ≥ 0 and let the initial data (U, Ψ, P)(0) be given as in the paragraph +above Theorem 3.3. Let T > 0 and let U, Ψ : P− × [0, T] → C and P : P− × [0, T] → R. +We say that (U, Ψ, P)(t) solves the Cauchy problem for the system (8) with gravity g in +the time interval [0, T] in the class SA (smoothly approximable solutions) if the following +holds: +(1) U, Ψ, P extend continuously to P− × [0, T] and (U, Ψ) ∈ C1(P− × (0, T)). Also for +each t ∈ [0, T] we have P(·, t) ∈ C1(P−). +(2) U(·, t), Ψ(·, t) are holomorphic maps for each t ∈ [0, T], Ψz′(z′, t) ̸= 0 for (z′, t) ∈ +P− × [0, T] and limz′→∞(U, Ψt, Ψz′, (∂x′ − i∂y′)P)(z′, t) → (0, 0, 1, ig) for any t ∈ +[0, T]. Moreover +1 +Ψz′ and Ψt +Ψz′ extend continuously to P−×[0, T] (and we will continue +to denote these extensions as +1 +Ψz′ and +Ψt +Ψz′ respectively). +(3) We have +sup +t∈[0,T] +� +sup +y′<0 +��U(· + iy′, t) +�� +H1(R,dα′) + sup +y′<0 +���� +1 +Ψz′ (· + iy′, t) − 1 +���� +H1(R,dα′) +� +< ∞ +(4) For g > 0 we have supt∈[0,T] E(t) < ∞ and for g = 0 we have supt∈[0,T] Eb(t) < ∞. +(5) (U, Ψ, P) solves (8) in P− × [0, T] with the given initial data and P solves (11). +Moreover for any �T ∈ [0, T], there exists T1, T2 such that 0 ≤ T1 ≤ �T ≤ T2 ≤ T satisfying +T1 < �T if �T > 0 and T2 < �T if �T < T, and a sequence of smooth functions (Z(n), Z(n) +t +) : +R × [T1, T2] → C for n ≥ 1 satisfying: +(a) For each n ∈ N we have Z(n) +,α′ (α′, t) ̸= 0 for all (α′, t) ∈ R×[T1, T2] and +� +Z(n) +t +, Z(n) +,α′ − +1, +1 +Z(n) +,α′ −1 +� +∈ Cl([T1, T2], Hs+ 1 +2−l(R), Hs−l(R), Hs−l(R)) for all s ≥ 4 and l = 0, 1. +(b) We have +sup +n≥1,t∈[T1,T2] +���Z(n) +t +�� +H1(t) + +���� +1 +Z(n) +,α′ +− 1 +���� +H1(t) +� +< ∞ + +66 +SIDDHANT AGRAWAL +(c) There exist holomorphic functions (U (n), Ψ(n))(·, t) : P− → C whose boundary val- +ues are (Z(n) +t +, Z(n))(·, t) and which satisfy for each n the property Ψ(n) +z′ (z′, t) ̸= +0 for all (z′, t) ∈ P− × [T1, T2]. +Moreover for each t ∈ [T1, T2] we have that +limz′→∞(U (n), Ψ(n) +t +, Ψ(n) +z′ )(z′, t) = (0, 0, 1). +Let P(n) : P− × [T1, T2] → R be the +function satisfying +∆P(n) = −2|U (n) +z′ |2 +on P−, +P(n) = 0 +on ∂P− +along with (∂x′ − i∂y′)P(n) → ig as z′ → ∞. +Then (U (n), Ψ(n), P(n), +1 +Ψ(n) +z′ ) → +(U, Ψ, P, +1 +Ψz′ ) uniformly on compact subsets of P − × [T1, T2] as n → ∞. +(d) If g > 0, then for each n ∈ N, (Z(n), Z(n) +t +) solves the system (10) with gravity g and +supn∈N,t∈[T1,T2] +� +En(t) + +��Z(n) +t,α′ +�� +2(t) +� +< ∞. If we fix the Lagrangian parametriza- +tions for these solutions by imposing h(n)(α′, �T) = h(α′, �T) for all α′ ∈ R and n ∈ N, +then supt∈[T1,T2] F∆((Z(n), Z(n) +t +), (Z, Zt))(t) → 0 as n → ∞. +If g = 0, then there exists a sequence of gn > 0 for n ≥ 1 with limn→∞ gn = 0 +such that for each n ∈ N, (Z(n), Z(n) +t +) solves the system (10) with gravity gn and +supn∈N,t∈[T1,T2] +� +(Eb)n(t) + +��Z(n) +t,α′ +�� +2(t) +� +< ∞ +Furthermore we say that (U, Ψ, P)(t) belongs in the class SA in the time interval [0, T), if +for any 0 < T1 < T the solution belongs in the class SA in the time interval [0, T1]. +Remark 4.4. We note here that the definition of the energy Eb(t) makes sense for these +singular solutions. +Indeed if U, Ψ : P− × [0, T] → C are holomorphic maps satisfying +conditions 1 − 3 in Definition 4.3, then we let the boundary values of U(·, t) be called +Zt(·, t) and the boundary value of +1 +Ψz′ (·, t) be called +1 +Z,α′ (·, t) by abuse of notation. Now as +U, Ψ satisfy conditions 1−3 in Definition 4.3, it is clear that Zt,α′(·, t), ∂α′ +1 +Z,α′ (·, t) ∈ L2(R) +and +1 +Z,α′ (·, t) ∈ L∞(R). From (45) and (46) we see that Ag ≥ 0 and from Proposition 6.4 +we then also see that Ag(·, t) ∈ L∞(R) as ∥Ag∥∞ ≲ g + +��Zt,α′ +��2 +2. Hence the energy Eb(t) is +now well defined. +The definition above of the class SA closely follows the definition of solutions given +in [35] with some minor changes. +First we allow the gravity g = 0 and have a corre- +sponding definition for that. +Then as the energy used in this paper is different from +the one used in [35], some changes are made in the definition such as the condition +supn∈N,t∈[T1,T2] +� +En(t) + +��Z(n) +t,α′ +�� +2(t) +� +< ∞ in condition (d). Finally as we are interested +in long time solutions, we allow the smooth approximations to hold only locally in time in +contrast to the global in time approximation used in the definition in [35]. +We have the following relations between the different energies: +Lemma 4.5. Let T > 0 and let (Z, Zt)(t) be a solution to the gravity water wave equation +(10) with gravity parameter g ≥ 0 in the time interval [0, T] with (Z,α′ − 1, +1 +Z,α′ − 1, Zt) ∈ +L∞([0, T], Hs(R) × Hs(R) × Hs+ 1 +2 (R)) for some s ≥ 4. Then we have the following: + +2D WATER WAVES +67 +(1) There exists a universal constant M1 > 0 such that E(t) ≤ M1E(t). Moreover there +exists a universal increasing function C1 : [0, ∞) → [0, ∞) such that for g > 0 we +have +E(t) ≤ C1 +� +E(t) + g + 1 +g + +��Zt,α′ +�� +2(t) +� +(80) +and also +sup +t∈[0,T] +E(t) ≤ C1 +� +sup +t∈[0,T] +E(t) + g + 1 +g + T + +��Zt,α′ +�� +2(0) +� +(81) +(2) There exists a universal constant M2 > 0 so that +1 +M2 Eb(t) ≤ E(t) ≤ M2Eb(t). +(3) Let c1 = +��Zt,α′ +��2 +2(0) + g and assume that c1 > 0. Then there exists a universal +increasing function C2 : [0, ∞) → [0, ∞) so that for all g ≥ 0 we have +sup +t∈[0,T] +� +∥Zt∥H1(t) + +���� +1 +Z,α′ − 1 +���� +H1 ++ +1 +��Zt,α′ +��2 +2(t) + g +� +≤ C2 +� +sup +t∈[0,T] +E(t) + T + g + 1 +c1 ++ ∥Zt∥H1(0) + +���� +1 +Z,α′ − 1 +���� +2 +(0) +� +(82) +(4) There exists a universal increasing function C3 : [0, ∞) → [0, ∞) such that for g > 0 +we have +sup +t∈[0,T] +� +∥Zt∥H2.5(t) + +��Z,α′ − 1 +�� +H2(t) + +���� +1 +Z,α′ − 1 +���� +H2 +(t) +� +≤ C3 +� +sup +t∈[0,T] +E(t) + g + 1 +g + T + ∥Zt∥H1(0) + +��Z,α′ +�� +∞(0) + +���� +1 +Z,α′ − 1 +���� +2 +(0) +� +(83) +Proof. Step 1: We first prove that E(t) ≲ E(t). Note that E1(t) ≲ E(t) as it is the same +as E1(t). Now from (61) we see that +��Dα′Zt +��2 +∞ ≲ E(t). Hence from (60) we see that +���Zt,α′ +�� +2 + √g +�����Dt∂α′ +1 +Z,α′ +���� +2 +≲ E(t) +Now using Proposition 6.8 with h = +� +Ag, f = ∂α′ +1 +Z,α′ and w = +1 +Z,α′ we get +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +≲ +��� +� +Ag +��� +∞ +����Dα′ +1 +Z,α′ +���� ˙H +1 +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +�����∂α′ +�� +Ag +Z,α′ +������ +2 ++ +��� +� +Ag +��� +∞ +����∂α′ +1 +Z,α′ +���� +2 +2 + +68 +SIDDHANT AGRAWAL +≲ +���Zt,α′ +�� +2 + √g +�����Dα′ +1 +Z,α′ +���� ˙H +1 +2 ++ +����∂α′ +1 +Z,α′ +���� +2 +� +��Dα′Zt +�� +L∞∩ ˙H +1 +2 ++ +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Hence we see that E2(t) +1 +2 ≲ E(t). The same argument as above also shows that +����� +� +Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 +≲ +���Zt,α′ +�� +2 + √g +����� +1 +Z,α′ D2 +α′Zt +���� ˙H +1 +2 ++ +��D2 +α′Zt +�� +2 +� +��Dα′Zt +�� +L∞∩ ˙H +1 +2 ++ +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +� +Now from (67), (68) and (69) and from the argument there, it is easy to see that +��DtD2 +α′Zt +�� +2 +≲ +����AgD2 +α′ +1 +Z,α′ +���� +2 ++ Ea(t) +1 +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +����∂α′ +1 +Z,α′ +���� +2 +≲ +���Zt,α′ +��2 +2 + g +�����D2 +α′ +1 +Z,α′ +���� +2 ++ Ea(t) +1 +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +����∂α′ +1 +Z,α′ +���� +2 +Now from this and previously proved estimates we therefore get E3(t) +1 +2 ≲ E(t) +3 +2 . Hence +proved. +Step 2: We now prove (80) and (81). Clearly E1(t) ≲ E(t) as it is the same as E1(t). +Now from (62) we see that +���Zt,α′ +�� +2 + √g +���D2 +α′Zt +�� +2 ≲ Ea(t) +1 +2 ≲ E(t) +1 +2 +Using Proposition 6.8 with h = +1 +√ +Ag , w = +1 +Z,α′ and f = +� +Ag∂α′ +1 +Z,α′ we get +����Dα′ +1 +Z,α′ +���� ˙H +1 +2 +≲ +����� +1 +� +Ag +����� +∞ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +�����∂α′ +� +1 +Z,α′� +Ag +������ +2 +���� +� +Ag∂α′ +1 +Z,α′ +���� +2 ++ +����� +1 +� +Ag +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 +2 +��� +� +Ag +��� +∞ + +2D WATER WAVES +69 +≲ +1 +√g +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ 1 +√g +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +2 ++ 1 +g +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +����Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +Hence we see that E2(t) +1 +2 ≲ C1(E(t) + g + 1 +g + +��Zt,α′ +�� +2(t)). Now by using the same proof +as above we also get the estimate +���� +1 +Z,α′ D2 +α′Zt +���� ˙H +1 +2 +≲ +1 +√g +����� +� +Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 ++ 1 +√g +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +��D2 +α′Zt +�� +2 ++ 1 +g +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +����Zt,α′ +�� +2 + √g +���D2 +α′Zt +�� +2 +Hence +���Zt,α′ +�� +2 + √g +���� +1 +Z,α′ D2 +α′Zt +��� ˙H +1 +2 ≲ C1(E(t) + g + 1 +g + +��Zt,α′ +�� +2(t)). Finally note that +we have already proved the estimate +����AgD2 +α′ +1 +Z,α′ +���� +2 +≲ +��DtD2 +α′Zt +�� +2 + Ea(t) +1 +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +����∂α′ +1 +Z,α′ +���� +2 +From this we easily see that +���Zt,α′ +�� +2 + √g +�����D2 +α′ +1 +Z,α′ +���� +2 +≲ 1 +g E(t) +3 +2 +Hence (80) now follows. To prove (81), we first observe from Lemma 4.2 that +d +dt +���Zt,α′ +��2 +2 + g +� +≲ ∥bα′∥∞ +��Zt,α′ +��2 +2 + +��Zt,α′ +�� +2 +��Ztt,α′ +�� +2 +≲ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +����Zt,α′ +��2 +2 + g +� +≲ Ea(t) +1 +2 +���Zt,α′ +��2 +2 + g +� +(84) + +70 +SIDDHANT AGRAWAL +Hence there exists a universal constant C4 > 0 such that +exp +� +−C4 +� t +0 +Ea(s) +1 +2 ds +����Zt,α′ +��2 +2(0) + g +� +≤ +��Zt,α′ +��2 +2(t) + g +≤ exp +� +C4 +� t +0 +Ea(s) +1 +2 ds +����Zt,α′ +��2 +2(0) + g +� +(85) +Hence combining this with (80) proves (81). +Step 3: We now prove E(t) ≲ Eb(t) ≲ E(t). Let us first prove Eb(t) ≲ E(t). First we +obviously have Eb,1(t) ≲ E(t). Next we see that +�����∂α′ +� +Zt,α′ +Z2 +,α′ +������ +2 +≲ +��D2 +α′Zt +�� +2 + +����∂α′ +1 +Z,α′ +���� +2 +��Dα′Zt +�� +∞ +Hence by using the estimates proved for E(t), we see that Eb,2(t) +1 +2 ≲ Ea(t) ≲ E(t). Now +observe that +�����∂α′ +� +Ag +Z2 +,α′ +∂α′ +1 +Z,α′ +� +− AgD2 +α′ +1 +Z,α′ +����� +2 +≲ +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 ++ +����� +Ag +��Z,α′ +�� +� +∂α′ +1 +Z,α′ +�� +∂α′ +1 +Z,α′ +������ +2 +≲ +����� +1 +��Z,α′ +��2 ∂α′Ag +����� +∞ +����∂α′ +1 +Z,α′ +���� +2 ++ +����� +Ag +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 +����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 +(86) +where in the last step we used Proposition 6.8 with f = g = ∂α′ +1 +Z,α′ and w = +Ag +|Z,α′|. We +also see that +����� +� +Ag +Z,α′ ∂α′ +� +Zt,α′ +Z2 +,α′ +� +− +� +Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 +≲ +����� +�� +Ag +Z,α′ ∂α′ +1 +Z,α′ +� +(Dα′Zt) +����� ˙H +1 +2 +≲ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +��Dα′Zt +�� +∞ + +���Zt,α′ +�� +2 + √g +���D2 +α′Zt +�� +2 +����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 +(87) + +2D WATER WAVES +71 +where in the last step we used Proposition 6.8 with f = ∂α′ +1 +Z,α′ , w = +√ +Ag +Z,α′ and h = Dα′Zt. +From these estimates and from previous estimates proved for E(t), we therefore see that +Eb,3(t) +1 +2 ≲ E(t) +3 +2 . Hence Eb(t) ≲ E(t). +Now let us prove E(t) ≲ Eb(t). First we clearly have E1(t) ≲ Eb(t). Next by modifying +the estimates (61), (62) and (63) by replacing +���Dt +� +∂α′ +1 +Z,α′ +���� +2 with +����∂α′ +� +Zt,α′ +Z2 +,α′ +����� +2 +we easily +get +��Dα′Zt +�� +L∞∩ ˙H +1 +2 ≲ +���∂α′ +1 +Z,α′ +��� +2 +��Zt,α′ +�� +2 + +�����∂α′ +� +Zt,α′ +Z2 +,α′ +������ +1 +2 +2 +��Zt,α′ +�� +1 +2 +2 +and also +����Dt +� +∂α′ +1 +Z,α′ +����� +2 +≲ +���∂α′ +1 +Z,α′ +��� +2 +2 +��Zt,α′ +�� +2 + +���∂α′ +1 +Z,α′ +��� +2 +��Dα′Zt +�� +∞ + +�����∂α′ +� +Zt,α′ +Z2 +,α′ +������ +2 +Hence from this we see that Ea(t) ≲ Eb(t). Now from (86) and (67), (68), (69) we see that +��DtD2 +α′Zt +�� +2 +≲ +�����∂α′ +� +Ag +Z2 +,α′ +∂α′ +1 +Z,α′ +������ +2 ++ Ea(t) +1 +2 +�����Dt +� +∂α′ +1 +Z,α′ +����� +2 ++ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +� ++ Ea(t) +����∂α′ +1 +Z,α′ +���� +2 +Also from (87) we obtain +����� +� +Ag +Z,α′ D2 +α′Zt +����� ˙H +1 +2 +≲ +����� +�Ag +Z,α′ ∂α′ +� +Zt,α′ +Z2 +,α′ +������ ˙H +1 +2 ++ +����� +�Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +��Dα′Zt +�� +∞ ++ +���Zt,α′ +�� +2 + √g +���D2 +α′Zt +�� +2 +����∂α′ +1 +Z,α′ +���� +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +���Zt,α′ +�� +2 + √g +�����∂α′ +1 +Z,α′ +���� +2 +�2����∂α′ +1 +Z,α′ +���� +2 +From this we therefore get E3(t) +1 +2 ≲ Eb(t) +3 +2. This proves E(t) ≲ Eb(t). +Step 4: We now prove (82). Let +L = sup +t∈[0,T] +E(t) + T + g + 1 +c1 ++ ∥Zt∥H1(0) + +���� +1 +Z,α′ − 1 +���� +2 +(0) + +72 +SIDDHANT AGRAWAL +Now from (85) we see that +��Zt,α′ +��2 +2(t) + g + +1 +��Zt,α′ +��2 +2(t) + g +≲L 1 +Hence we see that +sup +t∈[0,T] +���Zt,α′ +�� +2(t) + +��Dα′Zt +�� +∞(t) + +����∂α′ +1 +Z,α′ +���� +2 +(t) +� +≲L 1 +Now let +f(t) = +���� +1 +Z,α′ − 1 +���� +2 +2 ++ ∥Zt∥2 +2 + 1 +Therefore we see that f(0) ≲L 1. Now by following the proof of Lemma 6.2 in [2], we see +that ∂tf ≲L f. Hence (82) therefore follows. +Step 5: We now prove (83). Let +M = sup +t∈[0,T] +E(t) + g + 1 +g + T + ∥Zt∥H1(0) + +��Z,α′ +�� +∞(0) + +���� +1 +Z,α′ − 1 +���� +2 +(0) +As g > 0, by using the definition of E(t) and (80) we therefore see that +sup +t∈[0,T] +�����∂α′ +1 +Z,α′ +���� +2 +2 +(t) + +��D2 +α′Zt +��2 +2(t) + +����D2 +α′ +1 +Z,α′ +���� +2 +2 ++ +���� +1 +Z,α′ D2 +α′Zt +���� +2 +˙H +1 +2 +� +≲M 1 +(88) +Now we have +����Dα′ +1 +Z,α′ +���� +∞ +≲ +���� +1 +Ag +���� +∞ +����AgDα′ +1 +Z,α′ +���� +∞ +≲ 1 +g E(t) +Hence +sup +t∈[0,T] +����Dα′ +1 +Z,α′ +���� +∞ +≲M 1 +Now observe that +(∂t + b∂α′)Z,α′ = DtZ,α′ = Zt,α′ − bα′Z,α′ = Z,α′(Dα′Zt − bα′) +As ∥Dα′Zt∥∞ + ∥bα′∥∞ ≲ Ea(t) +1 +2, we see that for all t ∈ [0, T] we have +��Z,α′ +�� +∞(0) ≲M +��Z,α′ +�� +∞(t) ≲M +��Z,α′ +�� +∞(0) +Combining this with (88) we easily get +sup +t∈[0,T] +�����∂α′ +1 +Z,α′ +���� +H1 +(t) + +��∂α′Z,α′ +�� +H1(t) + +��Zt,α′ +�� +H1(t) +� +≲M 1 +Now observe that +1 +Z,α′ D2 +α′Zt = (Dα′Zt)Dα′ +1 +Z,α′ + +1 +Z3 +,α′ +∂α′Zt,α′ + +2D WATER WAVES +73 +From Proposition 6.5 we clearly have the estimate +����(Dα′Zt)Dα′ +1 +Z,α′ +���� ˙H +1 +2 +≲ +��Dα′Zt +�� +L∞∩ ˙H +1 +2 +����Dα′ +1 +Z,α′ +���� +L∞∩ ˙H +1 +2 +≲M 1 +This implies that for all t ∈ [0, T] +����� +1 +Z3 +,α′ +∂α′Zt,α′ +����� ˙H +1 +2 +(t) ≲M 1 +Hence again by using Proposition 6.5 we get for all t ∈ [0, T] +��∂α′Zt,α′ +�� +˙H +1 +2 (t) ≲ +��Z,α′ +��3 +∞(t) +����� +1 +Z3 +,α′ +∂α′Zt,α′ +����� ˙H +1 +2 +(t) + +��Z,α′ +��2 +∞(t) +��∂α′Z,α′ +�� +2(t) +��∂α′Zt,α′ +�� +2(t) +≲M 1 +So we are left to prove only the lower order estimates, namely proving the estimate +∥Zt∥2(t) + +��� +1 +Z,α′ − 1 +��� +2(t) + +��Z,α′ − 1 +�� +2(t) ≲M 1 for t ∈ [0, T]. This directly follows from +(82) as L ≤ M and from the fact that +��Z,α′ +�� +∞(t) ≲M 1. Hence proved. +□ +We now note down an existence result in Sobolev spaces. +Theorem 4.6 ([31]). Let g > 0 and let s ≥ 4. Assume that the initial data (Z, Zt)(0) +satisfies (Z,α′ − 1, +1 +Z,α′ − 1, Zt)(0) ∈ Hs(R) × Hs(R) × Hs+ 1 +2 (R). Then there exists a T > +0 such that on [0, T] the initial value problem for (10) has a unique solution (Z, Zt)(t) +satisfying (Z,α′ − 1, +1 +Z,α′ − 1, Zt) ∈ Cl([0, T], Hs−l(R) × Hs−l(R) × Hs+ 1 +2 −l(R)) for l = 0, 1. +Moreover if T ∗ is the maximal time of existence, then either T ∗ = ∞ or T ∗ < ∞ and +sup +t∈[0,T ∗) +���Z,α′ − 1 +�� +H2(t) + +���� +1 +Z,α′ − 1 +���� +H2 +(t) + ∥Zt∥H2+ 1 +2 (t) +� += ∞ +Proof. The existence part of the result is just a reformulation of Theorem 5.11 of [31]. The +result written there is for g = 1 but the same result holds for g > 0. Also the assumptions +on the initial data are given there in terms of Zt and Ztt, but from (13) we see that +Ztt = −i +� 1 +Z,α′ − 1 +� +Ag − iAg + ig +Now from (10) we see that Ag = g − Im[Zt, H]Zt,α′ ≥ g > 0. From this it is easy to see +that (Z,α′ − 1, +1 +Z,α′ − 1, Zt)(0) ∈ Hs(R) × Hs(R) × Hs+ 1 +2(R) is equivalent to (Zt, Ztt)(0) ∈ +Hs+ 1 +2 (R) × Hs(R) along with − ∂P +∂ˆn ◦ h−1(0) ≥ c > 0 for some c > 0 (Recall from (14) that +− ∂P +∂ˆn ◦h−1 = +Ag +|Z,α′|). The blow up criterion written here is not directly implied by Theorem +5.11 of [31] as there the blow up criterion is in terms of one higher derivative. However +by modifying the proof there one easily gets the above blow up criterion. Another way to + +74 +SIDDHANT AGRAWAL +see the blow up criterion is that it is implied by the blowup criterion of Theorem 3.6 in +[35]. +□ +Proof of Corollary 3.2. The existence and uniqueness of the solution follows directly from +Theorem 4.6. The fact that the time of existence is independent of g follows from Theo- +rem 3.1, Lemma 4.5 and the blow up criterion of Theorem 4.6. +□ +Proof of Theorem 3.3. Let 0 < ǫ < 1. Define +Ψǫ(z′, 0) = Ψ(z′ − iǫ, 0), +U ǫ(z′, 0) = U(z′ − iǫ, 0) +and +hǫ(α, 0) = α +As U(·, 0) and Ψ(·, 0) are holomorphic functions in P−, we see that the functions Zǫ(·, 0) +and Zǫ +t(·, 0) defined by +Zǫ(α′, 0) = Ψǫ(α′, 0) +and +Zǫ +t(α′, 0) = U ǫ(α′, 0) +are smooth functions on R for all 0 < ǫ < 1. Moreover as Ψz′ ̸= 0 in P−, we see that +Zǫ +,α′(α′, 0) ̸= 0 for all α′ ∈ R and 0 < ǫ < 1. Also from the initial data it is clear that for +all 0 < ǫ < 1 we have Zǫ +,α′(α′, 0) → 1 and +1 +Zǫ +,α′ (α′, 0) → 1 as |α′| → ∞. We also see from +the initial data (33) that for all 0 < ǫ < 1 we have +��Zǫ +t +�� +H1(0) + +����� +1 +Zǫ +,α′ +− 1 +����� +H1 +(0) ≤ c0 +(89) +Step 1: We first consider the case of g > 0. We observe that (Zǫ, Zǫ +t )(0) satisfies the +assumptions of Theorem 4.6 and therefore there exists a time Tǫ > 0 such that the initial +value problem to (10) has a unique smooth solution (Zǫ, Zǫ +t )(t) in the time interval [0, Tǫ] +so that for all s ≥ 4 we have +sup +t∈[0,Tǫ] +� +��Zǫ +,α′ − 1 +�� +Hs(t) + +����� +1 +Zǫ +,α′ +− 1 +����� +Hs +(t) + ∥Zǫ +t∥Hs+ 1 +2 (t) +� +< ∞ +(90) +Now from the definition of E(t), it is clear that E(Zǫ, Zǫ +t )(0) ≤ E(0). Also from Lemma 4.5 +there exists a universal constant M > 0 such that E(Zǫ, Zǫ +t )(0) ≤ ME(Zǫ, Zǫ +t )(0) ≤ ME(0). +Hence from (31), (83) and the blow up criterion of Theorem 4.6 we see that there exists +T, C1 > 0 depending only on E(0) with T ≳ +1 +√ +E(0) so that the smooth solution (Zǫ, Zǫ +t)(t) +exists in [0, T] and we have supt∈[0,T] E(Zǫ, Zǫ +t )(t) ≤ C1. Now from (81) we see that there +exists a constant C2 depending only on E(0), c0 and g so that supt∈[0,T] E(Zǫ, Zǫ +t )(t) ≤ C2. +Let Dǫ +α′ = +1 +Zǫ +,α′ ∂α′ and +��Dǫ +α′ +�� = +1 +|Zǫ +,α′|∂α′. Now from (85) and the definition of E(t) we see +that there exists a constant C3 > 0 depending only on E(0), c0 and g so that for all t ∈ [0, T] + +2D WATER WAVES +75 +we have +C3 ≥ +��Zǫ +t,α′ +��2 +2(t) + +�����∂α′ +1 +Zǫ +,α′ +����� +2 +2 +(t) + +��(Dǫ +α′)2Zǫ +t +��2 +2(t) + +�����(Dǫ +α′)2 1 +Zǫ +,α′ +����� +2 +2 +(t) ++ +����� +1 +Zǫ +,α′ +(Dǫ +α′)2Zǫ +t +����� +2 +˙H +1 +2 +(t) +By using H(Dǫ +α′Zǫ +t) = Dǫ +α′Zǫ +t we also get +��Dǫ +α′Zǫ +t +��2 +˙H +1 +2 = i +� +(Dǫ +α′Zǫ +t) +� +∂α′Dǫ +α′Zǫ +t +� +dα′ ≤ +��Zǫ +t,α′ +�� +2 +��(Dǫ +α′)2Zǫ +t +�� +2 ≤ C3 +Now as g > 0, we see from (82) that there exists a constant C4 depending only on E(0), c0 +and g so that for all t ∈ [0, T] we have +∥Zǫ +t∥2(t) + +����� +1 +Zǫ +,α′ +− 1 +����� +2 +(t) ≤ C4 +Hence we can now simply follow the proof of Theorem 3.9 of [35] to see that we have +a solution (U, Ψ, P)(t) to (8) in the time interval [0, T] with supt∈[0,T] E(t) ≤ C2. From +the proof of Theorem 3.9 of [35], we easily see that this solution satisfies essentially all +the properties of the class SA, except the final condition (d) in Definition 4.3. Now by +construction we see that F∆((Z(ǫ), Z(ǫ) +t ), (Z, Zt))(0) → 0 as ǫ → 0, and hence now by +Theorem 3.7 of [35], we see that supt∈[T1,T2] F∆((Z(ǫ), Z(ǫ) +t ), (Z, Zt))(t) → 0 as ǫ → 0. +Hence the solution (U, Ψ, P)(t) lies in the class SA. +For uniqueness, suppose we have two distinct solutions (U a, Ψa, Pa)(t) and (U b, Ψb, Pb)(t) +in the class SA with the same initial data. Define the time +Tu = inf +� +t ∈ [0, T] +��� ∃α′ ∈ R, such that (Za, Za +t )(α′, t) ̸= (Zb, Zb +t )(α′, t) +� +By the continuity of the functions Z, Zt we see that Tu ∈ [0, T) and that the two solutions +are equal at time Tu. Now from the definition of the class SA, we see that there exists +Tu < T2 ≤ T such that on the interval [Tu, T2] there exists smooth solutions (Za,(n), Za,(n) +t +) +and (Zb,(n), Zb,(n) +t +) converging to (Za, Za +t ) and (Zb, Zb +t ) respectively in the norm of F∆. +Now observe that if a sequence of smooth solutions (Z(n), Z(n) +t +) solves the system (10) with +gravity g > 0 in the time interval [T1, T2], supn∈N,t∈[T1,T2] +� +En(t) + +��Z(n) +t,α′ +�� +2(t) +� +≤ C, then +by using the definition of the energy E(t) and the argument used above, we see that for all + +76 +SIDDHANT AGRAWAL +n ∈ N and t ∈ [T1, T2] we have +1 ≳C,g +���Z(n) +t,α′ +��� +2 +2(t) + +������ +∂α′ +1 +Z(n) +,α′ +������ +2 +2 +(t) + +���(D(n) +α′ )2Z(n) +t +��� +2 +2(t) + +������ +(D(n) +α′ )2 +1 +Z(n) +,α′ +������ +2 +2 +(t) ++ +���D(n) +α′ Z(n) +t +��� +2 +˙H +1 +2 (t) + +������ +1 +Z(n) +,α′ +(D(n) +α′ )2Z(n) +t +������ +2 +˙H +1 +2 +(t) +where D(n) +α′ += +1 +Z(n) +,α′ ∂α′. +Hence we can now directly apply Theorem 3.7 of [35] to get +supt∈[Tu,T2] F∆((Za,(n), Za,(n) +t +), (Zb,(n), Zb,(n) +t +))(t) → 0 as n → ∞ thereby proving that +(U a, Ψa, Pa)(t) = (U b, Ψb, Pb)(t) for all t ∈ [Tu, T2]. This proves uniqueness. +Step 2: Let us now consider the case of g = 0. Observe that if +��Zt,α′ +�� +2(0) = 0, then as +Zt(0) ∈ H1(R), we see that Zt(α′, 0) = 0 for all α′ ∈ R. As g = 0, this implies that the trivial +solution namely Z(α′, t) = Z(α′, 0) and Zt(α′, t) = 0 for all α′ ∈ R, t ∈ [0, ∞), is a global +solution to the water wave equation. Observe that in this case we have E(t) = Eb(t) = 0 +for all t ≥ 0. Hence assume that +��Zt,α′ +�� +2(0) = c2 > 0. We mollify the initial data in the +same way as in the g > 0 case and so we still have (89). It is clear that +��Zǫ +t,α′ +�� +2(0) → c2 +as ǫ → 0, and so let 0 < ǫ0 ≤ min +� +c2 +2, 1 +� +be small enough so that for all 0 < ǫ ≤ ǫ0 we +have c2/2 ≤ +��Zǫ +t,α′ +�� +2(0) ≤ c2. Let gǫ = ǫ. Hence for the initial data (Zǫ, Zǫ +t)(0) we have +a unique smooth solution to (10) with gravity gǫ in a time interval Tǫ > 0 satisfying (90). +By the choice of ǫ0 and gǫ, it is clear that for all 0 < ǫ ≤ ǫ0 we have +c2 +2 +4 ≤ +��Zǫ +t,α′ +��2 +2(0) + gǫ ≤ 2c2 +2 +and hence E(Zǫ, Zǫ +t)(0) ≲ E(0). Hence from the same argument as above for g > 0, we see +that there exists T, C5 > 0 depending only on E(0) with T ≳ +1 +√ +E(0) so that the smooth +solution (Zǫ, Zǫ +t)(t) exists in [0, T] and we have supt∈[0,T] E(Zǫ, Zǫ +t )(t) ≤ C5. Now from +Lemma 4.5 we see that Eb(Zǫ, Zǫ +t)(t) ≲ C5 for all t ∈ [0, T]. +Now from (82) we see that for all t ∈ [0, T] and 0 < ǫ ≤ ǫ0 we have +∥Zǫ +t∥H1(t) + gǫ + +����� +1 +Zǫ +,α′ +− 1 +����� +H1 +(t) + +1 +��Zǫ +t,α′ +��2 +2(t) + gǫ +≲c0,c2,E(0) 1 +(91) +Let bǫ and Aǫ +g be given by (10) with Zt, Z,α′ and g replaced by Zǫ +t, Zǫ +,α′ and gǫ respectively. +Let Dǫ +t = ∂t + bǫ∂α′. From the quantities controlled in §4.3 and we see that for all t ∈ [0, T] +and 0 < ǫ ≤ ǫ0 we have +��Aǫ +g +�� +∞(t) + ∥Dǫ +α′Zǫ +t∥∞(t) + ∥bǫ +α′∥∞(t) + +���� +Dǫ +tAǫ +g +Aǫg +���� +∞ +(t) + +����� +1 +�Aǫg +|Dǫ +α′|Aǫ +g +����� +2 +(t) ≲c2,E(0) 1 + +2D WATER WAVES +77 +Now from (43) we see that +�����Dǫ +t +1 +Zǫ +,α′ +����� +∞ +≲ +����� +1 +Zǫ +,α′ +����� +∞ +(∥Dǫ +α′Zǫ +t∥∞ + ∥bǫ +α′∥∞) +and from (13) we obtain +∥Dǫ +tZǫ +tt∥∞ ≲ +��Aǫ +g +�� +∞ +�����Dǫ +t +1 +Zǫ +,α′ +����� +∞ ++ +����� +1 +Zǫ +,α′ +����� +∞ +��Aǫ +g +�� +∞ +���� +Dǫ +tAǫ +g +Aǫg +���� +∞ +Hence we see that for all t ∈ [0, T] and 0 < ǫ ≤ ǫ0 we have +1 ≳c0,c2,E(0) ∥Zǫ +t ∥∞(t) + +��Zǫ +t,α′ +�� +2(t) + ∥Zǫ +tt∥∞(t) + +��Zǫ +tt,α′ +�� +2(t) + ∥Dǫ +tZǫ +tt∥∞(t) ++ +����� +1 +Zǫ +,α′ +����� +∞ +(t) + +�����∂α′ +1 +Zǫ +,α′ +����� +2 +(t) + +�����Dǫ +t +1 +Zǫ +,α′ +����� +∞ +(t) +Now from (10) it is easy to see that +∥bǫ∥H1 ≲ ∥Zǫ +t∥H1 +����� +1 +Zǫ +,α′ +− 1 +����� +H1 +From (10), the relation HZt = −Zt and the fact that +1 +Zǫ +,α′ → 1 as |α′| → ∞, we see that +bǫ = Re[Zǫ +t, H] +� 1 +Zǫ +,α′ +− 1 +� ++ 2Re(Zǫ +t ) +Now using Proposition 6.1 we see that +∥Dǫ +tbǫ∥∞ ≲ ∥Zǫ +tt∥∞ + +�����[Zǫ +tt, H] +� 1 +Zǫ +,α′ +− 1 +������ +∞ ++ +�����[Zǫ +t, H]∂α′ +� +bǫ +� 1 +Zǫ +,α′ +− 1 +������� +∞ ++ +����� +� +bǫ, Zǫ +t; +� 1 +Zǫ +,α′ +− 1 +������� +∞ +Therefore from Proposition 6.4 and Proposition 6.6 we obtain +∥Dǫ +tbǫ∥∞ ≲ ∥Zǫ +tt∥∞ + +��Zǫ +tt,α′ +�� +2 +����� +1 +Zǫ +,α′ +− 1 +����� +2 ++ +��Zǫ +t,α′ +�� +2∥bǫ +α′∥H1 +����� +1 +Zǫ +,α′ +− 1 +����� +H1 +Hence we see that for all t ∈ [0, T] and 0 < ǫ ≤ ǫ0 we have +∥bǫ∥∞(t) + ∥bǫ +α′∥2(t) + ∥Dǫ +tbǫ∥∞(t) ≲c0,c2,E(0) 1 +Now we can follow the proof of Theorem 3.9 of [35] to see that there exists a solution +(U, Ψ, P)(t) to (8) with zero gravity belonging to the class SA. Let Z(α′, t) = Ψ(α′, t) and +Zt(α′, t) = U(α′, t). We are only left to show that supt∈[0,T] Eb(Z, Zt)(t) < ∞. + +78 +SIDDHANT AGRAWAL +As +1 +Ψz′ extends continuously to P − × [0, T], by abuse of notation we write +1 +Z,α′ (α′, t) = +1 +Ψz′ (α′, t) for (α′, t) ∈ R × [0, T]. From the proof of Theorem 3.9 of [35] we moreover see +that there exists a subsequence ǫn → 0 and a function Ztt : R × [0, T] → C so that +1 +Zǫn +,α′ +→ +1 +Z,α′ , +Zǫn +t +→ Zt +and +Zǫn +tt → Ztt +(92) +uniformly on compact subsets of R × [0, T]. Now as +��Zǫn +t,α′ +�� +2(t) are uniformly bounded, we +see that for any fixed t ∈ [0, T] +(Zǫn +t Zǫn +t,α′)(t) → (ZtZt,α′)(t) +weakly in L2(R) +If we define A0 = −Im[Zt, H]Zt,α′ = −Im +� +(I − H)(ZtZt,α′) +� +, then we therefore see that for +any fixed t ∈ [0, T] +Aǫn +g (t) − gǫn → A0(t) +weakly in L2(R) +This in particular implies that +Aǫn +g (t) +Zǫn +,α′ +− gǫn +Zǫn +,α′ +→ A0(t) +Z,α′ +weakly in L2(R) +However we know that Zǫn +tt → Ztt uniformly on compact subsets of R × [0, T] and we also +have the relation (13). This therefore implies that +Aǫn +g (t) +Zǫn +,α′ +→ A0(t) +Z,α′ +uniformly on compact subsets of R +By a similar logic we see that +Aǫn +g (t) +(Zǫn +,α′)2(t) → A0(t) +Z2 +,α′(t) +and +� +Aǫn +g (t) +��Zǫn +,α′ +��(t) → +� +A0(t) +��Z,α′ +��(t) +uniformly on compact subsets of R. Now as +1 +Ψz′ (t) is a bounded continuous function on +P − and +1 +Ψz′ → 1 as z′ → ∞ and Ψz′(z′) ̸= 0 for z′ ∈ P−, we see that the set S(t) = +� +α′ ∈ R +��� +1 +Ψz′ = 0 +� +is a compact set of measure zero (see Theorem 17.18 in [27]). We also +see that ωǫn → ω uniformly on Oβ,t = +� +α′ ∈ R +���� +1 +|Z,α′|(α′, t) > β +� +for any β > 0. This then +implies that +� +Aǫn +g (t) +Zǫn +,α′(t) +→ +� +A0(t) +Z,α′(t) +uniformly on compact subsets of R +We can now show that supt∈[0,T] Eb(Z, Zt)(t) < ∞. Clearly supt∈[0,T] Eb,1(Z, Zt)(t) < ∞ +from (91). To control supt∈[0,T] Eb,2(Z, Zt)(t), observe that supt∈[0,T] Eb(Zǫn, Zǫn +t )(t) ≲ C5, + +2D WATER WAVES +79 +and therefore from (91) we obtain +���Zǫn +t,α′ +��� +2(t) + +�����∂α′ +1 +Zǫn +,α′ +����� +2 +(t) + +�����∂α′ +� +Zǫn +t,α′ +(Zǫn +,α′)2 +������ +2 +(t) + +����� +� +Aǫn +g +Zǫn +,α′ +∂α′ +1 +Zǫn +,α′ +����� ˙H +1 +2 +(t) ≲c0,c2,E(0) 1 +These estimates along with (92) easily gives us supt∈[0,T] Eb,2(Z, Zt)(t) < ∞. The argument +for Eb,3(Z, Zt)(t) is similar. +Step 3: In Step 3 and Step 4 we prove the blowup criterion for g > 0. Let (U, Ψ, P)(t) +be a solution to (8) with gravity g in [0, T] in the class SA with supt∈[0,T] E(t) < ∞. To +prove the blowup criterion we need to prove a version of (30), so first we need to be able +to make sense of the energy Ea(t) for this singular solution. Fix t ∈ [0, T] and as usual let +Zt(α′, t) = U(α′, t) and +1 +Z,α′ (α′, t) = +1 +Ψz′ (α′, t) be the boundary values of U and +1 +Ψz′ . From +the definition of the class SA, we see that (Zt, +1 +Z,α′ − 1)(·, t) ∈ H1(R) × H1(R). Hence +all quantities in Ea(t) except for Dt∂α′ +1 +Z,α′ are well defined. We now give an equivalent +definition of Dt∂α′ +1 +Z,α′ which makes sense for this singular solution and which agrees with +the standard definition for smooth solutions. +For smooth solutions, a straightforward calculation using (43) gives us +Dt∂α′ +1 +Z,α′ = −ω2 +� +∂α′ +1 +Z,α′ +� +Dα′Zt − 2ω2 +� +∂α′ +1 +Z,α′ +� +Dα′Zt + +� +∂α′ +1 +Z,α′ +� +Dα′Zt +ω|Dα′| +� +bα′ − Dα′Zt − Dα′Zt +� ++ ω2∂α′ +� 1 +Z,α′ Dα′Zt +� +(93) +where from (58) and (59) we see that +|Dα′| +� +bα′ − Dα′Zt − Dα′Zt +� += Re +� +ω(I − H)Dα′(bα′ − Dα′Zt − Dα′Zt) +� +− Re +� +ω +� 1 +Z,α′ , H +� +∂α′(bα′ − Dα′Zt − Dα′Zt) +� += Re +� +ω(I − H) +� +−Dα′Dα′Zt + (Dα′Zt) +� +∂α′ +1 +Z,α′ +��� ++ Re +� +ω +� +PA +� Zt +Z,α′ +� +, H +� +∂α′ +� +∂α′ +1 +Z,α′ +�� +− Re +� +ω +� 1 +Z,α′ , H +� +∂α′(bα′ − Dα′Zt − Dα′Zt) +� +(94) +Here from (57) we have +bα′ − Dα′Zt − Dα′Zt = Re +�� 1 +Z,α′ , H +� +Zt,α′ + [Zt, H] +� +∂α′ +1 +Z,α′ +�� +(95) + +80 +SIDDHANT AGRAWAL +and a straightforward computation gives us +Dα′Dα′Zt = −2ω2 +� +∂α′ +1 +Z,α′ +� +Dα′Zt + +� +∂α′ +1 +Z,α′ +� +Dα′Zt + ω2∂α′ +� 1 +Z,α′ Dα′Zt +� +(96) +With these equation we can now make sense of Dt∂α′ +1 +Z,α′ for a solution with E(t) < ∞. To +see this, observe that +1 +Z,α′ Dα′Zt is the boundary value of +1 +Ψ2 +z′ ∂zU and by the using the fact +that E(t) < ∞ we see that +1 +Z,α′ Dα′Zt(·, t) ∈ H1(R). Now define +S(t) = +� +α′ ∈ R +���� +1 +Ψz′ (α′, t) = 0 +� +and +NS(t) = R\S(t) +Now as shown in [1], we see that S(t) is a closed bounded set of measure zero and hence +ω is a well defined function on NS(t) with |ω| = 1. It was also shown there that +1 +Ψz′ Uz +and +1 +Ψz′ U z extend continuously to the boundary and hence Dα′Zt and Dα′Zt are well +defined bounded continuous functions. From the definition of the class SA, we clearly have +Zt,α′(·, t), ∂α′ +1 +Z,α′ (·, t) ∈ L2(R) and it is also easy to see that the estimate (56) holds for +these solutions. Hence all terms are now justified and hence Dt∂α′ +1 +Z,α′ (·, t) is a well defined +function in L2(R). +Claim: Consider a solution in time [0, T] in the class SA with supt∈[0,T] E(t) < ∞ and let +M > 0 be such that +sup +t∈[0,T] +���Dα′Zt +�� +L∞∩ ˙H +1 +2 (t) + +���Zt,α′ +�� +2(t) + √g +�����∂α′ +1 +Z,α′ +���� +2 +(t) +� +≤ M +Then there exists a universal constant C > 0 such that for all t ∈ [0, T] we have Ea(t) ≤ +eCMtEa(0). +Let us now prove this claim. Assume that L > 0 is such that supt∈[0,T] E(t) ≤ L. Assume +that this claim is proved for all 0 ≤ t ≤ T1 and we will now prove it for t ∈ [T1, T1 + δ] +where δ depends only on L and M. Let (Zǫ, Zǫ +t)(α′, T1) = (Ψ, U)(α′ − iǫ, T1) and consider +the smooth solutions (Zǫ, Zǫ +t)(t) for t ≥ T1. From the definition of E(t), it is clear that +E(Zǫ, Zǫ +t)(T1) ≤ E(T1) ≤ L and hence there exists a δ1 > 0 depending only on L such that +these smooth solutions exist in [T1, T1 + δ1] with supt∈[T1,T1+δ1] E(Zǫ, Zǫ +t )(t) ≤ C(L), for +some constant C(L) depending only on L. +We now show that limǫ→0 Ea(Zǫ, Zǫ +t )(T1) = Ea(T1). To see this, we see from the defini- +tion of Zǫ, Zǫ +t and the energy E(t) that +Zǫ +t,α′(·, T1) → Zt,α′, +∂α′ +1 +Zǫ +,α′ +(·, T1) → ∂α′ +1 +Z,α′ (·, T1) +in L2(R) +∂α′ +� +1 +Zǫ +,α′ +Dǫ +α′Zǫ +t +� +(·, T1) → ∂α′ +� 1 +Z,α′ Dα′Zt +� +(·, T1) +in L2(R) +1 +Zǫ +,α′ +∂α′ +1 +Zǫ +,α′ +(·, T1) → +1 +Z,α′ ∂α′ +1 +Z,α′ (·, T1) +in ˙H +1 +2 (R) + +2D WATER WAVES +81 +This already shows that E1(Zǫ, Zǫ +t )(T1) → E1(T1). Now as mentioned before, +1 +Ψz′ Uz extends +continuously to the boundary and so Dα′Zt is a continuous function. Also as Dα′Zt(α′, t) → +0 as |α′| → ∞, this implies that Dα′Zt(·, T1) is a uniformly continuous function and so +Dǫ +α′Zǫ +t(·, T1) → Dα′Zt(·, T1) in L∞(R). Now the same argument used in [1] used to prove +the continuity of Dα′Zt also works exactly in the same was for the function +1 +Z,α′ ∂α′ +1 +Z,α′ +and hence we also get that +1 +Zǫ +,α′ ∂α′ +1 +Zǫ +,α′ (·, T1) → +1 +Z,α′ ∂α′ +1 +Z,α′ (·, T1) in L∞(R). Using the fact +that Zǫ +t,α′(·, T1) → Zt,α′(·, T1) in L2(R) and the fact that Aǫ +g ≥ g > 0, it is easy to see +from Proposition 6.4, Proposition 6.5 and Proposition 6.3 that �Aǫg(·, T1) → +� +Ag(·, T1) in +L∞ ∩ ˙H +1 +2 (R). Hence from Proposition 6.5 we get +�Aǫg +Zǫ +,α′ +∂α′ +1 +Zǫ +,α′ +(·, T1) → +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +in ˙H +1 +2 (R) +As +1 +Ψz′ (·, T1) is continuous on P −, it is clear that +1 +Zǫ +,α′ (·, T1) → +1 +Z,α′ (·, T1) in L∞(R), and +hence ωǫ → ω in L∞(R) on the set Oβ,T1 = +� +α′ ∈ R +���� +1 +|Z,α′|(α′, T1) > β +� +for any β > 0. +As Dα′Zt(α′, t) = 0 for α′ ∈ S(t) and as S(t) is a compact set of measure zero, this then +also implies that Dǫ +α′Zǫ +t(·, T1) → Dα′Zt(·, T1) in L∞(R). Now using the fact that ωǫ → ω +pointwise on NS(t) and by using dominated convergence theorem, it is easy to see that +Dǫ +t∂α′ +1 +Zǫ +,α′ (·, T1) → Dt∂α′ +1 +Z,α′ (·, T1) in L2(R). Therefore limǫ→0 Ea(Zǫ, Zǫ +t )(T1) = Ea(T1). +Now observe that for all ǫ > 0, we have +��Dǫ +α′Zǫ +t +�� +L∞∩ ˙H +1 +2 (T1) + +���Zǫ +t,α′ +�� +2(T1) + √g +������∂α′ +1 +Zǫ +,α′ +����� +2 +(T1) ≤ M +Using Lemma 4.2 and Proposition 6.9 we see that for all t ∈ [T1, T1 + δ1) we have +d +dt +� +��Dǫ +α′Zǫ +t +�� +L∞∩ ˙H +1 +2 + +���Zǫ +t,α′ +�� +2 + √g +������∂α′ +1 +Zǫ +,α′ +����� +2 +� +≲ ∥bǫ +α′∥∞ +� +��Dǫ +α′Zǫ +t +�� +L∞∩ ˙H +1 +2 + +���Zǫ +t,α′ +�� +2 + √g +������∂α′ +1 +Zǫ +,α′ +����� +2 +� ++ +��Dǫ +tDǫ +α′Zǫ +t +�� +L∞∩ ˙H +1 +2 + +��Dǫ +tZǫ +t,α′ +�� +2 +�����∂α′ +1 +Zǫ +,α′ +����� +2 ++ +���Zǫ +t,α′ +�� +2 + √g +������Dǫ +t∂α′ +1 +Zǫ +,α′ +����� +2 +≤ C(L) +Hence there exists an 0 < δ < δ1 depending only on M and L so that for all t ∈ [T1, T1 + δ] +and ǫ > 0 we have +��Dǫ +α′Zǫ +t +�� +L∞∩ ˙H +1 +2 (t) + +���Zǫ +t,α′ +�� +2(t) + √g +������∂α′ +1 +Zǫ +,α′ +����� +2 +(t) ≤ 2M + +82 +SIDDHANT AGRAWAL +Hence using (30) we see that there exists a universal constant C > 0 so that for all t ∈ +[T1, T1 + δ] and ǫ > 0 we have +Ea(Zǫ, Zǫ +t )(t) ≤ eCM(t−T1)Ea(Zǫ, Zǫ +t)(T1) +Let T2 = T1 + δ. As limǫ→0 Ea(Zǫ, Zǫ +t )(T1) = Ea(T1), to prove the claim it is therefore +enough to show that we have lim supǫ→0 Ea(Zǫ, Zǫ +t )(T2) ≥ Ea(T2). +Now from the proof of Theorem 3.9 of [35], we see that +∥Zǫ +t − Zt∥ ˙H +1 +2 (T2) + +����� +1 +Zǫ +,α′ +− +1 +Z,α′ +����� ˙H +1 +2 +(T2) → 0 +(97) +as ǫ → 0 and moreover there exists a subsequence ǫn → 0 such that +1 +Zǫn +,α′ +(·, T2) → +1 +Z,α′ (·, T2) +and +Zǫn +t (·, T2) → Zt(·, T2) +uniformly on compact subsets of R. As all the quantities in the energy Ea(Zǫ, Zǫ +t)(T2) +are uniformly bounded, we can assume by going to a subsequence if necessary that the +sequences +���Zǫn +t,α′ +��� +2, +�����∂α′ +1 +Zǫn +,α′ +����� +2 +, +�����Dǫn +t ∂α′ +1 +Zǫn +,α′ +����� +2 +and +����� +� +Aǫn +g +Zǫn +,α′ +∂α′ +1 +Zǫn +,α′ +����� ˙H +1 +2 +are all convergent sequences. Now as Zǫn +t,α′(·, T2) → Zt,α′(·, T2) and also ∂α′ +1 +Zǫn +,α′ (·, T2) → +∂α′ +1 +Z,α′ (·, T2) in distributions, this then implies that they converge weakly in L2(R) and +hence we have lim supn→∞ E1(Zǫn, Zǫn +t )(T2) ≥ E1(T2). Now by the same argument used in +Step 2 of this proof, we see that +� +Aǫn +g +Zǫn +,α′ +∂α′ +1 +Zǫn +,α′ +(·, T2) → +� +Ag +Z,α′ ∂α′ +1 +Z,α′ (·, T2) +weakly in ˙H +1 +2 (R). Hence +lim sup +n→∞ +����� +� +Aǫn +g +Zǫn +,α′ +∂α′ +1 +Zǫn +,α′ +����� ˙H +1 +2 +(T2) ≥ +����� +� +Ag +Z,α′ ∂α′ +1 +Z,α′ +����� ˙H +1 +2 +(T2) +Now observe that the set Oβ,T2 = +� +α′ ∈ R +���� +1 +|Z,α′|(α′, T2) > β +� +is an open set and that the +sequences +���∂α′Zǫn +t,α′ +��� +L2(Oβ,T2)(T2) and +�����∂2 +α′ +1 +Zǫn +,α′ +����� +L2(Oβ,T2) +(T2) +are uniformly bounded. Hence by the Arzela Ascoli theorem, there exists a subsequence +(which by abuse of notation we still call ǫn) so that +Zǫn +t,α′(·, T2) → Zt,α′(·, T2) +and ∂α′ +1 +Zǫn +,α′ +→ ∂α′ +1 +Z,α′ (·, T2) + +2D WATER WAVES +83 +uniformly on compact subsets of Oβ,T2. Now as NS(T2) = ∪n∈NOβn,T2 for βn = 1 +n, and by +a diagonalization argument we get a subsequence (again calling it ǫn) so that +Zǫn +t,α′(·, T2) → Zt,α′(·, T2) +and ∂α′ +1 +Zǫn +,α′ +→ ∂α′ +1 +Z,α′ (·, T2) +uniformly on compact subsets of NS(T2). Also clearly ωǫn → ω uniformly on compact +subsets of NS(T2). From this it is clear that +(ωǫn)2 +� +∂α′ +1 +Zǫn +,α′ +� +Dǫn +α′Zǫn +t (·, T2) → ω2 +� +∂α′ +1 +Z,α′ +� +Dα′Zt(·, T2) +weakly in L2(NS(T2)). However as the measure of S(T2) is zero, this implies that this +sequence is convergent in fact on L2(R) (As all the terms are uniformly bounded on +L2(R)). +The same argument works for all terms of (96) and (93) except for the term +ω|Dα′| +� +bα′ − Dα′Zt − Dα′Zt +� +. +For this term, first observe that from (94) and the above argument, it is easy to see that +ωǫn(I − H)Dǫn +α′ (bǫn +α′ − Dǫn +α′ Zǫn +t +− Dǫn +α′Zǫn +t )(·, T2) → ω(I − H)Dα′(bα′ − Dα′Zt − Dα′Zt)(·, T2) +weakly in L2(R). For the last term of (94), we first note that +� +bǫn +α′ − Dǫn +α′ Zǫn +t +− Dǫn +α′Zǫ +t +� +(·, T2) → +� +bα′ − Dα′Zt − Dα′Zt +� +(·, T2) +(98) +in L2(R) from (97). Now using (98), (97) and the uniform bounds on +��bǫn +α′ − Dǫn +α′ Zǫn +t +− Dǫn +α′Zǫ +t +�� +˙H +1 +2 ∩L∞(T2) +and +�����∂α′ +1 +Zǫn +,α′ +����� +2 +(T2) +it is easy to see that +�� +1 +Zǫn +,α′ +, H +� +∂α′(bǫn +α′ − Dǫn +α′ Zǫn +t +− Dǫn +α′Zǫn +t ) +� +(·, T2) +→ +�� 1 +Z,α′ , H +� +∂α′(bα′ − Dα′Zt − Dα′Zt) +� +(·, T2) +in distributions. However as the sequence is uniformly bounded on L2(R), we see that the +convergence is in fact weakly in L2(R). A similar argument works for the second term on the +right hand side of (94) as well and hence by combining this with the uniform convergence +of ωǫn → ω on compact subsets of NS(T2), we see that +Dǫn +t ∂α′ +1 +Zǫn +,α′ +(·, T2) → Dt∂α′ +1 +Z,α′ (·, T2) +weakly in L2(R). Hence this shows that lim supn→∞ E2(Zǫn, Zǫn +t )(T2) ≥ E2(T2), thereby +proving the claim. + +84 +SIDDHANT AGRAWAL +Step 4: We are now ready to prove the blow up criterion for g > 0. We will prove this via +contradiction. Assume that 0 < T ∗ < ∞ is the maximal time of existence and that +sup +t∈[0,T ∗) +���Dα′Zt +�� +L∞∩ ˙H +1 +2 (t) + +���Zt,α′ +�� +2(t) + √g +�����∂α′ +1 +Z,α′ +���� +2 +(t) +� +≤ M < ∞ +From the definition of a solution in the class SA in time [0, T ∗), we see that for any δ > 0 +small enough, supt∈[0,T ∗−δ] E(t) < ∞, and hence by the claim proved in Step 3 we see that +there exists a universal constant C > 0 so that for all t ∈ [0, T ∗) we have Ea(t) ≤ eCMtEa(0). +Define +Q = eCMT ∗Ea(0) < ∞ +Let T ∈ [0, T ∗) and as usual let (Zǫ, Zǫ +t )(α′, T) = (Ψ, U)(α′−iǫ, T) and consider the smooth +solutions (Zǫ, Zǫ +t )(t) for t ≥ T. By Step 3 we know that limǫ→0 Ea(Zǫ, Zǫ +t )(T) = Ea(T) and +hence let ǫ0 > 0 be small enough so that for all 0 < ǫ ≤ ǫ0 we have +Ea(Zǫ, Zǫ +t)(T) ≤ 2Ea(T) ≤ 2Q +Now by the estimates (30), (31), (83) and the blow up criterion of Theorem 4.6, there exists +δ1 > 0 depending only on Q such that the smooth solutions (Zǫ, Zǫ +t )(t) exist in [T, T + δ1] +and +Ea(Zǫ, Zǫ +t )(t) ≤ 4Q +for all t ∈ [T, T + δ1] +Now from (31) and part 1 of Lemma 4.5 we see that there exists a constant Q1 > 0 +depending only on Q so that +E(Zǫ, Zǫ +t )(t) ≤ Q1E(Zǫ, Zǫ +t )(T) ≤ Q1E(T) +for all t ∈ [T, T + δ1] +Now from (85) and an approximation argument, we see that there exists a constant Q2 +depending only on Q, T ∗, g and c0 so that +��Zǫ +t,α′ +�� +2(t) ≤ Q2 +for all t ∈ [T, T + δ1] +Hence from (80) we see that +E(Zǫ, Zǫ +t)(t) ≤ C1 +� +Q1E(T) + g + 1 +g + Q2 +� +for all t ∈ [T, T + δ1] +Therefore by the uniqueness of solutions, we get that for all t ∈ [T, T + δ1] we have +E(t) ≤ lim inf +ǫ→0 +E(Zǫ, Zǫ +t)(t) ≤ C1 +� +Q1E(T) + g + 1 +g + Q2 +� +(99) +Define a function J : [0, ∞) → [0, ∞) given by +J(x) = C1 +� +Q1x + g + 1 +g + Q2 +� +Using (99) repeatedly we see that for all t ∈ [0, T ∗) we have +E(t) ≤ J(N)(E(0)) + +2D WATER WAVES +85 +where N is an integer such that N > T ∗ +δ1 and J(N) is the function J composed with itself N +times. From this and the lower bound on the time of existence from Step 1, we clearly see +that we can extend the solution beyond T ∗. This contradicts the maximality of T ∗. Hence +the blow up criterion is proven. +Proof of Corollary 3.2. From Theorem 4.6 we see that there exists a time Tg > 0 such that +there exists a unique solution (Z, Zt)(t) in [0, Tg] satisfying satisfying (Z,α′ −1, +1 +Z,α′ −1, Zt) ∈ +Cl([0, T], Hs−l(R) × Hs−l(R) × Hs+ 1 +2−l(R)) for l = 0, 1. Now from +□ +□ +5. Proof of the main results for bounded domain case +5.1. Derivation of equations +In this subsection we derive the main equations and the various formulae used. +We +closely follow [10] though there are some important differences in the formulae we derive. +Note that from Lemma 2.2, a function f : S1 → C satisfies Hf = f if and only if +f = Tr(F) where F : D → C is holomorphic. Using this and the definition of �H we see +that a function f : S1 → C satisfies �Hf = f if and only if f = Tr(F) where F : D → C +is a holomorphic function with F(0) = 0. Let us now write down some common functions +which satisfy these properties: +(1) As Zt(α′, t) = U(eiα′, t), we see that Zt(α′, t) = Tr(U(z, t)). Hence HZt = Zt. +(2) We first observe that since Φ(Z(α′, t), t) = eiα′, by taking a derivative we obtain +Φz(Z(α′, t), t)Z,α′(α′, t) = ieiα′ +(100) +Now as Φ(Ψ(z, t), t) = z for all z ∈ D, we see that +Φz(Ψ(z, t), t) = +1 +Ψz(z, t) +(101) +Hence +1 +Z,α′(α′, t) = +1 +ieiα′Ψz(eiα′, t) = Tr +� +1 +izΨz(z, t) +� +(102) +Therefore we see that +eiα′ +Z,α′(α′, t) = Tr +� +1 +iΨz(z, t) +� +and consequently H +� +eiα′ +Z,α′ +� += eiα′ +Z,α′ . Using this we also see that H +� +e−iα′∂α′ +� +eiα′ +Z,α′ +�� += +e−iα′∂α′ +� +eiα′ +Z,α′ +� +. Now observe that +e−iα′∂α′ +� +eiα′ +Z,α′ +� += ∂α′ +1 +Z,α′ + +i +Z,α′ + +86 +SIDDHANT AGRAWAL +Therefore by using the fact that Av +� +∂α′ +1 +Z,α′ +� += 0, we obtain +�H +� +∂α′ +1 +Z,α′ + i +� 1 +Z,α′ − Av +� 1 +Z,α′ +��� += ∂α′ +1 +Z,α′ + i +� 1 +Z,α′ − Av +� 1 +Z,α′ +�� +(103) +(3) We see that +Zt − Av(Zt) +Z,α′ +(α′, t) = Tr +�U(z, t) − U(0, t) +izΨz(z, t) +� +Hence we have H +� +Zt−Av(Zt) +Z,α′ +� += Zt−Av(Zt) +Z,α′ +. +(4) As Zt(α′, t) = U(eiα′, t), we see that +Zt,α′(α′, t) = ieiα′Uz(eiα′, t) = Tr(izUz(z, t)) +Therefore �HZt,α′ = Zt,α′. +(5) From the above calculations we observe that +(Dα′Zt)(α′, t) = Zt,α′ +Z,α′ (α′, t) = Tr +� Uz(z, t) +Ψz(z, t) +� +Hence H(Dα′Zt) = Dα′Zt. +Let us now derive some important equations. As z(·, t) is a counterclockwise parametriza- +tion of ∂Ω(t), the unit outward normal is ˆn = −i zα +|zα|. Now as P = 0 on ∂Ω(t) we see that +∇P(z, t) = iazα, where +a = − 1 +|zα| +∂P +∂ˆn +Hence we see that the Euler equations on the boundary can be written as +ztt = −iazα +Taking a time derivative and complex conjugate, we get +zttt − iaztα = iatzα +Define A = (ahα) ◦ h−1 and A0 = A +��Z,α′ +��2. Hence by precomposing the above equations +with h−1 we obtain +Ztt = i A0 +Z,α′ +(104) +and +� +D2 +t − i +A0 +��Z,α′ +��2 ∂α′ +� +Zt = i +�at +a ◦ h−1� A0 +Z,α′ +(105) + +2D WATER WAVES +87 +5.1.1. Formulae of A0 and at +a ◦ h−1. We know that zt(α, t) = v(z(α, t), t) and hence we +have the following identities: +(1) Dαzt(α, t) = vz(z(α, t), t) and D2 +αzt(α, t) = vzz(z(α, t), t) +(2) ztt(α, t) = vt(z(α, t), t) + vz(z(α, t), t)zt(α, t) and hence +(ztt − (Dαzt)zt)(α, t) = vt(z(α, t), t) +(3) Dα(ztt − (Dαzt)zt)(α, t) = vtz(z(α, t), t) +(4) We also have +zttt(α, t) = vtt(z(α, t), t) + 2vtz(z(α, t), t)zt(α, t) + vzz(z(α, t), t)z2 +t (α, t) ++ vz(z(α, t), t)ztt(α, t) +Precomposing the second identity above with h−1 we get +(Ztt − (Dα′Zt)Zt)(α′, t) = Tr(vt(Ψ(z, t), t)) +Now multiplying both sides with Z,α′ and using (104) and (102) we obtain +(iA0 − ZtZt,α′)(α′, t) = Tr(izΨz(z, t)vt(Ψ(z, t), t)) +Note that the quantity inside the trace is a holomorphic function which vanishes at z = 0 +and that A0 is real valued. Hence applying I− �H to the above equation and taking imaginary +part we get the formula for A0 +A0 = Im +� +Zt, �H +� +Zt,α′ +(106) +Now precomposing the last identity above for zttt with h−1 and using the previous iden- +tities we get +Zttt = vtt ◦ Z + 2Dα′(Ztt − (Dα′Zt)Zt)Zt + (D2 +α′Zt)Z2 +t + (Dα′Zt)Ztt +Hence using (105) and (104) we get +i +�at +a ◦ h−1� +A0 += Z,α′Zttt − i A0 +Z,α′ Zt,α′ += Z,α′Zttt + ZttZt,α′ += Z,α′vtt ◦ Z + 2Zt∂α′(Ztt − (Dα′Zt)Zt) + Z2 +t (∂α′Dα′Zt) + 2ZttZt,α′ +(107) +Now as Z,α′(α′, t) = Tr{izΨz(z, t)} from (102) we have +(1) (Z,α′vtt ◦ Z)(α′, t) = Tr{izΨz(z, t)vtt(Ψ(z, t), t)}. Hence we see that +(I − �H)(Z,α′vtt ◦ Z) = 0 +(2) ∂α′(Ztt − (Dα′Zt)Zt)(α′, t) = Tr{izΨz(z, t)vtz(Ψ(z, t), t)}. Hence we see that +(I − �H)∂α′(Ztt − (Dα′Zt)Zt) = 0 + +88 +SIDDHANT AGRAWAL +(3) ∂α′Dα′Zt = Tr{izΨz(z, t)vzz(Ψ(z, t), t)}. Hence we see that +(I − �H)(∂α′Dα′Zt) = 0 +(4) Zt,α′ = Tr{izΨz(z, t)vz(Ψ(z, t), t)}. Hence we see that +(I − �H)(Zt,α′) = 0 +Hence applying (I − �H) to (107) and using Proposition 6.1 we obtain +(I − �H) +� +i +�at +a ◦ h−1� +A0 +� += 2 +� +Zt, �H +� +∂α′(Ztt − (Dα′Zt)Zt) + +� +Z2 +t , �H +� +∂α′Dα′Zt + 2 +� +Ztt, �H +� +Zt,α′ += 2 +� +Zt, �H +� +Ztt,α′ + 2 +� +Ztt, �H +� +Zt,α′ + +� +Z2 +t , �H +� +∂α′Dα′Zt − 2 +� +Zt, �H +� +∂α′((Dα′Zt)Zt) += 2 +� +Zt, �H +� +Ztt,α′ + 2 +� +Ztt, �H +� +Zt,α′ − [Zt, Zt; Dα′Zt] +Taking imaginary part, we get +at +a ◦ h−1 = Im +� +2 +� +Zt, �H +� +Ztt,α′ + 2 +� +Ztt, �H +� +Zt,α′ − [Zt, Zt; Dα′Zt] +� +A0 +(108) +5.1.2. Formula of b. Let us first note some useful formulae +(1) From the normalization of the conformal map, we know that Φ(X(x0, t), t) = 0 for +all t ≥ 0. Hence taking a time derivative we get +Φt(X(x0, t), t) + Φz(X(x0, t), t)v(X(x0, t), t) = 0 +Now we see that v(X(x0, t), t) = U(0, t). As U is holomorphic in D with boundary +value Zt, we see that +v(X(x0, t), t) = Av(Zt) +(109) +As X(x0, t) = Ψ(0, t) we obtain using (101) +Φt(Ψ(0, t), t) + Av(Zt) +Ψz(0, t) = 0 +(110) +(2) Again by the normalization of the conformal map, we know that Φz(Ψ(0, t), t) > 0 +for all t ≥ 0. Hence taking a time derivative we get +Im{Φzt(Ψ(0, t), t) + Φzz(Ψ(0, t), t)Ψt(0, t)} = 0 +Now as Ψ(0, t) = X(x0, t), we see that Ψt(0, t) = v(X(x0, t), t) = Av(Zt). Hence +we have +Im{Φzt(Ψ(0, t), t) + Φzz(Ψ(0, t), t)Av(Zt)} = 0 +Now using (101) we get +Im +� +Φzt(Ψ(0, t), t) + Av(Zt) +� 1 +Ψz +∂z +1 +Ψz +� +(0, t) +� += 0 +(111) + +2D WATER WAVES +89 +Now from (15) we see that +h(α, t) = −i log(Φ(z(α, t), t)) +Taking a time derivative and precomposing with h−1 we obtain +(ht ◦ h−1)(α′, t) = +−i +Φ(Z(α′, t), t) +� +Φt(Z(α′, t), t) + Φz(Z(α′, t), t)Zt(α′, t) +� +Hence using (100), Φ(Z(α′, t), t) = eiα′ and (102) we get +b(α′, t) = Φt(Z(α′, t), t) +ieiα′ ++ Zt(α′, t) +Z,α′(α′, t) += +�Φt(Z(α′, t), t) +ieiα′ ++ +Av(Zt) +Z,α′(α′, t) +� ++ Zt(α′, t) − Av(Zt) +Z,α′(α′, t) += Tr +�Φt(Ψ(z, t), t) +iz ++ +Av(Zt) +izΨz(z, t) +� ++ Zt(α′, t) − Av(Zt) +Z,α′(α′, t) +Now using (110) we obtain +b(α′, t) = Tr +�Φt(Ψ(z, t), t) − Φt(Ψ(0, t), t) +iz ++ Av(Zt) +iz +� +1 +Ψz(z, t) − +1 +Ψz(0, t) +�� ++ Zt(α′, t) − Av(Zt) +Z,α′(α′, t) +(112) +As the term inside Tr is a holomorphic function on D, applying Re(I − �H) to the above +equation we obtain +b = Re +� +Av Tr +�Φt(Ψ(z, t), t) − Φt(Ψ(0, t), t) +iz ++ Av(Zt) +iz +� +1 +Ψz(z, t) − +1 +Ψz(0, t) +��� ++ Re(I − �H) +�Zt − Av(Zt) +Z,α′ +� +Now as the term inside Tr is holomorphic, we see that the average value of the trace equals +the value of the holomorphic function at z = 0. Hence by the definition of the complex +derivative we get +b = Re +�1 +i Φtz(Ψ(0, t))Ψz(0, t) + Av(Zt) +i +� +∂z +1 +Ψz +� +(0, t) +� ++ Re(I − �H) +�Zt − Av(Zt) +Z,α′ +� += Re +�Ψz(0, t) +i +� +Φtz(Ψ(0, t)) + Av(Zt) +� 1 +Ψz +∂z +1 +Ψz +� +(0, t) +�� ++ Re(I − �H) +�Zt − Av(Zt) +Z,α′ +� + +90 +SIDDHANT AGRAWAL +As Ψz(0, t) > 0, therefore from (111) we see that the first term vanishes. Hence we obtain +the formula for b +b = Re(I − �H) +�Zt − Av(Zt) +Z,α′ +� +(113) +Note that we have now derived the system (21). +5.1.3. Some useful identities. We now write down some useful identities similar to §4.1. +First we observe that (36), (41), (42) and (43) hold here as well. Also it is easy to see that +the analog of (37) and (38) is +(I + �H)(Ref) = f − iIm(I − �H)f +(114) +(I + �H)(iImf) = f − Re(I − �H)f +(115) +for any complex valued function on S1. Now using (114) and the formula (106) we see that +A0 = Im +� +Zt, �H +� +Zt,α′ = Im +� +(I − �H)(ZtZt,α′) +� += −iZtZt,α′ + i(I + �H) +� +Re(ZtZt,α′) +� +(116) +Now using (115) and (113) we obtain +b = Re(I − �H) +�Zt − Av(Zt) +Z,α′ +� += Zt − Av(Zt) +Z,α′ +− i(I + �H)Im +�Zt − Av(Zt) +Z,α′ +� +Hence +bα′ = Dα′Zt + (Zt − Av(Zt)) +� +∂α′ +1 +Z,α′ +� +− i(I + �H)∂α′Im +�Zt − Av(Zt) +Z,α′ +� += Dα′Zt + (Zt − Av(Zt)) +� +∂α′ +1 +Z,α′ + i +� 1 +Z,α′ − Av +� 1 +Z,α′ +��� +− i(Zt − Av(Zt)) +� 1 +Z,α′ − Av +� 1 +Z,α′ +�� +− i(I + �H)∂α′Im +�Zt − Av(Zt) +Z,α′ +� +Therefore +bα′ − 2Re(Dα′Zt) += −Dα′Zt + (Zt − Av(Zt)) +� +∂α′ +1 +Z,α′ + i +� 1 +Z,α′ − Av +� 1 +Z,α′ +��� +− i(Zt − Av(Zt)) +� 1 +Z,α′ − Av +� 1 +Z,α′ +�� +− i(I + �H)∂α′Im +�Zt − Av(Zt) +Z,α′ +� +(117) +Applying Re(I − �H) we get +bα′ − 2Re(Dα′Zt) += Re +� +− +� +1 +Z,α′ , �H +� +Zt,α′ + +� +Zt − Av(Zt), �H +�� +∂α′ +1 +Z,α′ + i +� 1 +Z,α′ − Av +� 1 +Z,α′ +��� +− i(I − �H) +� +(Zt − Av(Zt)) +� 1 +Z,α′ − Av +� 1 +Z,α′ +���� +(118) + +2D WATER WAVES +91 +Now similar to the calculation of (46), we see that +A0(α′) = (Im[Zt, �H]Zt,α′)(α′) += Im +� 1 +2πi +� 2π +0 +(Zt(α′) − Zt(β′)) cot +�β′ − α′ +2 +� +Zt,α′(β′) dβ′ +� += 1 +8π +� 2π +0 +����� +Zt(α′) − Zt(β′) +sin +� +α′−β′ +2 +� +����� +2 +dβ′ += 1 +8π +������ +Zt(α′) − Zt(β′) +sin +� +α′−β′ +2 +� +������ +2 +L2([0,2π],dβ′) += − i +2 +� +Zt, Zt; 1 +� +(119) +From this it is clear that if Zt is continuous and not a constant function, then there exists +c > 0 such that A0 ≥ c > 0 on [0, 2π]. Here c depends on the profile of Zt and changes with +time in general. +We now give an analogue of Lemma 4.1. +Lemma 5.1. We have +DtA0 +A0 += Im +� +2 +� +Zt, Ztt; 1 +� +− +� +Zt, Zt; Dα′Zt +�� +Ag ++ 2Re(Dα′Zt) − bα′ += 2Re +�� +Zt, +1 +Z,α′ ; 1 +� +(α′) +� ++ 1 +A0 +Im +� +2i +� +Zt, A0; +1 +Z,α′ +� +(α′) − +� +Zt, Zt; Dα′Zt +� +(α′) +� ++ 2Re(Dα′Zt) − bα′ +Except for some sign changes due to the difference of parametrization of the interface, +the formulae are exactly the same as Lemma 4.1 and the proof is also identical. +5.1.4. Equations of motion. Let us now derive the main equations. This will be very similar +to §4.2. We define +�J0 = Dt(bα′ − Dα′Zt − Dα′Zt) +(120) +Using (43) we get +Dt +� +eiα′ +Z,α′ +� += eiα′ +Z,α′ +� +(bα′ − Dα′Zt − Dα′Zt) + Dα′Zt + ib +� +Now following a similar calculation as in §4.2 we get +� +D2 +t − i +A0 +��Z,α′ +��2 ∂α′ +�� +eiα′ +Z,α′ +� += eiα′ +Z,α′ ( �J0 + �Q0) + +92 +SIDDHANT AGRAWAL +where +�Q0 = (bα′ − Dα′Zt + ib)2 − (Dα′Zt)2 + +i +��Z,α′ +��2 ∂α′A0 + iDtb + +A0 +��Z,α′ +��2 +(121) +Now applying ∂α′ to the above equation and doing a similar calculation as in §4.2 we get +the main equation as +� +D2 +t − i +A0 +��Z,α′ +��2 ∂α′ +� +∂α′ +� +eiα′ +Z,α′ +� += eiα′Dα′ �J0 + �R0 +(122) +where +�R0 = +� +∂α′ +� +eiα′ +Z,α′ +�� +( �J0 + �Q0) + eiα′Dα′ �Q0 − bα′ +� +∂α′Dt +� +eiα′ +Z,α′ +� ++ Dt∂α′ +� +eiα′ +Z,α′ +�� +− (Dtbα′) +� +∂α′ +� +eiα′ +Z,α′ +�� ++ 2iA0 +� +|Dα′| +1 +��Z,α′ +�� +�� +∂α′ +� +eiα′ +Z,α′ +�� ++ i +� +1 +��Z,α′ +��2 ∂α′A0 +�� +∂α′ +� +eiα′ +Z,α′ +�� +(123) +The other main equation is also derived in a very similar manner. We define +�J1 = DtA0 + A0 +� +bα′ − Dα′Zt − Dα′Zt +� +(124) +Following the same calculations as in §4.2 we get +� +D2 +t − i +A0 +��Z,α′ +��2 ∂α′ +� +Zt = i +�J1 +Z,α′ +The formula (44) also holds here and hence by following the same argument as in §4.2 we +obtain +� +D2 +t − i +A0 +��Z,α′ +��2 ∂α′ +� +D2 +α′Zt = �R1 + i ω3 +��Z,α′ +��∂α′ +� +1 +��Z,α′ +��2 ∂α′ �J1 +� +(125) +where +�R1 = −2(D2 +α′Ztt)(Dα′Zt) − 4(Dα′Ztt)(D2 +α′Zt) − 2(D2 +α′Zt)(DtDα′Zt) +− 2(Dα′Zt)(Dα′DtDα′Zt) − 2(Dα′Zt)(DtD2 +α′Zt) + i(Dα′ �J1) +� +Dα′ +1 +Z,α′ +� ++ i �J1D2 +α′ +1 +Z,α′ + 2iω(Dα′ω) +� +1 +��Z,α′ +��2 ∂α′ �J1 +� +(126) +Note the equation is the same as (54) except for the minus sign in front of A0 and terms +involving �J1. + +2D WATER WAVES +93 +5.2. The a priori estimate +We define the energies +�E1(t) = +��Zt,α′ +��2 +2 +�����∂α′ +1 +Z,α′ +���� +2 +2 ++ +���� +1 +Z,α′ +���� +2 +2 +� +�E2(t) = +��Zt,α′ +��2 +2 + + + +�����Dt∂α′ +� +eiα′ +Z,α′ +������ +2 +2 ++ +����� +√A0 +Z,α′ ∂α′ +� +eiα′ +Z,α′ +������ +2 +˙H +1 +2 + + + +�E3(t) = +��Zt,α′ +��2 +2 +� +��DtD2 +α′Zt +��2 +2 + +���� +√A0 +Z,α′ D2 +α′Zt +���� +2 +˙H +1 +2 +� +and also define +�Ea(t) = +� +�E1(t)2 + �E2(t) +� 1 +2 +(127) +�E(t) = +� +�E1(t)3 + �E2(t) +3 +2 + �E3(t) +� 1 +3 +(128) +Note that the energy is essentially the same as for R except for some small differences. +The first difference is the addition of +��Zt,α′ +��2 +2 +��� +1 +Z,α′ +��� +2 +2 in �E1(t) which is a lower order term +(note that this is very minor change as we are on a bounded domain). The other change +is replacing ∂α′ +1 +Z,α′ with ∂α′ +� +eiα′ +Z,α′ +� +in �E2(t). This is done as ∂α′ +1 +Z,α′ is no longer boundary +value of a holomorphic function whereas ∂α′ +� +eiα′ +Z,α′ +� +is (see (102) and (103)). +For this energy we have the following energy estimate. +Theorem 5.2. Let T > 0 and let (Z, Zt) be a solution to the water wave equation (21) in +the time interval [0, T) with (Z,α′ − 1, +1 +Z,α′ − 1, Zt) ∈ L∞([0, T], Hs(R) × Hs(R) × Hs+ 1 +2(R)) +for some s ≥ 4. Then E(t) < ∞ for all t ∈ [0, T) and there exists a universal constant +c > 0 so that for all t ∈ [0, T) we have +d �Ea(t) +dt +≤ c +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +�� +�Ea(t) ≤ c �Ea(t)3/2 +and also +d �E(t) +dt +≤ c �Ea(t) +1 +2 �E(t) ≤ c �E(t)3/2 +The proof of this theorem is essentially the same as the one for Theorem 3.1 with only a +few changes with most of them related to some modifications to §4.3. Some of the common +changes are: +(a) Replacing occurrences of Zt with Zt − Av(Zt). +(b) Replacing +���∂α′ +1 +Z,α′ +��� +2 with +����∂α′ +1 +Z,α′ +��� +2 + +��� +1 +Z,α′ +��� +2 +� +on the right hand side of the +estimates. + +94 +SIDDHANT AGRAWAL +(c) Replacing Dt +� +∂α′ +1 +Z,α′ +� +and +√ +Ag +Z,α′ ∂α′ +1 +Z,α′ with Dt +� +∂α′ +� +eiα′ +Z,α′ +�� +and +√A0 +Z,α′ ∂α′ +� +eiα′ +Z,α′ +� +re- +spectively. +We will give a few examples of these common changes and also explain any other non +standard changes. +1) +The estimates for ∥A0∥L∞ and ∥A0∥ ˙H +1 +2 remain the same. Similarly for +����∂α′ +1 +|Z,α′| +���� +2 +and ∥|Dα′|ω∥2. Now observe that +2∂α′PA +�Zt − Av(Zt) +Z,α′ +� += (I − H) +� +Dα′Zt + (Zt − Av(Zt))∂α′ +1 +Z,α′ +� += 2Dα′Zt − (I + H)Dα′Zt + (I − H) +� +(Zt − Av(Zt))∂α′ +1 +Z,α′ +� += 2Dα′Zt + +� 1 +Z,α′ , H +� +Zt,α′ + [Zt − Av(Zt), H] +� +∂α′ +1 +Z,α′ + i +� 1 +Z,α′ − Av +� 1 +Z,α′ +��� +− i(I − H) +� +(Zt − Av(Zt)) +� 1 +Z,α′ − Av +� 1 +Z,α′ +��� +Hence using Proposition 6.4 and Proposition 6.3 we get +����∂α′PA +�Zt − Av(Zt) +Z,α′ +����� +∞ +≲ +��Dα′Zt +�� +∞ + +����∂α′ +1 +Z,α′ +���� +2 +��Zt,α′ +�� +2 + +����(Zt − Av(Zt)) +� 1 +Z,α′ − Av +� 1 +Z,α′ +������ +H1 +≲ +��Dα′Zt +�� +∞ + +����∂α′ +1 +Z,α′ +���� +2 +��Zt,α′ +�� +2 +≲ +��Dα′Zt +�� +∞ + +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +���Zt,α′ +�� +2 +Hence the estimate (56) gets modified by the changes mentioned above. +The estimate for ∥bα′∥∞ now follows similar to the above calculation by using (118). +We also note from (21) and Proposition 6.3 that +∥b∥∞ ≲ ∥b∥H1 ≲ +���� +Zt − Av(Zt) +Z,α′ +���� +H1 +≲ +��Zt,α′ +�� +2 +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +� +2) +To get the estimate for +��|Dα′|(bα′ − Dα′Zt − Dα′Zt) +�� +2 we follows as in the R case and +see that +|Dα′|(bα′ − Dα′Zt − Dα′Zt) = Re +� ω +Z,α′ (I − �H)∂α′(bα′ − Dα′Zt − Dα′Zt) +� + +2D WATER WAVES +95 +Note that to get this equality we had to use I− �H instead of I−H (we are again using the +fact that bα′−Dα′Zt−Dα′Zt is real valued). However we see that (I− �H)∂α′ = (I−H)∂α′ +and hence we get +|Dα′|(bα′ − Dα′Zt − Dα′Zt) = Re +� ω +Z,α′ (I − H)∂α′(bα′ − Dα′Zt − Dα′Zt) +� +Now following the same proof as in the R case and using (117) instead of (40) we get +the estimate +��|Dα′|(bα′ − Dα′Zt − Dα′Zt) +�� +2 +≲ +����∂α′ +1 +Z,α′ +��� +2 + +���� +1 +Z,α′ +���� +2 +�2��Zt,α′ +�� +2 + +����∂α′ +1 +Z,α′ +��� +2 + +���� +1 +Z,α′ +���� +2 +���Dα′Zt +�� +∞ +We follow this same approach for estimating the term +���|Dα′| �J0 +��� +2 and also for the term +����|Dα′| +� +1 +|Z,α′| +2∂α′ �J1 +����� +2 +. +3) +The proof of the estimate for +��Dα′Dα′Zt +�� +2 follows the same as for the R case and we +get the estimate +��Dα′Dα′Zt +�� +2 ≲ +����∂α′ +1 +Z,α′ +��� +2 + +���� +1 +Z,α′ +���� +2 +�2��Zt,α′ +�� +2 + +����∂α′ +1 +Z,α′ +��� +2 + +���� +1 +Z,α′ +���� +2 +���Dα′Zt +�� +∞ ++ +�����Dt +� +∂α′ +� +eiα′ +Z,α′ +������� +2 +Similarly the estimates for +��Dα′Zt +�� +L∞∩ ˙H +1 +2 , +��Dα′Zt +�� +L∞∩ ˙H +1 +2 and most other terms +follow in the same manner as R. From now on we will concentrate on the non-standard +changes. +4) +Similar to the R case, we easily obtain the estimates +�����ωn +√A0 +��Z,α′ +��∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +�����e−iα′ωn +√A0 +��Z,α′ +��∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 +≲n +����� +√A0 +Z,α′ ∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +�2��Zt,α′ +�� +2 ++ +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +���Dα′Zt +�� +L∞∩ ˙H +1 +2 +Now observe that +e−iα′ωn +√A0 +��Z,α′ +��∂α′ +� +eiα′ +Z,α′ +� += ωn +√A0 +��Z,α′ +��∂α′ +1 +Z,α′ + iωn +√A0 +��Z,α′ +��Z,α′ + +96 +SIDDHANT AGRAWAL +We can easily control the second term +�����iωn +√A0 +��Z,α′ +��Z,α′ +����� ˙H +1 +2 +≲ +�����iωn +√A0 +��Z,α′ +��Z,α′ +����� +H1 +≲ +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +�2��Zt,α′ +�� +2 + +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +���Dα′Zt +�� +L∞∩ ˙H +1 +2 +Therefore we get control for the first term. Now again by following the proof in the R +case, we get the estimates +�����ωn +√A0 +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +�����ωn +√A0 +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 ++ +�����ωn +√A0 +��Z,α′ +��2 ∂α′ω +����� ˙H +1 +2 +≲n +����� +√A0 +Z,α′ ∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +�2��Zt,α′ +�� +2 ++ +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +���Dα′Zt +�� +L∞∩ ˙H +1 +2 +Following this same logic, we also get the estimates +�����ωn A0 +��Z,α′ +��∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +�����e−iα′ωn A0 +��Z,α′ +��∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +�����ωn A0 +��Z,α′ +��∂α′ +1 +Z,α′ +����� ˙H +1 +2 ++ +�����ωn A0 +��Z,α′ +��∂α′ +1 +��Z,α′ +�� +����� ˙H +1 +2 ++ +�����ωn +A0 +��Z,α′ +��2 ∂α′ω +����� ˙H +1 +2 +≲n +��Zt,α′ +�� +2 +����� +√A0 +Z,α′ ∂α′ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +���Dα′Zt +�� +L∞∩ ˙H +1 +2 + +��Zt,α′ +�� +2 +�����∂α′ +1 +Z,α′ +���� +2 ++ +���� +1 +Z,α′ +���� +2 +��2 +5) +We need a slight modification to the proof of the estimate of +����|Dα′| +� +1 +|Z,α′| +2∂α′A0 +����� +2 +. +This is because we no longer have the property that Hf = f then H +� +1 +Z2 +,α′ ∂α′f +� += + +2D WATER WAVES +97 +1 +Z2 +,α′ ∂α′f, which is true in R. To remedy this, we write +|Dα′| +� +1 +��Z,α′ +��2 ∂α′A0 +� += Re +�ω3ω3e−iα′eiα′ +��Z,α′ +�� +(I − �H)∂α′ +� +1 +��Z,α′ +��2 ∂α′A0 +�� += Re +�ω3ω3e−iα′eiα′ +��Z,α′ +�� +(I − H)∂α′ +� +1 +��Z,α′ +��2 ∂α′A0 +�� +Now +ω3eiα′ +��Z,α′ +�� (I − H)∂α′ +� +1 +��Z,α′ +��2 ∂α′A0 +� += (I − H) +� +Dα′ +� eiα′ +Z2 +,α′ +∂α′A0 +� +− 2 +� +ω +��Z,α′ +��2 ∂α′A0 +� +(Dα′ω)eiα′ − ieiα′ω2 +Z,α′ +� +1 +��Z,α′ +��2 ∂α′A0 +�� +− +� +ω3eiα′ +��Z,α′ +�� , H +� +∂α′ +� +1 +��Z,α′ +��2 ∂α′A0 +� +Now we can follow the same proof as in the R by using the property that if Hf = f, +then H(Dα′f) = Dα′f and H +� +eiα′ +Z2 +,α′ ∂α′f +� += eiα′ +Z2 +,α′ ∂α′f. +6) +The estimate for +���DtA0 +A0 +��� +∞ follows essentially the same was as in the R with some +minor modifications. First we obviously have to use Lemma 5.1 instead of Lemma 4.1. +The main difference comes in the calculation of (65). To remedy this use the following +identity +cot +�s − α′ +2 +� +− cot +�s − β′ +2 +� += +sin +� +α′−β′ +2 +� +sin +� α′−s +2 +� +sin +� +β′−s +2 +� +Then by noting the difference between the formulae of (119) and (46), instead of (65) +we get +A0(α′) − A0(β′) +sin +� +α′−β′ +2 +� += Im + + + +1 +2πi +� 2π +0 +1 +sin +� +β′−s +2 +� +� +Zt(α′) − Zt(s) +sin +�α′−s +2 +� +� +Zt,α′(s) ds + Zt(α′) − Zt(β′) +sin +� +α′−β′ +2 +� +Zt,α′(β′) + + + +The rest of the proof is the same as the one in R. +7) +In the bounded domain case, we estimate the term +���(I − H)D2 +t +� +∂α′ +� +eiα′ +Z,α′ +����� +2 instead +of +���(I − H)D2 +t +� +∂α′ +1 +Z,α′ +���� +2 and similarly replace +����(I − H) +� +i +Ag +|Z,α′| +2 ∂α′ +� +∂α′ +1 +Z,α′ +������ +2 +with + +98 +SIDDHANT AGRAWAL +����(I − H) +� +i +A0 +|Z,α′| +2 ∂α′ +� +∂α′ eiα′ +Z,α′ +������ +2 +. The replacement of ∂α′ +1 +Z,α′ with ∂α′ +� +eiα′ +Z,α′ +� +is nat- +ural because the main equation (122) is in terms ∂α′ +� +eiα′ +Z,α′ +� +. The proof to estimate +the term +����(I − H) +� +i +A0 +|Z,α′| +2 ∂α′ +� +∂α′ eiα′ +Z,α′ +������ +2 +remains the same, however the proof for +the term +���(I − H)D2 +t +� +∂α′ +� +eiα′ +Z,α′ +����� +2 gets slightly modified as the formula (66) does not +hold exactly. It gets modified as follows: Let f = ∂α′ +� +eiα′ +Z,α′ +� +then we still have Hf = f +and using (18) and Proposition 6.1 we get +(I − H)D2 +t f += [Dt, H]Dtf + Dt[Dt, H]f += [b, H]∂α′Dtf + Dt[b, H]∂α′f += [b, H]∂α′Dtf + Dt[b, �H]∂α′f + Dt[b, Av]∂α′f += [b, H]∂α′Dtf + [b, �H]∂α′Dtf + [Dtb, �H]∂α′f − [b, b; ∂α′f] + Dt[b, Av]∂α′f +Now all terms except for the last term are handled as in the R case. For the last term +we observe that +Dt[b, Av]∂α′f += −∂t +� 1 +2π +� 2π +0 +b(s)fα′(s) ds +� += − 1 +2π +� 2π +0 +(∂tb)(s)fα′(s) + b(s)∂α′(∂tf)(s) ds += − 1 +2π +� 2π +0 +(Dtb)(s)fα′(s) + b(s)∂α′(Dtf)(s) ++ 1 +2π +� 2π +0 +(bbα′)(s)fα′(s) + b(s)∂α′(bfα′)(s) ds += 1 +2π +� 2π +0 +(∂α′Dtb)(s)f(s) + bα′(s)(Dtf)(s) +where in the last step we integrated by parts. Hence from Proposition 6.3 we see that +∥Dt[b, Av]∂α′f∥2 ≲ ∥∂α′Dtbα′∥2∥f∥2 + ∥bα′∥2∥Dtf∥2 +≲ ∥∂α′Dtbα′∥BMO∥f∥2 + ∥bα′∥∞∥Dtf∥2 +≲ ∥∂α′Dtbα′∥ ˙H +1 +2 ∥f∥2 + ∥bα′∥∞∥Dtf∥2 +Hence ∥Dt[b, Av]∂α′f∥2 can be controlled in the same way as is done in the R case. +Therefore the term +���(I − H)D2 +t +� +∂α′ +� +eiα′ +Z,α′ +����� +2 is controlled. By the same process, we + +2D WATER WAVES +99 +also control the term +��(I − H)D2 +t D2 +α′Zt +�� +2. The proof for +����(I − H) +� +i +Ag +|Z,α′| +2 ∂α′D2 +α′Zt +����� +2 +remains the same as in the R case. +This concludes the changes needed with respect to §4.3 and §4.4. There are essentially no +changes needed to §4.5 and hence the proof of Theorem 5.2 is complete. +5.3. Proof of Theorem 3.6 and Theorem 3.8 +As in the unbounded case, we need to define the class SA i.e. the space of smoothly +approximable solutions. The definition is very similar to the unbounded case except for +some minor changes. We now give the definition. +Let (Z, Zt)a and (Z, Zt)b be two solutions of the water wave equation (21) and define �h +and �U in the same way as (76). Similarly define ∆(f) as in (77). We define +�F∆((Z, Zt)a, (Z, Zt)b)(t) += ∥∆(Zt)∥ ˙H +1 +2 + ∥∆(Ztt)∥ ˙H +1 +2 + +�����∆ +� +eiα′ +Z,α′ +������ ˙H +1 +2 ++ +����hα′ − 1 +��� +2 + ∥∆(Dα′Zt)∥2 ++ ∥∆(A0)∥2 + ∥∆(bα′)∥2 + |Av(∆(Zt))| + +�����Av +� +∆ +� +eiα′ +Z,α′ +������� +(129) +As compared to the definition (78), here we have the addition of two terms which takes +into account the difference of the averages of Zt and eiα′ +Z,α′ . We can now define the class SA +for the bounded case: +Definition 5.3. Let the initial data (U, Ψ, P)(0) be given as in the paragraph above The- +orem 3.6. Let T > 0 and let U, Ψ : D × [0, T] → C and P : D × [0, T] → R. We say +(U, Ψ, P)(t) solves the Cauchy problem for the system (17) in the time interval [0, T] in the +class SA (smoothly approximable solutions) if the following holds: +(1) U, Ψ, P extend continuously to D×[0, T] and (U, Ψ) ∈ C1(D×(0, T)). Also for each +t ∈ [0, T] we have P(·, t) ∈ C1(D). +(2) U(·, t), Ψ(·, t) are holomorphic maps for each t ∈ [0, T], Ψz′(z′, t) ̸= 0 for (z′, t) ∈ +D × [0, T] and Ψz′(0, t) > 0 for any t ∈ [0, T]. +Moreover +1 +Ψz′ and +Ψt +Ψz′ extend +continuously to D × [0, T] (and by abuse of notation we will continue to denote +these extensions as +1 +Ψz′ and +Ψt +Ψz′ respectively). +(3) We have +sup +t∈[0,T] +� +sup +0 0 and T2 < �T if �T < T, and a sequence of smooth 2π functions (Z(n), Z(n) +t +) : +R × [T1, T2] → C for n ≥ 1 satisfying: +(a) For each n ∈ N we have Z(n) +,α′ (α′, t) ̸= 0 for all (α′, t) ∈ R × [T1, T2]. +(b) We have +sup +n≥1,t∈[T1,T2] +���Z(n) +t +�� +H1(t) + +���� +1 +Z(n) +,α′ +���� +H1(t) +� +< ∞ +(c) There exist holomorphic functions (U (n), Ψ(n))(·, t) : D → C whose boundary values +are (Z(n) +t +, Z(n))(·, t) and which satisfy for each n the property Ψ(n) +z′ (z′, t) ̸= 0 for all +(z′, t) ∈ D × [T1, T2] and Ψ(n) +z′ (0, t) > 0 for t ∈ [T1, T2]. Let P(n) be the function +satisfying +∆P(n) = −2|U (n) +z′ |2 +on D, +P(n) = 0 +on ∂D +Then (U (n), Ψ(n), P(n), +1 +Ψ(n) +z′ +) → (U, Ψ, P, +1 +Ψz′ ) uniformly on D × [T1, T2] as n → ∞. +(d) For each n ∈ N, (Z(n), Z(n) +t +) solves the system (21) and we have the uniform bound +supn∈N,t∈[T1,T2] +� +�En(t) + +���Z(n) +t,α′ +��� +2(t) +� +< ∞. If we fix the Lagrangian parametriza- +tions for these solutions by imposing h(n)(α′, �T) = h(α′, �T) for all α′ ∈ R and n ∈ N, +then supt∈[T1,T2] �F∆((Z(n), Z(n) +t +), (Z, Zt))(t) → 0 as n → ∞. +Furthermore we say that (U, Ψ, P)(t) belongs in the class SA in the time interval [0, T), if +for any 0 < T1 < T the solution belongs in the class SA in the time interval [0, T1]. +Similar to Lemma 4.5 we have the following relations between the different energies. +Lemma 5.4. Let T > 0 and let (Z, Zt)(t) be a solution to the water wave equation(21) in +the time interval [0, T] with (Z,α′, +1 +Z,α′ , Zt) ∈ L∞([0, T], Hs(S1) × Hs(S1) × Hs+ 1 +2 (S1)) for +some s ≥ 4. Then we have the following: +(1) There exists a universal constant M1 > 0 such that �E(t) ≤ M1 �E(t). Moreover there +exists a universal increasing function C1 : [0, ∞) → [0, ∞) such that if +��Zt,α′ +�� +2(0) > +0, then we have +�E(t) ≤ C1 +� +�E(t) + +���� +1 +√A0 +���� +∞ +(t) + +��Zt,α′ +�� +2(t) +� +(130) +and also +sup +t∈[0,T] +�E(t) ≤ C1 +� +sup +t∈[0,T] +�E(t) + +���� +1 +√A0 +���� +∞ +(0) + T + +��Zt,α′ +�� +2(0) +� +(131) + +2D WATER WAVES +101 +(2) Assume that +��Zt,α′ +�� +2(0) > 0. +Then there exists a universal increasing function +C2 : [0, ∞) → [0, ∞) so that we have +sup +t∈[0,T] +� +∥Zt∥H1(t) + +���� +1 +Z,α′ +���� +H1 ++ +1 +��Zt,α′ +��2 +2(t) +� +≤ C2 +� +sup +t∈[0,T] +�E(t) + T + ∥Zt∥H1(0) + +1 +��Zt,α′ +��2 +2(0) +� +(132) +and also +sup +t∈[0,T] +� +∥Zt∥H2.5(t) + +��Z,α′ +�� +H2(t) + +���� +1 +Z,α′ +���� +H2 +(t) +� +≤ C2 +� +sup +t∈[0,T] +�E(t) + +���� +1 +√A0 +���� +∞ +(0) + T + ∥Zt∥H1(0) + +1 +��Zt,α′ +��2 +2(0) ++ +��Z,α′ +�� +∞(0) +� +(133) +Proof. The proof of this lemma is essentially the same as the proof of Lemma 4.5. The +only main difference is that instead of the assumption of g > 0 in Lemma 4.5, we have +the condition of having the term +��� +1 +√A0 +��� +∞(0) on the right hand side. Now assume that +��Zt,α′ +�� +2(0) > 0. Note that this implies that ∥Zt − Av(Zt)∥2(0) > 0. From (119) we have +A0(α′, t) = 1 +8π +� 2π +0 +����� +Zt(α′, t) − Zt(β′, t) +sin +� +α′−β′ +2 +� +����� +2 +dβ′ +≳ +� 2π +0 +��Z(α′, t) − Zt(β′, t) +��2 dβ′ +≳ ∥Zt − Av(Zt)∥2 +2(t) +(134) +Hence +���� +1 +√A0 +���� +∞ +(0) ≲ +1 +∥Zt − Av(Zt)∥2(0) < ∞ +Now we also have the same estimate as in (85) here as well, namely that there exists a +universal constant C4 > 0 such that +exp +� +−C4 +� t +0 +�Ea(s) +1 +2 ds +���Zt,α′ +��2 +2(0) ≤ +��Zt,α′ +��2 +2(t) ≤ exp +� +C4 +� t +0 +�Ea(s) +1 +2 ds +���Zt,α′ +��2 +2(0) +(135) +Similarly by using the fact that +���DtA0 +A0 +��� +∞ is controlled by �Ea, we also easily get the estimate +exp +� +−C4 +� t +0 +�Ea(s) +1 +2 ds +����� +1 +√A0 +���� +∞ +(0) ≤ +���� +1 +√A0 +���� +∞ +(t) ≤ exp +� +C4 +� t +0 +�Ea(s) +1 +2 ds +����� +1 +√A0 +���� +∞ +(0) +(136) + +102 +SIDDHANT AGRAWAL +With these bounds, we can now follow the proof of Lemma 4.5 to complete the proof of +this lemma and we leave the details to the reader. +□ +Similar to Theorem 4.6 we have the following existence result in Sobolev spaces. +Theorem 5.5. Let s ≥ 4. Assume that the initial data (Z, Zt)(0) satisfies the condition +(Z,α′, +1 +Z,α′ , Zt)(0) ∈ Hs(S1)×Hs(S1)×Hs+ 1 +2(S1) and there exists a c > 0 such that A0(·, 0) ≥ +c > 0. Then there exists a T > 0 such that on [0, T] the initial value problem for (10) has +a unique solution (Z, Zt)(t) satisfying (Z,α′, +1 +Z,α′ , Zt) ∈ Cl([0, T], Hs−l(S1) × Hs−l(S1) × +Hs+ 1 +2 −l(S1)) for l = 0, 1. Moreover if T ∗ is the maximal time of existence, then either +T ∗ = ∞ or T ∗ < ∞ and +sup +t∈[0,T ∗) +���Z,α′ − 1 +�� +H2(t) + +���� +1 +Z,α′ − 1 +���� +H2 +(t) + ∥Zt∥H2+ 1 +2 (t) + +���� +1 +A0 +���� +∞ +(t) +� += ∞ +The proof of this theorem is very similar to Theorem 4.6 and so we skip it. +There +are some minor differences of this theorem with respect to Theorem 4.6 which we now +explain. +First we have imposed the condition A0(·, 0) ≥ c > 0 on the initial data. +If +∥Zt − Av(Zt)∥2(0) = 0, then the initial velocity is constant and hence the trivial solution +namely the solution where the domain moves with constant speed for all time is clearly the +unique global solution. If ∥Zt − Av(Zt)∥2(0) > 0, then from (134) we do get A0(·, 0) ≥ c > 0 +and hence the condition is satisfied. Moreover this condition is equivalent to the condition +that the Taylor sign condition is satisfied at t = 0 from (24). The main difference of this +theorem with Theorem 4.6 is in regards to the blow up criterion where there is an additional +term of +��� 1 +A0 +��� +∞. This term is necessary to ensure that the Taylor sign condition is satisfied +as again can be seen from (24). In Theorem 4.6 we do not such a term because as gravity +g > 0, we already have Ag ≥ g everywhere from (45) and (46) and therefore +��� 1 +Ag +��� +∞ ≤ 1 +g. +One of the important features of Theorem 3.6 is that we remove this condition of +��� 1 +A0 +��� +∞ +in the blow up criterion. +We now write down a result which allows us to prove uniqueness of solutions in the class +SA. +Theorem 5.6. Let s ≥ 4 and T > 0. Let (Z, Zt)a and (Z, Zt)b be two solutions of the water +wave equation (21) in [0, T] satisfying (Z,α′, +1 +Z,α′ , Zt)a, (Z,α′, +1 +Z,α′ , Zt)b ∈ Cl([0, T], Hs−l(S1)× +Hs−l(S1) × Hs+ 1 +2−l(S1)) for l = 0, 1. Also assume that there exists a constant c1 > 0 such +that +c1 ≥ +��(Zt,α′)a +�� +2(0) + +��(Zt,α′)b +�� +2(0) + +1 +��(Zt,α′)a +�� +2(0) + +1 +��(Zt,α′)b +�� +2(0) ++ +���� +1 +(A0)a +���� +∞ +(0) + +���� +1 +(A0)b +���� +∞ +(0) + +2D WATER WAVES +103 +Then there exists a constant L > 0 depending only on c1, T, supt∈[0,T] �E((Z, Zt)a)(t) and +supt∈[0,T] �E((Z, Zt)b)(t) such that +sup +t∈[0,T] +�F∆((Z, Zt)a, (Z, Zt)b)(t) ≤ L +� +�F∆((Z, Zt)a, (Z, Zt)b)(0) + +����∆ +� 1 +Z,α′ +����� +∞ +(0) +� +Proof. The proof of this is very similar to the proof of Theorem 3.7 in [35] and so we will +only describe the important differences. In the following L > 0 will be a constant depending +only on c1, T, supt∈[0,T] �E((Z, Zt)a)(t) and supt∈[0,T] �E((Z, Zt)b)(t). Now note that from (135) +and (136) we see that for all t ∈ [0, T] +1 ≳L +��(Zt,α′)a +�� +2(t) + +��(Zt,α′)b +�� +2(t) + +1 +��(Zt,α′)a +�� +2(t) + +1 +��(Zt,α′)b +�� +2(t) ++ +���� +1 +(A0)a +���� +∞ +(t) + +���� +1 +(A0)b +���� +∞ +(t) +Hence from the definition of �E we see that +1 ≳L +���� +� 1 +Z,α′ +� +a +���� +H1 +(t) + +���� +� 1 +Z,α′ +� +b +���� +H1 +(t) + +��� +D2 +α′Zt +� +a +�� +2(t) + +��� +D2 +α′Zt +� +b +�� +2(t) ++ +����� +� +D2 +α′ +� +eiα′ +Z,α′ +�� +a +����� +2 +(t) + +����� +� +D2 +α′ +� +eiα′ +Z,α′ +�� +b +����� +2 +(t) + +����� +� +eiα′ +Z,α′ D2 +α′Zt +� +a +����� ˙H +1 +2 +(t) ++ +����� +� +eiα′ +Z,α′ D2 +α′Zt +� +b +����� ˙H +1 +2 +(t) +From the energy estimate we also get +1 ≳L +��� +Dα′Zt +� +a +�� +L∞∩ ˙H +1 +2 (t) + +��� +Dα′Zt +� +b +�� +L∞∩ ˙H +1 +2 (t) +With these estimate we now have control of all quantities which are required to follow the +proof of Theorem 3.7 in [35]. There are only slight differences in the energy estimate which +are as follows. Define +κ = +� +(A0)a +�U(A0)b +�hα′ +and the energies +F0(t) = +����hα′ − 1 +��� +2 +2 + |Av(∆(Zt))|2 + +�����Av +� +∆ +� +eiα′ +Z,α′ +������� +2 +F1(t) = +� 2π +0 +κ +(A0)a +��∆(Z,α′Ztt) +��2 dα′ − i +� 2π +0 +∂α′(∆(Zt))∆(Zt) dα′ +F2(t) = +� 2π +0 +κ +(A0)a +�����∆ +� +Z,α′Dt +� +eiα′ +Z,α′ +������� +2 +dα′ − i +� 2π +0 +∂α′ +� +∆ +� +eiα′ +Z,α′ +�� +∆ +� eiα′ +Z,α′ +� +dα′ + +104 +SIDDHANT AGRAWAL +F3(t) = +� 2π +0 +κ +(A0)a +��∆(Z,α′Zttt) +��2 dα′ − i +� 2π +0 +∂α′(∆(Ztt))∆(Ztt) dα′ +The main difference from the energy used in Theorem 3.7 in [35] is that we have the addition +of two terms in F0(t) and in F2(t) we have used the function +� +eiα′ +Z,α′ +� +instead of +� +1 +Z,α′ +� +. The +modification of F0(t) is necessary as we need to control the difference of average values of +the solutions in a bounded domain as there is no corresponding decay condition at infinity +as in the unbounded case. In F2(t) we use +� +eiα′ +Z,α′ +� +as +� +1 +Z,α′ +� +is no longer the boundary value +of a holomorphic function. +We then define the energy +F(t) = MF0(t) + F1(t) + F2(t) + M−1F3(t) +where M > 0 is a constant taken large enough and depending only on values of c1, +T, supt∈[0,T] �E((Z, Zt)a)(t) and supt∈[0,T] �E((Z, Zt)b)(t). +One can then show that F(t) is +equivalent to the energy �F∆((Z, Zt)a, (Z, Zt)b)(t) and then prove the following estimate +F(t) + +� t +0 +F(s) ds ≤ L +� +F(0) + +����∆ +� 1 +Z,α′ +����� +2 +∞ +(0) +� +for all t ∈ [0, T] +The proof of this estimate follows in the same way as in Theorem 3.7 in [35] and so we skip +it. +□ +We are now ready to prove Theorem 3.6. +Proof of Theorem 3.6. The proof of this result is very similar to Theorem 3.3 and so we +only highlight the main differences. We will first consider the existence part of the result. +Let 1 +2 < ǫ ≤ 1 and define +Ψǫ(z′, 0) = Ψ(ǫz′, 0), +U ǫ(z′, 0) = U(ǫz′, 0) +and +hǫ(α, 0) = α +We again define their boundary values as +Zǫ(α′, 0) = Ψǫ(α′, 0) +and +Zǫ +t(α′, 0) = U ǫ(α′, 0) +which are now smooth for 1 +2 < ǫ < 1. It is clear that for all 1 +2 < ǫ < 1, Ψǫ(·, 0), U ǫ(·, 0) +satisfy the conditions on the initial data and we have +sup +0 0. This in particular implies that c3 = ∥Zt − Av(Zt)∥2(0) > +0. Hence by the definition of Zǫ +t we see that there exists 1 +2 < ǫ0 < 1 such that for all +ǫ0 ≤ ǫ < 1 we have +c2 +2 ≤ +��Zǫ +t,α′ +�� +2(0) ≤ c2 +and +c3 +2 ≤ ∥Zǫ +t − Av(Zǫ +t )∥2(0) ≤ c3 +From (134) this means that for all ǫ0 ≤ ǫ < 1 we have +����� +1 +� +Aǫ +0 +����� +∞ +(0) ≲ 1 +c3 +Now the proof of existence of solution in the class SA follows in the same was as Theo- +rem 3.3. +Let us now prove the uniqueness of solutions in the class SA. We first need to prove a +basic property: +Claim: Consider a solution (U, Ψ, P)(t) in the class SA in the time interval [0, T]. If there +exists t0 ∈ [0, T] such that +��Zt,α′ +�� +2(t0) > 0, then there exists c1, c2 > 0 such that for all +α′ ∈ R and t ∈ [0, T] we have A0(α′, t) ≥ c1 and +��Zt,α′ +�� +2(t) ≥ c2. +To prove this claim, first note that by the definition of the class SA and the compactness +of the interval [0, T], there exists N ≥ 1 and 0 = T0 < T1 < · · · < TN = T such that for each +interval [Ti, Ti+1] for 0 ≤ i ≤ N − 1, we have a sequence of smooth solutions (Zi,(n), Zi,(n) +t +) +converging to the solution (Z, Zt) with supt∈[Ti,Ti+1] �F∆((Zi,(n), Zi,(n) +t +), (Z, Zt))(t) → 0 as +n → ∞. Suppose 0 ≤ j ≤ N − 1 is such that t0 ∈ [Tj, Tj+1]. Now as +��Zt,α′ +�� +2(t0) > 0, +this implies that ∥Zt − Av(Zt)∥2(t0) > 0. Now �F∆((Zj,(n), Zj,(n) +t +), (Z, Zt))(t0) → 0 implies +that +��Zj,(n) +t +− Zt +�� +˙H +1 +2 (t0) → 0 and hence +��(Zj,(n) +t +− Av(Zj,(n) +t +)) − (Zt − Av(Zt)) +�� +2(t0) → 0. +Hence from (134) we see that there exists cj > 0 and Nj ∈ N such that for n ≥ Nj we have +Aj,(n) +0 +(·, t0) ≥ cj > 0. Now from (136) we see that there exists c∗ +j > 0 such that for all t ∈ +[Tj, Tj+1] and n ≥ Nj we have Aj,(n) +0 +(·, t) ≥ c∗ +j > 0. Now as �F∆((Zj,(n), Zj,(n) +t +), (Z, Zt))(t) → +0 as n → ∞, this implies that +���Aj,(n) +0 +− A0 +��� +2(t) → 0. As A0(·, t) is a continuous function +(by a similar argument as Lemma 6.10) we see that A0(·, t) ≥ c∗ +j. This directly implies that +c∗ +j ≤ A0(·, t) ≲ +��Zt,α′ +��2 +2(t) for all t ∈ [Tj, Tj+1]. Now following this same proof for the other +intervals, we prove the claim. +Let us now complete the proof of uniqueness. If +��Zt,α′ +�� +2(0) = 0, then by the above +claim, we see that +��Zt,α′ +�� +2(t) = 0 for all t ∈ [0, T]. Hence b(α′, t) = A0(α′, t) = 0 for all +α′ ∈ R and t ∈ [0, T] and therefore Ztt(α′, t) = 0 for all α′ ∈ R and t ∈ [0, T]. Hence there +exists a constant C ∈ C such that Zt(α′, t) = C for all α′ ∈ R and t ∈ [0, T]. Therefore +Z(α′, t) = Z(α′, 0) + Ct. This proves uniqueness for this case. +If +��Zt,α′ +�� +2(0) > 0, then by the above claim we see that there exists c1, c2 > 0 such that +for all α′ ∈ R and t ∈ [0, T] we have A0(α′, t) ≥ c1 and +��Zt,α′ +�� +2(t) ≥ c2. Now on each +interval [Ti, Ti+1] as in the proof of the claim, we can use Theorem 5.6 to prove uniqueness +of the solution in the interval [Ti, Ti+1]. Hence we are done. + +106 +SIDDHANT AGRAWAL +□ +Proof of Theorem 3.8. Consider an initial domain Ω(0) of the form Figure 1 with corners +of angle νπ with 0 ≤ ν < 1 +2 (here ν = 0 corresponds to cusps) which is symmetric with +respect to the y-axis and 0 ∈ Ω(0). Let Ψ(·, 0) : D → Ω(0) be the conformal map with +Ψ(0, 0) = 0 and which is also symmetric with respect to the y-axis. Hence we see that +for z′ ∈ [−1, 1], we have Ψ(z′, 0) ∈ R, Ψz′(0, 0) > 0 and that d = |Z(0, 0) − Z(π, 0)| > 0. +Consider the initial velocity U(z′, 0) = −z′. Note that v = |Zt(0, 0) − Zt(π, 0)| > 0. Now it +is easy to check that this initial condition satisfies the conditions of the theorem and also +has 0 < �E(0) < ∞ (see [1] for details on computations regarding the conformal map when +the domain has a corner or cusp). +Now let T ∗ > 0 be the maximal time of existence in the class SA with this initial data. +Clearly by Theorem 3.6 we have that T ∗ ≥ +c +√ �E(0). We argue by contradiction and assume +that T ∗ > d +v. Let T0 = d +v and so we have a unique solution in the class SA in the time +interval [0, T0] with supt∈[0,T0] �E(t) < ∞. As the initial data is symmetric with respect to +the y-axis, by using the approximations of the solution by smooth solutions given by the +class SA, we see that for each t ∈ [0, T0] the solution is also symmetric with respect to the +y-axis. +By the proof of Theorem 3.6 we see that there exists a constant C1 > 0 such that +��Zt,α′ +�� +2(t) ≥ C1 for all t ∈ [0, T0]. As supt∈[0,T0] �E(t) < ∞, this implies that there exists a +constant C2 > 0 such that for all z′ ∈ D and t ∈ [0, T0] we have +���� +1 +Ψz′ (z′, t) +���� ≤ C2 +=⇒ +��Ψz′(z′, t) +�� ≥ 1 +C2 +> 0 +(137) +Now by the assumptions we have 0, π ∈ S(0) and as the solution is symmetric with +respect to the y-axis, we see from Theorem 3.7 that 0, π ∈ S(t) for all t ∈ [0, T0]. Also from +Theorem 3.7 we see that Ztt(0, t) = Ztt(π, t) = 0. Hence we have that Zt(0, t) = Zt(0, 0) +and Zt(π, t) = Zt(π, 0) for all t ∈ [0, T0]. +Therefore Z(0, t) = Z(0, 0) + Zt(0, 0)t and +Z(π, t) = Z(π, 0) + Zt(π, 0)t and so we have Z(0, T0) = Z(π, T0). Consider the function +g : [−1, 1] → C given by g(z′) = Ψ(z′, T0). By the fact that Ψ(·, T0) is symmetric with +respect to the y-axis implies that g is real valued. By the fact that the solution lies in the +class SA, we see that g is continuous on [−1, 1] and C1 on (−1, 1). As Z(0, T0) = Z(π, T0) +this implies that g(−1) = g(1). Therefore by Rolle’s theorem, there exists z0 ∈ (−1, 1) such +that g′(z0) = 0. In particular we have Ψz′(z0, T0) = 0 which directly contradicts (137). +Therefore T ∗ ≤ d +v < ∞. The blow up condition follows from Theorem 3.6 and hence we +are done. +□ +6. Appendix +Let Dt = ∂t + b∂α′ and let D∗ +t be defined as D∗ +t f = ∂tf + ∂α′(bf). We have + +2D WATER WAVES +107 +Proposition 6.1. Let n ≥ 1 and let f1, · · · , fn, g, h ∈ S(R). +Consider the function +[f1, · · · , fn; g] : R → C defined by (9). Then we have the following identities +h∂α′[f1, · · · , fn; g] = [h∂α′f1, · · · , fn; g] + · · · + [f1, · · · , h∂α′fn; g] ++ [f1, · · · , fn; ∂α′(hg)] − n[h, f1, · · · , fn; g] +and +Dt[f1, · · · , fn; g] = [Dtf1, · · · , fn; g] + · · · + [f1, · · · , Dtfn; g] ++ [f1, · · · , fn; D∗ +t g] − n[b, f1, · · · , fn; g] +If we have functions f1, f2, f3, g, h ∈ C∞(S1) and we consider the function [f1, · · · , fn; g] : +S1 → C for n = 1, 2, 3 given by (18), (19) and (20), then we have the same two identities +as above for n = 1, 2. In addition we have the identity +[f 2, �H]∂α′g − 2[f, �H]∂α′(fg) = −[f, f; g] +Proof. First we consider the case of functions on R. We see that +h(α′)∂α′[f1, · · · , fn; g] += h(α′)∂α′ +� 1 +iπ +� �f1(α′) − f1(β′) +α′ − β′ +� +· · · +�fn(α′) − fn(β′) +α′ − β′ +� +g(β′) dβ′ +� += h(α′)f ′ +1(α′) +� 1 +iπ +� +1 +α′ − β′ +�f2(α′) − f2(β′) +α′ − β′ +� +· · · +�fn(α′) − fn(β′) +α′ − β′ +� +g(β′) dβ′ +� ++ · · · + h(α′)f ′ +n(α′) +� 1 +iπ +� +1 +α′ − β′ +�f1(α′) − f1(β′) +α′ − β′ +� +· · · +�fn−1(α′) − fn−1(β′) +α′ − β′ +� +g(β′) dβ′ +� +− n +iπ +� �h(α′) − h(β′) +α′ − β′ +��f1(α′) − f1(β′) +α′ − β′ +� +· · · +�fn(α′) − fn(β′) +α′ − β′ +� +g(β′) dβ′ +− n +iπ +� +1 +α′ − β′ +�f1(α′) − f1(β′) +α′ − β′ +� +· · · +�fn(α′) − fn(β′) +α′ − β′ +� +h(β′)g(β′) dβ′ += [h∂α′f1, · · · , fn; g] + · · · + [f1, · · · , h∂α′fn; g] + [f1, · · · , fn; ∂α′(hg)] − n[h, f1, · · · , fn; g] +This proves the first identity. The second identity follows directly from this. The proof of +these identities for the case of S1 is identical and so we skip it. Now let us prove the last +identity. For f, g ∈ C∞(S1) we observe from the first identity proved that +f∂α′[f; g] = +� +ff ′; g +� ++ [f; ∂α′(fg)] − [f, f; g] +(138) +On the other hand by using the definition of [f; g] (18) and by expanding the commutator +we obtain +f∂α′[f; g] = f∂α′ +� +f, �H +� +g += f +� +f ′, �H +� +g + f +� +f, �H +� +∂α′g += +� +ff ′, �H +� +g − +� +f, �H +� +g∂α′f + +� +f 2, �H +� +∂α′g − +� +f, �H +� +f∂α′g + +108 +SIDDHANT AGRAWAL += +� +ff ′, �H +� +g − +� +f, �H +� +∂α′(fg) + +� +f 2, �H +� +∂α′g +Combining this with (138) gives us the required identity. +□ +Proposition 6.2. Fix m ∈ N. Let H ∈ C1(R), Ai ∈ C1(R) for i = 1, · · · m and F ∈ +C∞(R). Define +C1(H, A, f)(x) = p.v. +� +F +�H(x) − H(y) +x − y +�Πm +i=1(Ai(x) − Ai(y)) +(x − y)m+1 +f(y) dy +C2(H, A, f)(x) = p.v. +� +F +�H(x) − H(y) +x − y +�Πm +i=1(Ai(x) − Ai(y)) +(x − y)m +∂yf(y) dy +Then there exists constants c1, c2, c3, c4 depending only on F and ∥H′∥∞ so that +(1) ∥C1(H, A, f)∥2 ≤ c1∥A′ +1∥∞ · · · ∥A′ +m∥∞∥f∥2 +(2) ∥C1(H, A, f)∥2 ≤ c2∥A′ +1∥2∥A′ +2∥∞ · · · ∥A′ +m∥∞∥f∥∞ +(3) ∥C2(H, A, f)∥2 ≤ c3∥A′ +1∥∞ · · · ∥A′ +m∥∞∥f∥2 +(4) ∥C2(H, A, f)∥2 ≤ c4∥A′ +1∥2∥A′ +2∥∞ · · · ∥A′ +m∥∞∥f∥∞ +Now let Ai ∈ C1(S1) for i = 1, · · · m. Define +B1(A, f)(α′) = p.v. +� 2π +0 +Πm +i=1(Ai(α′) − Ai(β′)) +� +sin +� +α′−β′ +2 +��m+1 +f(β′) dβ′ +B2(A, f)(α′) = p.v. +� 2π +0 +Πm +i=1(Ai(α′) − Ai(β′)) +� +sin +� +α′−β′ +2 +��m +∂β′f(β′) dβ′ +Then we have the following estimates +(1) ∥B1(A, f)∥2 ≲ ∥A′ +1∥∞ · · · ∥A′ +m∥∞∥f∥2 +(2) ∥B1(A, f)∥2 ≲ ∥A′ +1∥2∥A′ +2∥∞ · · · ∥A′ +m∥∞∥f∥∞ +(3) ∥B2(A, f)∥2 ≲ ∥A′ +1∥∞ · · · ∥A′ +m∥∞∥f∥2 +(4) ∥B2(A, f)∥2 ≲ ∥A′ +1∥2∥A′ +2∥∞ · · · ∥A′ +m∥∞∥f∥∞ +Proof. Let us first consider the real line case. The first estimate is a theorem by Coifman, +McIntosh and Meyer [13]. See also chapter 9 of [25]. Proof of the second estimate can +be found in [33]. The third and fourth estimates can be obtained from the first two by +integration by parts. +The estimates for S1 can be obtained from the real line case by some modifications. +For the functions B1(A, f) and B2(A, f), we first replace sin +� +α′−β′ +2 +� +by eiα′ − eiβ′ from +the formula (16). +Now using Proposition 2.2 and Proposition 2.3 of [10] the estimates +follow. +□ +Proposition 6.3. Let f ∈ S(R). Then we have +(1) ∥f∥∞ ≲ ∥f∥Hs if s > 1 +2 and for s = 1 +2 we have ∥f∥BMO ≲ ∥f∥ ˙H +1 +2 +(2) +�����sup +β′ +���� +f(α′) − f(β′) +α′ − β′ +���� +����� +L2(R,dα′) +≲ +��f ′�� +2 + +2D WATER WAVES +109 +(3) +� ���� +f(α′) − f(β′) +α′ − β′ +���� +2 +dβ′ ≲ +��f ′��2 +2 +(4) ∥f∥2 +˙H +1 +2 = 1 +2π +�� ���� +f(α′) − f(β′) +α′ − β′ +���� +2 +dβ′ dα′ +Now let f ∈ C∞(S1). Then we have +(1) ∥f∥∞ ≲ ∥f∥Hs if s > +1 +2 and for s = +1 +2 we have ∥f∥BMO ≲ ∥f∥ ˙H +1 +2 . Moreover +∥f − Av(f)∥p ≲p ∥f∥BMO for any 1 ≤ p < ∞. We also have +∥f − Av(f)∥2 ≲ ∥f − Av(f)∥∞ ≲ +��f ′�� +1 ≲ +��f ′�� +2 +(2) +�����sup +β′ +����� +f(α′) − f(β′) +sin +�β′−α′ +2 +� +����� +����� +L2([0,2π],dα′) +≲ +��f ′�� +2 +(3) +� 2π +0 +����� +f(α′) − f(β′) +sin +�β′−α′ +2 +� +����� +2 +dβ′ ≲ +��f ′��2 +2 +(4) ∥f∥2 +˙H +1 +2 = 1 +8π +� 2π +0 +� 2π +0 +����� +f(α′) − f(β′) +sin +�β′−α′ +2 +� +����� +2 +dβ′ dα′ +Proof. See [2] for a proof of the results on R. For functions on S1 we have: +(1) The first two estimates follow from standard Sobolev embedding results and properties +of BMO functions. For the last estimate we only need to prove the middle inequality. +Observe that +f(x) − f(y) = +� x +y +f ′(s) ds +Now averaging over y gives +f(x) − Av(f) = 1 +2π +� 2π +0 +�� x +y +f ′(s) ds +� +dy +from which we easily get ∥f(x) − Av(f)∥∞ ≲ ∥f ′∥1. +(2) We identify f with the 2π periodic function ˜f : R → C defined as ˜f(x) = f(eix). Now +we define f ∗ : R → C as +f ∗ = +� +f ′ +on [−π, 3π) +0 +otherwise +Now for any fixed α′ ∈ [0, 2π) we see that +sup +β′∈[α′−π,α′+π) +����� +f(α′) − f(β′) +sin +� β′−α′ +2 +� +����� ≲ +sup +β′∈[α′−π,α′+π) +���� +f(α′) − f(β′) +α′ − β′ +���� ≲ M(f ∗)(α′) +where M is the uncentered Hardy Littlewood maximal operator on R. As the maximal +operator is bounded on L2(R), the estimate follows. +(3) This is a consequence of the second inequality. + +110 +SIDDHANT AGRAWAL +(4) The proof of this identity is the same as the one for the real line case. +□ +Proposition 6.4. Let f, g ∈ S(R) with s, a ∈ R and m, n ∈ Z. Then we have the following +estimates +(1) +��|∂α′|s[f, H](|∂α′|ag) +�� +2 ≲ +��|∂α′|s+af +�� +BMO∥g∥2 +for s, a ≥ 0 +(2) +��|∂α′|s[f, H](|∂α′|ag) +�� +2 ≲ +��|∂α′|s+af +�� +2∥g∥BMO +for s ≥ 0 and a > 0 +(3) +��� +f, |∂α′| +1 +2 � +g +�� +2 ≲ +��|∂α′| +1 +2f +�� +BMO∥g∥2 +(4) +��� +f, |∂α′| +1 +2 � +(|∂α′| +1 +2g) +�� +2 ≲ +��|∂α′|f +�� +BMO∥g∥2 +(5) ∥[f, H]g∥L∞∩ ˙H +1 +2 ≲ +��f ′�� +2∥g∥2 +(6) +��∂m +α′[f, H]∂n +α′g +�� +2 ≲ ∥∂(m+n) +α′ +f∥∞∥g∥2 +for m, n ≥ 0 +(7) +��∂m +α′[f, H]∂n +α′g +�� +2 ≲ ∥∂(m+n) +α′ +f∥2∥g∥∞ +for m ≥ 0 and n ≥ 1 +(8) ∥[f, H]g∥2 ≲ ∥f ′∥2∥g∥1 +The same estimates also hold for f, g ∈ C∞(S1). In addition for f, g ∈ C∞(S1) the above +estimates except (3) and (4) also hold with H replaced with �H. +Proof. See [2] for a proof for the estimates on R. +Now let f, g ∈ C∞(S1). +All of the +required estimates except for the L∞ estimate of (5) and the estimate (8) follow from +relatively straightforward modifications to the proof in the R case. For (5) we see that +([f, �H]g)(α′) = +1 +2πi +� 2π +0 +(f(α′) − f(β′)) cot +�β′ − α′ +2 +� +g(β′) dβ′ +Hence using Proposition 6.3 and the Cauchy Schwarz inequality we get ∥[f, �H]g∥∞ ≲ +∥f ′∥2∥g∥2. Now observe that +[f, Av]g = fAv(g) − Av(fg) = (f − Av(f))Av(g) − Av((f − Av(f))g) +Hence from Proposition 6.3 we get +∥[f, Av]g∥∞ ≲ ∥f − Av(f)∥∞∥g∥1 ≲ +��f ′�� +2∥g∥1 ≲ +��f ′�� +2∥g∥2 +As H = �H + Av, we therefore obtain ∥[f, H]g∥∞ ≲ ∥f ′∥2∥g∥2. The proof for (8) is similar +to this computation. +□ +Proposition 6.5. Let f, g, h ∈ S(R) with s, a ∈ R and m, n ∈ Z. +Then we have the +following estimates +(1) ∥|∂α′|s(fg)∥2 ≲ ∥|∂α′|sf∥2∥g∥∞ + ∥f∥∞∥|∂α′|sg∥2 +for s > 0 +(2) ∥fg∥ ˙H +1 +2 ≲ ∥f∥ ˙H +1 +2 ∥g∥∞ + ∥f∥∞∥g∥ ˙H +1 +2 +(3) ∥fg∥ ˙H +1 +2 ≲ ∥f ′∥2∥g∥2 + ∥f∥∞∥g∥ ˙H +1 +2 +The same estimates also hold for f, g, h ∈ C∞(S1). +Proof. See [2] for a proof for the estimates on R and the proof for functions on S1 is very +similar to the R case. +□ +Proposition 6.6. Let f, g, h ∈ S(R) . Then we have the following estimates + +2D WATER WAVES +111 +(1) ∥[f, g; h]∥2 ≲ ∥f ′∥2∥g′∥2∥h∥2 +(2) ∥[f, g; h′]∥2 ≲ ∥f ′∥∞∥g′∥∞∥h∥2 +(3) ∥[f, g; h]∥L∞∩ ˙H +1 +2 ≲ ∥f ′∥∞∥g′∥2∥h∥2 +The same estimates hold if instead we have f, g, h ∈ C∞(S1). +Proof. See [2] for a proof for the R case. For functions on S1, the proofs are essentially +identical to the real line case, with the only main difference being that for the second +estimate we have to use the S1 estimates from Proposition 6.2. +□ +Proposition 6.7. Let f ∈ S(R) and let w be a smooth non-zero weight with w, 1 +w ∈ L∞(R) +and w′ ∈ L2(R). Then +∥f∥2 +L∞∩ ˙H +1 +2 ≲ +���� +f +w +���� +2 +��wf ′�� +2 + +���� +f +w +���� +2 +2 +��w′��2 +2 +If f, w, 1 +w ∈ C∞(S1), then we have +∥f∥2 +L∞∩ ˙H +1 +2 ≲ +���� +f +w +���� +2 +��wf ′�� +2 + +���� +f +w +���� +2 +2 +��w′��2 +2 + ∥f∥2 +2 +Proof. See [2] for a proof for the R case (note that it does not matter whether we have +∥wf ′∥2 or ∥(wf)′∥2 on the right hand side). Now for the S1 case, the proof of the +˙H +1 +2 +estimate is the same as for R and we don’t actually need the extra ∥f∥2 +2 on the right hand +side. For the L∞ estimate, observe that +f 2(α′) − f 2(β′) = 2 +� α′ +β′ +� f +w +� +(wf ′) ds +Hence by averaging in β′, we see that +��f 2(α′) − Av(f 2) +�� ≲ +���� +f +w +���� +2 +��wf ′�� +2 +Therefore we see that +∥f∥2 +∞ ≲ +���� +f +w +���� +2 +��wf ′�� +2 + +��Av(f 2) +�� ≲ +���� +f +w +���� +2 +��wf ′�� +2 + ∥f∥2 +2 +□ +Proposition 6.8. Let f, g ∈ S(R) and let w, h ∈ L∞(R) be smooth functions with w′, h′ ∈ +L2(R). Then +∥fwh∥ ˙H +1 +2 ≲ ∥fw∥ ˙H +1 +2 ∥h∥∞ + ∥f∥2 +��(wh)′�� +2 + ∥f∥2 +��w′�� +2∥h∥∞ +If in addition we assume that w is real valued then +∥fgw∥2 ≲ ∥fw∥ ˙H +1 +2 ∥g∥2 + ∥gw∥ ˙H +1 +2 ∥f∥2 + ∥f∥2∥g∥2 +��w′�� +2 +If f, g, h, w ∈ C∞(S1), then the first estimate also holds and the second estimate gets mod- +ified to +∥fgw∥2 ≲ ∥fw∥ ˙H +1 +2 ∥g∥2 + ∥gw∥ ˙H +1 +2 ∥f∥2 + ∥f∥2∥g∥2 +� +∥w∥2 + +��w′�� +2 +� + +112 +SIDDHANT AGRAWAL +Proof. See [2] for a proof for the real line case. The proof of the ˙H +1 +2 estimate on S1 case is +identical to the real line case. Similarly for the L2 estimate, by following the proof we see +that the same estimate holds if f and g have zero mean and hence +∥(f − Av(f))(g − Av(g))w∥2 +≲ ∥(f − Av(f))w∥ ˙H +1 +2 ∥g − Av(g)∥2 + ∥(g − Av(g))w∥ ˙H +1 +2 ∥f − Av(f)∥2 ++ ∥f − Av(f)∥2∥g − Av(g)∥2 +��w′�� +2 +≲ ∥fw∥ ˙H +1 +2 ∥g∥2 + ∥gw∥ ˙H +1 +2 ∥f∥2 + ∥f∥2∥g∥2 +� +∥w∥2 + +��w′�� +2 +� +Now observe that +fgw = (f − Av(f))(g − Av(g))w + Av(f)gw + Av(g)fw − Av(f)Av(g)w +By taking the L2 norms on both side, we get the required estimate. +□ +Proposition 6.9. Let f ∈ C3([0, T), H3(R)). Then for any t ∈ [0, T) we have +lim sup +s→0+ +∥f(·, t + s)∥∞ − ∥f(·, t)∥∞ +s +≤ ∥∂tf(·, t)∥∞ +The same estimate holds for f ∈ C3([0, T), H3(T)) as well. +Proof. See [2] for a proof for the R case and the T case is proved in the same manner. +□ +Lemma 6.10. Let f ∈ H1(R) and consider the function g : R → R +g(α′) = +� ���� +f(α′) − f(β′) +α′ − β′ +���� +2 +dβ′ +Then g is a continuous function and g(α′) → 0 as |α′| → ∞. +Proof. From Proposition 6.3 it is clear that ∥g∥∞ ≲ ∥f ′∥2 +2. Let fn : R → C be a sequence of +smooth functions with compact support such that fn → f in H1(R). Consider the functions +gn(α′) = +� ���� +fn(α′) − fn(β′) +α′ − β′ +���� +2 +dβ′ +We again have the estimate ∥gn∥∞ ≲ ∥f ′ +n∥2 +2 and it is clear that for each fixed n, we have +gn is a continuous function and gn(α′) → 0 as |α′| → ∞. Now by Proposition 6.3 we see +that for n ∈ N +∥gn − g∥∞ ≲ +��f ′ +n − f ′�� +2 +���f ′ +n +�� +2 + +��f ′�� +2 +� +Therefore gn → g in L∞(R) and hence we are done. +□ + +2D WATER WAVES +113 +References +1. Siddhant Agrawal, Rigidity of singularities of 2D gravity water waves, J. Differential Equations 268 +(2020), no. 3, 1220–1249. MR 4029004 +2. +, Angled crested like water waves with surface tension: wellposedness of the problem, Comm. +Math. Phys. 383 (2021), no. 3, 1409–1526. MR 4244258 +3. 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MR 2410409 +Instituto de Ciencias Matem´aticas, ICMAT, Madrid, Spain +Email address: siddhant.govardhan@icmat.es + diff --git a/vNE3T4oBgHgl3EQfkgr7/content/tmp_files/load_file.txt b/vNE3T4oBgHgl3EQfkgr7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c7a6aa6d9e80cdd319f9f3cce2b1dcef1f863a9 --- /dev/null +++ b/vNE3T4oBgHgl3EQfkgr7/content/tmp_files/load_file.txt @@ -0,0 +1,3657 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf,len=3656 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='04599v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='AP] 11 Jan 2023 UNIFORM IN GRAVITY ESTIMATES FOR 2D WATER WAVES SIDDHANT AGRAWAL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We consider the 2D gravity water waves equation on an infinite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We prove a local wellposedness result which allows interfaces with corners and cusps as initial data and which is such that the time of existence of solutions is uniform even as the gravity parameter g → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For g > 0, we prove an improved blow up criterion for these singular solutions and we also prove an existence result for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Moreover the energy estimate used to prove this result is scaling invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' As an application of this energy estimate, we then consider the water wave equation with no gravity where the fluid domain is homeomorphic to the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We prove a local wellposedness result which allows for interfaces with angled crests and cusps as initial data and then by a rigidity argument, we show that there exists initial interfaces with angled crests for which the energy blows up in finite time, thereby proving the optimality of this local wellposedness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For smooth initial data, this local wellposedness result gives a longer time of existence as compared to previous results when the initial velocity is small and we also improve upon the blow up criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Contents 1 Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 2 2 Notation, preliminaries and equations of motion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 24 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='4 Quantities controlled by the energy E(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 93 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 113 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Introduction We are concerned with the motion of a fluid in dimension two with a free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In this work we will identify 2D vectors with complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The fluid is assumed to be inviscid, incompressible and irrotational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The fluid domain Ω(t) ⊂ C and the air region are separated by an interface ∂Ω(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The air and the fluid are assumed to have constant densities of 0 and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We will consider two different models: In the first model we assume that the interface ∂Ω(t) is homeomorphic to R and tends to real line at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The fluid is below the air region and there is no bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The fluid is subject to a uniform gravitational field −gi acting in the downward direction (here g ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The motion of the fluid is then governed by the Euler equation vt + (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='∇)v = −gi − ∇P on Ω(t) div v = 0, curl v = 0 on Ω(t) P = 0 on ∂Ω(t) (1, v) is tangent to the free surface (t, ∂Ω(t)) (1) along with the decay conditions v → 0, ∇P → −gi as |(x, y)| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We will consider this problem for all values of the gravity parameter g ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For the second model we will consider the problem when the domain Ω(t) is bounded and is homeomorphic to the unit disc D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In this case we assume that there is no gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' With these assumptions, the equations become vt + (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='∇)v = −∇P on Ω(t) div v = 0, curl v = 0 on Ω(t) P = 0 on ∂Ω(t) (1, v) is tangent to the free surface (t, ∂Ω(t)) (2) The earliest results on local well-posedness for the Cauchy problem are for small data in 2D and were obtained by Nalimov [26], Yoshihara [36, 37] and Craig [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In the case of zero surface tension and gravity g = 1, Wu [31, 32] obtained the proof of local well-posedness for arbitrary data in Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This result has been extended in various directions, see the works in [12, 24, 23, 15, 38, 11, 6, 22, 20, 19, 7, 17, 3, 5, 4, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' An important quantity related to the well-posedness of the problem in the zero surface tension case is the Taylor sign condition proposed in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This says that there should exist a constant c > 0 such that −∂P ∂n ≥ c > 0 on ∂Ω(t) where n is the outward unit normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In [18], Ebin gave an example of initial data with non- zero vorticity not satisfying the Taylor sign condition, for which the problem is ill posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 2D WATER WAVES 3 In [9] the authors proved that the linearized problem around a solution is wellposed if the Taylor sign condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In [31] for gravity g = 1, Wu proved that the Taylor sign condition is satisfied for the infinite bottom case if the interface is C1,α for α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This was later shown to be true for flat bottoms and with perturbations to flat bottom by Lannes [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' See also [20, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In [11] the authors proved the existence of splash and splat singularities for the water wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In order to study solutions with non C1 interfaces, Kinsey and Wu [21] proved an a priori estimate for angled crested water waves in the case of zero surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Using this, Wu [35] proved a local wellposedness result which allows the initial data to have interfaces with corners and cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In [1] the author proved that the singular solutions constructed in [35] are rigid and in particular the angle of the corner does not change with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In a recent work [14] for a zero gravity model, the authors construct solutions with interfaces which have corners and cusps, with the property that the angle of the corner changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Note that at the corners and cusps in [35, 14] the Taylor sign condition is not satisfied and one has − ∂P ∂n = 0 at those singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In this paper we explore the question of local wellposedness when the gravity parameter g → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For example consider the following question: For some fixed initial data, does one have a uniform time of existence for the solutions as g → 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' All the above results give a time of existence T → 0 as g → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We now give a heuristic argument to illustrate one of the main difficulties of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Consider the following simplified model of the water wave equation D2 t f + � −∂P ∂n � |∂α′|f = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='t Here Dt is the material derivate, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='t are lower order terms and f is some variable such as the velocity on the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' To prove energy estimates for this equation, one can naturally take the following energy E(t) = � |Dtf|2 + � � −∂P ∂n ����|∂α′| 1 2 f ��� 2 and take the time derivative to get d dtE(t) ≈ � Dt � −∂P ∂n ����|∂α′| 1 2 f ��� 2 + errors (3) Now if g > 0 and fixed, and the interface is C1,α, then as shown by Wu [31], we have a positive lower bound on − ∂P ∂n and hence the energy E(t) controls the ˙H 1 2 norm of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Then one shows that we can control ��Dt � − ∂P ∂n ��� ∞ from the energy and hence the first term on the right hand side of (3) is controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Now as we show in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1, − ∂P ∂n → g at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' So as g → 0, the energy E(t) no longer controls the ˙H 1 2 norm of f uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In the extreme case of g = 0, we show that if the interface is C1,α and the velocity is in H1 and is not identically zero, then − ∂P ∂n > 0 everywhere but − ∂P ∂n → 0 at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence to control the first term of (3), it is no longer 4 SIDDHANT AGRAWAL enough to control ��Dt � − ∂P ∂n ��� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' To solve this issue, we prove the following new estimate ����� Dt � − ∂P ∂n � � − ∂P ∂n � ����� ∞ ≲ ����� Zt,α′ Z,α′ ����� L∞∩ ˙H 1 2 + ��Zt,α′ �� 2 ����∂α′ 1 Z,α′ ���� 2 Here Z and Zt are the interface and the velocity on the boundary in conformal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Note that the term Zt,α′ Z,α′ is nothing but the boundary value of ∂zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This estimate shows that if one has control of the right hand side of the above inequality, then Dt � − ∂P ∂n � decays at least as fast as � − ∂P ∂n � at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This estimate is in fact also scaling invariant with respect to the 2 parameter family of scaling transformations (28) and the only term on the right hand side which controls the interface namely ���∂α′ 1 Z,α′ ��� 2, allows corners of angle less than 90◦ and cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Using this and several other similar estimates, we prove an energy estimate which works uniformly in gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let us now summarize the main results of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1 which are the results for model (1): (1) We construct an energy E(t) which is uniform in gravity and prove an a priori estimate for it in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The estimate is scaling invariant with respect to the 2 parameter family of scaling transformations (28) and allows both smooth initial data and initial interfaces with angled crests and cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' (2) In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='3, we prove a local wellposedness result which allows for both smooth initial data and initial data with interfaces having angled crests and cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The time of existence of solutions is shown to be uniform in gravity g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The result also establishes an existence result for g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We also prove a blow up criterion which is an improvement from previous one in [35] for these singular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' By modifying the energy estimate for the energy E(t), we then also prove an energy estimate for the model (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let us now summarize the main results of §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1 which are the results for model (2): (1) Similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='3, we prove a local wellposedness result Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='6 which allows for both smooth initial data and initial data with interfaces having angled crests and cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For smooth initial data, this result gives a longer time of existence as compared to previous results in the case where the initial velocity is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We also prove a blow up criterion which improves upon previous blow up criterions for both smooth and singular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' (2) Using a rigidity argument Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='7, we prove that there exists initial data with singular interfaces for which the energy blows up in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This shows the optimality of the local wellposedness result Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This also shows that the time of existence obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='6 is optimal in a suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This blow up result does not say anything about blow up for smooth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We now remark that if one were to prove a local wellposedness result in an exactly analogous manner to [35] without using our new energy estimate, then by the same rigidity argument one can indeed show that there is a finite maximal time of existence for these singular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' However what is not clear is whether the energy blows up 2D WATER WAVES 5 or not at the maximal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The blow up criterion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='6, which is proved using the new energy estimate, ensures that the energy indeed blows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' See §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='2 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In this paper we follow the framework of Wu [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We also use several identities and estimates proved in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The paper is organized as follows: In §2 we introduce the notation and write down the system of equations in conformal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In §3 we explain our main results in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In §4 we prove the results for the unbounded domain case stated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1 and in §5 we prove the results for the bounded domain case stated in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Finally in §6 we collect some of the identities and estimates used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Acknowledgment: The author was supported by the National Science Foundation un- der Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' DMS-1928930 while participating in a program hosted by MSRI during the Spring 2021 semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The author also received funding from the European Research Coun- cil (ERC) under the European Union’s Horizon 2020 research and innovation programme through the grant agreement 862342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Notation, preliminaries and equations of motion In this paper we write a ≲ b if there exists a universal constant C > 0 so that a ≤ Cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We write a ≲M b if there exits a constant C(M) depending only on M so that a ≤ C(M)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Similar definitions for a ≲M1,M2 b etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For singular integrals, all integrals will be understood in the principle value sense and we will suppress writing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' in front of the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Notation for the unbounded domain case In this section we recall the notation used in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The Fourier transform is defined as ˆf(ξ) = 1 √ 2π � R e−ixξf(x) dx We will denote by S(R) the Schwartz space of rapidly decreasing functions and S′(R) is the space of tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' A Fourier multiplier with symbol a(ξ) is the operator Ta defined formally by the relation � Taf = a(ξ) ˆf(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For s ≥ 0 the operators |∂α′|s and ⟨∂α′⟩s are defined as the Fourier multipliers with symbols |ξ|s and (1 + |ξ|2) s 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The Sobolev space Hs(R) for s ≥ 0 is the space of functions with ∥f∥Hs = ∥⟨∂α′⟩sf∥L2(dx) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The homogenous Sobolev space ˙H 1 2(R) is the space of functions modulo constants with ∥f∥ ˙H 1 2 = ∥|∂α′| 1 2 f∥L2(dx) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' From now on compositions of functions will always be in the spatial variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We write f = f(·, t), g = g(·, t), f ◦ g(·, t) := f(g(·, t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Define the operator Ug as given by Ugf = f ◦ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Observe that UfUg = Ug◦f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let [A, B] := AB − BA be the commutator of the operators A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' If A is an operator and f is a function, then (A + f) will represent the addition of the operators A and the multiplication operator Tf where Tf(g) = fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We denote the convolution of f and g by f ∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We will denote the spacial coordinates in Ω(t) with z = x + iy, whereas z′ = x′ + iy′ will denote the coordinates in the lower half plane 6 SIDDHANT AGRAWAL P− = � (x, y) ∈ R2 �� y < 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' As we will frequently work with holomorphic functions, we will use the holomorphic derivatives ∂z = 1 2(∂x − i∂y) and ∂z′ = 1 2(∂x′ − i∂y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In this paper all norms will be taken in the spacial coordinates unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For example for a function f : R × [0, T] → C we write ∥f∥2 = ∥f(·, t)∥2 = ∥f(·, t)∥L2(R,dα′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Also for a function f : P− → C we write supy′<0∥f∥L2(R,dx′) = supy′<0∥f(·, y′)∥L2(R,dx′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The Poisson kernel is given by Kǫ(x) = ǫ π(ǫ2 + x2) for ǫ > 0 (4) Let the interface be parametrized in Lagrangian coordinates by z(·, t) : R → Σ(t) satis- fying zα(α, t) ̸= 0 for all α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence zt(α, t) = v(z(α, t), t) is the velocity of the fluid on the interface and ztt(α, t) = (vt + (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='∇)v)(z(α, t), t) is the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let Ψ(·, t) : P− → Ω(t) be conformal maps satisfying limz→∞ Ψz(z, t) = 1 and that limz→∞ Ψt(z, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' With this, the only ambiguity left in the definition of Ψ is that of the choice of translation of the conformal map at t = 0, which does not play any role in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let Φ(·, t) : Ω(t) → P− be the inverse of the map Ψ(·, t) and define h(·, t) : R → R as h(α, t) = Φ(z(α, t), t) (5) hence h(·, t) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' As we use both Lagrangian and conformal parameter- izations, we will denote the Lagrangian parameter by α and the conformal parameter by α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let h−1(·, t) be its spacial inverse i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' h(h−1(α′, t), t) = α′ From now on, we will fix our Lagrangian parametrization at t = 0 by imposing h(α, 0) = α for all α ∈ R Hence the Lagrangian parametrization is the same as conformal parametrization at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Define the variables Z(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = z ◦ h−1(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = ∂α′Z(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) Hence ( zα hα ) ◦ h−1 = Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ Zt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = zt ◦ h−1(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = ∂α′Zt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) Hence (ztα hα ) ◦ h−1 = Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ Ztt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = ztt ◦ h−1(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) Ztt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = ∂α′Ztt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) Hence (zttα hα ) ◦ h−1 = Ztt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ Hence Z(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Zt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) and Ztt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) are the parameterizations of the boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' the veloc- ity and the acceleration in conformal coordinates and in particular Z(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) is the boundary value of the conformal map Ψ(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Note that as Z(α′, t) = z(h−1(α′, t), t) we see that ∂tZ ̸= Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Similarly ∂tZt ̸= Ztt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The substitute for the time derivative is the material 2D WATER WAVES 7 derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Define: Dt = material derivative = ∂t + b∂α′ where b = ht ◦ h−1 Dα′ = 1 Z,α′ ∂α′ Dα′ = 1 Z,α′ ∂α′ |Dα′| = 1 ��Z,α′ ��∂α′ H = Hilbert transform = Fourier multiplier with symbol − sgn(ξ) Hf(α′) = 1 iπ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' � 1 α′ − β′ f(β′) dβ′ PH = Holomorphic projection = I + H 2 PA = Antiholomorphic projection = I − H 2 |∂α′| = iH∂α′ = √ −∆ = Fourier multiplier with symbol |ξ| |∂α′|1/2 = Fourier multiplier with symbol |ξ|1/2 ω = Z,α′ ��Z,α′ �� (6) Now we have DtZ = Zt and DtZt = Ztt and more generally Dt(f(·, t)◦h−1) = (∂tf(·, t))◦h−1 or equivalently ∂t(F(·, t) ◦ h) = (DtF(·, t)) ◦ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' This means that Dt = U −1 h ∂tUh i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Dt is the material derivative in conformal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Define U : P − × [0, T) → C and P : P − × [0, T) → R as U = v ◦ Ψ P = P ◦ Ψ (7) and observe that U(·, t) is a holomorphic function on P−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Also note that its boundary value is given by Zt(α′, t) = U(α′, t) for all α′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence we can write the Euler equations (1) as equations on P− Ut − Ψt Uz′ Ψz′ + U Uz′ Ψz′ = − 1 Ψz′ (∂x′ − i∂y′)P + ig on P− U(·, t) is holomorphic on P− P = 0 on ∂P− Trace of 1 Ψz′ (U − Ψt) is real valued on ∂P− (8) along with the condition that Ψ(·, t) is conformal and the decay conditions U → 0, (∂x′ − i∂y′)P → ig, Ψz′ → 1 and Ψt → 0 as z′ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The condition that the particles on the boundary stay on the boundary is equivalent to saying that the trace of 1 Ψz′ (U − Ψt) is real valued and in particular � 1 Ψz′ (U − Ψt) ���� ∂P− = b where Dt = ∂t + b∂α′ is the material derivative on the boundary ∂P−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Also it should be noted that the process of obtaining (8) from (1) is reversible so long as the interface ∂Ω(t) = {Z(α′, t) | α′ ∈ R} is non-self intersecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' See [1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The Hilbert transform defined above in (6) satisfies the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 8 SIDDHANT AGRAWAL Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1 ([30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let 1 < p < ∞ and let F : P− → C be a holomorphic function in the lower half plane with F(z) → 0 as z → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Then the following are equivalent (1) sup y<0 ∥F(· + iy)∥p < ∞ (2) F(z) has a boundary value f, non-tangentially almost everywhere with f ∈ Lp and H(f) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In particular this says if U decays appropriately at infinity, then the boundary value of U namely Zt will satisfy HZt = Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Similarly as Ψz → 1 as z → ∞, we have that H � 1 Z,α′ − 1 � = 1 Z,α′ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Also note that as product of holomorphic functions is holomorphic, we can use the above lemma to conclude that several other functions on the boundary also satisfy similar identities e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' H( Zt Z,α′ ) = Zt Z,α′ , H(Dα′Zt) = Dα′Zt etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' For n ≥ 1 and for functions f1, · · · , fn, g : R → C we define the function [f1, · · · , fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' g] : R → C as [f1, · · · , fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' g](α′) = 1 iπ � �f1(α′) − f1(β′) α′ − β′ � · · �fn(α′) − fn(β′) α′ − β′ � g(β′) dβ′ (9) Hence with this notation we see that [f, H]g = 1 iπ � f(α′) − f(β′) α′ − β′ g(β′) dβ′ = [f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' g] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The system of equations on the boundary for the unbounded domain case To solve the system (1), in [2] we obtained a system for the variables (Z,α′, Zt) which we then solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The system is as follows: b = Re(I − H) � Zt Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ � Ag = g − Im[Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' H]Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ (∂t + b∂α′)Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ = Zt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ − bα′Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ (∂t + b∂α′)Zt = ig − i Ag Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ (10) along with the condition that their harmonic extensions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' namely Ψz′(· + iy) = K−y ∗ Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′ and U(· + iy) = K−y ∗ Zt for all y < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 1 are holomorphic functions on P− and satisfy 2 lim c→∞ sup |z′|≥c ���Ψz′(z′) − 1 �� + ��U(z′) ��� = 0 and Ψz′(z′) ̸= 0 for all z′ ∈ P− After solving the above system one can obtain Z(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) by the formula Z(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' t) = Z(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 0) + � t 0 � Zt(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' s) − b(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' s)Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='α′(α′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' s) � ds 1here K−y is the Poisson kernel (4) 2We observe that for such a Ψz′ we can uniquely define log(Ψz′) : P− → C such that log(Ψz′) is a continuous function with Ψz′ = exp{log(Ψz′)} and (log(Ψz′))(z′) → 0 as z′ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' 2D WATER WAVES 9 and hence (∂t + b∂α′)Z = DtZ = Zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence one can view the system being in variables (Z, Zt) instead of the variables (Z,α′, Zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Note that once one has a solution to the above system, we can recover a solution to the system (8) by letting Ψ and U be defined as above and defining P as the unique solution to the equation ∆P = −2|Uz′|2 on P−, P = 0 on ∂P− (11) along with the condition (∂x′ − i∂y′)P → ig as z′ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' See [35] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We observe that the above system allows self intersecting interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' However if the interface is self-intersecting then it becomes nonphysical and so its relation to the Euler equation (1) is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' See [2] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Now to get the function h(α, t) (recall the definition (5)), we solve the ODE dh dt = b(h, t) h(α, 0) = α (12) Observe that as long as sup[0,T]∥bα′∥∞(t) < ∞ we can solve this ODE uniquely and for any t ∈ [0, T] we have that h(·, t) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence it makes sense to talk about the functions z = Z ◦h, zt = Zt ◦h which are Lagrangian parameterizations of the interface and the velocity on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We also note that the last equation in (10) can be written as Ztt − ig = −i Ag Z,α′ (13) We note here that from the calculations in [31], we see that the gradient of the pressure on the boundary in conformal coordinates is (here ˆn is the outward unit normal) −∂P ∂ˆn ◦ h−1 = Ag ��Z,α′ �� (14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Notation for the bounded domain case Let the unit disc be D = � (x, y) ∈ R2 �� x2 + y2 < 1 � and let S1 = ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In the following we will identify functions f : S1 → C with their pullbacks ˜f : R → C, where ˜f(α′) = f(eiα′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We will frequently abuse notation and for a function f : S1 → C we will usually write f(α′) instead of f(eiα′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In particular if there exists a function F : D → C whose boundary value is f : S1 → C, then by abuse of notation we will also say that the boundary value of F is ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We will denote the boundary value of F by Tr(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' If g : R → C is a 2π periodic function, then in this paper whenever we use the Lp or Sobolev norms of g, what we mean is that we are computing the norms by looking at g as a function on S1 and not as a function on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We define the Fourier transform for a function f : S1 → C as ˆf(n) = 1 2π � 2π 0 f(α′)e−inα′ dα′ 10 SIDDHANT AGRAWAL and the inverse Fourier transform as f(α′) = ∞ � n=−∞ ˆf(n)einα′ The Lp norm of f is defined as ∥f∥p = �� 2π 0 |f(α′)|p dα′ � 1 p and the Sobolev norms of f are defined in the same way as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence in particular we observe that ∥f∥2 ˙H 1 2 = ��|∂α′| 1 2f ��2 2 = � 2π 0 f|∂α′|f dα′ Similar to §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1, compositions of functions are again only taken in spacial coordinates and we maintain the notation for the operator Ug and convolution of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We keep the notation z = x + iy for z ∈ Ω(t) and z′ = x′ + iy′ for z′ ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We have the same definitions for ∂z, ∂z′ and we also suppress the time variables when writing norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The Poisson kernel for the disc is given by Pr(θ) = 1 − r2 1 − 2r cos(θ) + r2 for 0 ≤ r < 1 Let the interface be parametrized in Lagrangian coordinates by a 2π periodic counter- clockwise parametrization z(·, t) : R → ∂Ω(t) satisfying zα(α, t) ̸= 0 for all α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence we see that zt(α, t) = v(z(α, t), t) is the velocity of the fluid on the interface and ztt(α, t) = (vt + (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='∇)v)(z(α, t), t) is the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We fix a point x0 ∈ Ω(0) and let X(x0, t) be the Lagrangian trajectory of this particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence ∂tX(x0, t) = v(X(x0, t), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let Ψ(·, t) : D → Ω(t) be conformal maps satisfying Ψ(0, t) = X(x0, t) and Ψz′(0, t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let Φ(·, t) : Ω(t) → D be the inverse of the map Ψ(·, t) and observe that Φ(z(α, t), t) ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Define h(·, t) : R → R as h(α, t) = −i log(Φ(z(α, t), t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' (15) where the ambiguity in the definition of log(z) is removed by ensuring that h is a continuous function on R × [0, T) and that h(0, 0) ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' From this it is easy to see that h(·, t) is an increasing function and is a homeomorphism on R satisfying h(α+2π, t) = h(α, t)+2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' As we use both Lagrangian and conformal parameterizations, we will denote the Lagrangian parameter by α and the conformal parameter by α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let h−1(·, t) be its spacial inverse i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' h(h−1(α′, t), t) = α′ From now on, we will fix our Lagrangian parametrization at t = 0 by imposing h(α, 0) = α for all α ∈ R Hence the Lagrangian parametrization is the same as conformal parametrization at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The functions Z, Zt, Ztt are defined as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Hence the functions Z(·, t) : R → ∂Ω(t) and Zt(·, t), Ztt(·, t) : R → C are 2π periodic functions and are the parameterizations of the boundary, the velocity and the acceleration in conformal coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' In particular we 2D WATER WAVES 11 have Z(α′, t) = Ψ(eiα′, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' We define the material derivative Dt and the weighted derivatives Dα′, Dα′, |Dα′| in the same way as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' The variable ω is also defined in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Define Av(f) = 1 2π � 2π 0 f(β′) dβ′ (�Hf)(α′) = 1 2πi � 2π 0 f(β′) cot �β′ − α′ 2 � dβ′ Observe that we have Av(f) = ˆf(0) � (�Hf)(n) = sgn(n) ˆf(n) |∂α′| = −i�H∂α′ where sgn(n) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 1 if n ≥ 1 0 if n = 0 −1 if n ≤ −1 We define the Hilbert transform as H = �H + Av The projection operators PH and PA are defined in the same way as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Using the identity eiα′ − eiβ′ = 2iei � α′+β′ 2 � sin �α′ − β′ 2 � (16) it is easily seen that (Hf)(α′) = 1 2πi � 2π 0 f(β′) cot �β′ − α′ 2 � dβ′ + 1 2π � 2π 0 f(β′) dβ′ = e−i α′ 2 2πi � 2π 0 f(β′) sin � β′−α′ 2 �ei β′ 2 dβ′ = 1 iπ � 2π 0 f(β′) eiβ′ − eiα′ ieiβ′ dβ′ The Hilbert transform H defined above has the following property similar to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='1 and is proved in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Let 1 < p < ∞ and let F : D → C be a holomorphic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNE3T4oBgHgl3EQfkgr7/content/2301.04599v1.pdf'} +page_content=' Then the following are equivalent (1) sup00 +Ekϵk . +(8) +The task is to design an N-segmented gate U such that +E = O(ϵn) for a given n. In many practical realizations, +it is sufficient to take n = 2. +Note that since E(ϵ) is +a function of a random variable instance, removing the +linear order term is not simply removing the expectation +value of ϵ. +Let us take for example U = X, and construct the gate +up to an overall phase, that is, iX. We need to find N +and {Ωk}N +k=1, {∆k}N +k=1, {tk}N +k=1 such that U (N) (0) = iX, +∂E(N)(ϵ) +∂ϵ +��� +ϵ=0 = 0 , +∂2E(N)(ϵ) +∂ϵ2 +��� +ϵ=0 = 0, etc. Using the uni- +tarity of each propagator one can simplify the equations +and reduce their complexity. In Appendix A we present +analytical robust solutions of the equations for the gates +of the form (iX) +1 +n , where n is a positive integer, employ- +ing three segments, and two solutions of the iX gate using +four segments with the same coupling constant. We also +compare in this appendix our solutions’ fidelity to that +of a single segment for n = 1, 2, 3. +2. +Non-Perturbative Method +In the non-perturbative approach, we search for a max- +imum of a cost function (minimum of a loss function) by +optimization. In order to simplify the optimization pro- +cess, we split the loss function into two subfunctions, the +invalid range loss subfunction and the robust fidelity loss +subfunction. The former ensures that the parameters we +obtain are within their allowed range (for instance, the +length of the waveguides cannot be negative) by strongly +penalizing deviations from it. +The robust fidelity loss +subfunction calculates the fidelity for a range of N error +values between −3σ to 3σ, σ being the standard devia- +tion, and weighs these fidelities according to the assumed +error distribution (for instance, a normal distribution). +An overall minus sign is added in order for the algorithm + +4 +to minimize this value and thus maximize the overall fi- +delity. For example, the loss function used for errors that +have a normal distribution is: +Loss = 1 − +3σ +� +x=−3σ +e− x2 +2σ2 +Distsum +· F(Uideal, Uoptimization)(x) ++µ · +N−1 +� +i=0 +m−1 +� +j=0 +max(0, pj,i − pMax +j +) + max(0, pMin +j +− pj,i) , +(9) +The former sum in the loss function is discrete: be- +tween each pair of subsequent x values there’s an interval +of 1 +n, where n + 1 is the number of samples used to esti- +mate the integral. The value Distsum = �x=3σ +x=−3σ e− x2 +2σ2 is +used to normalize the distribution function, which guar- +antees that the robust fidelity loss subfunction’s minimal +value is 0. µ defines the weight ratio between the loss +subfunctions (set to be in the range of [5, 100]). There +are N segments used, with m parameters per segment +(for instance, in this model, m = 3 because there are +three parameters: Ω, ∆, t). p is the selected parameter +values. pMin +j +and pMax +j +are chosen by physical limitations +(for instance, tMin = 0, since the length of the waveguides +cannot be negative). +By minimizing these two subfunctions, we obtain phys- +ically feasible parameters which minimize the fidelity loss +for errors between −3σ to 3σ weighted by the given error +distribution. Furthermore, the optimizer we used is the +Adam optimizer with a learning rate of 10−3. +Examples of non-perturbative solutions for the detun- +ing error model and their simulations can be seen in Ap- +pendix B. We show in Fig. 1(a),(b) one solution on the +Bloch sphere compared to the uniform gate, as well as +how errors affect the result of the gate for two different +initial states for each case (uniform and composite). +Further details regarding the optimization process are +described in Appendix C. +III. +ROBUST SEGMENTED GATES IN +INTEGRATED PHOTONICS +As there are several realizations of quantum gates and +each one has an appropriate statistical model of errors, +we choose, the following, to apply our methods to the +photonic realm [42–44]. This realm, which utilizes pho- +tons as excellent low-noise carriers of quantum informa- +tion, requires that unitary gate comply with a target de- +sign to a fourth decimal point accuracy[41]. +(c) +( +d +) +𝑤2 +𝑔 +𝑡 +𝑡1 +𝑡2 +𝑡3 +𝑔 +𝑤𝑎,1 +𝑤𝑎,2 +𝑤𝑏,1 +𝑤𝑏,2 +𝑤𝑐,1 +𝑤𝑐,2 +𝑤1 +FIG. 1. Na¨ıve vs. composite gate. (a-b) Bloch sphere repre- +sentation of a robust composite −iX Gate. The plot provides +a schematic description of two different states on Bloch sphere +and the trajectories they follow under the uniform −iX gate +(in continuous red), and under the segmented gate presented +in Table V (in continuous blue, turquoise, and purple). In +dashed lines, we show the trajectories the states follow when +an error ϵ = 0.17Ω occurs simultaneously in all detunings. In +black, we depict the torque vector of the uniform gate around +which the states rotate under the −iX operation. One can +see the robustness of the segmented gate against such errors +compared to the uniform regular gate. We show this for two +different initial states, |0⟩ and cos +� π +8 +� +|0⟩ + sin +� π +8 +� +|1⟩, to em- +phasize that the whole gate is robust, not only the complete +population transfer between |0⟩ and |1⟩. In other words, the +−iX gate is robust for any initial state; i.e. not only the mag- +nitude of the element U12 of the representative matrix of the +operation is robust (against errors in the physical system) but +also its phase, as well as the phase of the element U11. (c-d) +Dual rail photonic realization of unitary gates. Schematic 2D +top view illustrations of standard and composite gates respec- +tively, based on the directional couplers realization of gates +in integrated photonics. +We denote the waveguides widths +w1, w2, the length t and the gap g. +A. +Directional Couplers as Gates and their Error +Model +According to the coupled-mode theory, the propaga- +tion of the pair of electrical fields E1,2 in a directional +coupler of a fixed cross-section is described exactly by +Eqs. (1) and (3), where the actual matrix elements that +describe the dynamics along the two waveguides are the +mode propagation constants’ mismatch ∆β and the inter- +action coupling κ between the two waveguides [45]. The +coupling coefficient κ between the waveguides is equiv- +alent to the off-diagonal term Ω. The mode mismatch +between the mode index ∆β = β1 − β2 is equivalent to +the diagonal term ∆, and the propagation length z, is +equivalent to the evolution time t. +The coupling κ is +largely determined by the distance between the cores. + +(b) +(a) +
    0 and ∆ are free parameters. Examples of this +solution are presented in Table I, and their fidelities are shown +in Fig. 7(a). One can see clearly how the fidelity improves +with the composite design. +Composite iX gate in 3 segments +solution +Ω1, ∆1, t1 +Ω2, ∆2, t2 +Ω3, ∆3, t3 +∆1 = 0.5Ω +1.00, 0.500, 2.81 +0.625, 0, 5.03 +1.00, -0.500, 2.81 +∆1 = 0.75Ω +1.00, 0.750, 2.51 +0.781, 0, 4.02 +1.00, -0.750, 2.51 +∆1 = 1Ω +1.00, 1.00, 2.22 +1.00, 0, 3.14 +1.00, -1.00, 2.22 +∆1 = 1.1Ω +1.00, 1.1, 2.11326 1.105, 0, 2.84307 1.00, -1.1, 2.11326 +∆1 = 1.2Ω +1.00, 1.2, 2.0112 +1.22, 0, 2.57508 +1.00, -1.2, 2.0112 +TABLE I. Examples of iX gate generated by Eq. (A1), such +that the couplings and detunings are of the same order. The +fidelity of these gates compared to the one-segments iX gate +are shown in Fig. 7(a). +2. +(iX) +1 +n gate in 3 segments +A first-order solution for the (iX) +1 +n gate in 3 segments is +given by: +Ω1 = Ω, ∆1 = 0, t1 = θ +Ω, +(A2a) +Ω2 = +sin +� +θ + +π +2n +� +sin +� +θ + +π +2n +� +− sin +� π +2n +�Ω, ∆2 = 0, +t2 = 2 +� +2πm − θ − +π +2n +� +Ω2 +, +(A2b) +Ω3 = Ω, ∆3 = 0, t3 = θ +Ω, +(A2c) +where Ω > 0 and θ are free real parameters and m is a free +integer parameter (with the constraint t2 > 0). Examples of +this solution for n = 2, 3 are presented in Tables II,III,and +their fidelities are shown in Figs. +7(b)+(c), where m = 1 +where chosen for all of them. One can see clearly how the +fidelity improves with the composite design. +3. +Other families of solutions for (iX) +1 +n +We found additional first-order solutions for (iX) +1 +n : +1. +Ω1 = Ω, ∆1 = 0, t1 = π +Ω, +(A3a) + +11 +Composite (iX) +1 +2 gate in 3 segments +solution +Ω1, ∆1, t1 +Ω2, ∆2, t2 +Ω3, ∆3, t3 +θ = +π +2.2 +1.00, 0, 1.428 8.56794, 0, 0.950004 1.00, 0, 1.428 +θ = +π +2.4 +1.00, 0, 1.309 +5.44949, 0, 1.53731 +1.00, 0, 1.309 +θ = +π +2.6 +1.00, 0, 1.2083 4.45279, 0, 1.92665 1.00, 0, 1.2083 +θ = +π +2.8 +1.00, 0, 1.122 +3.98639, 0, 2.19537 +1.00, 0, 1.122 +θ = π +3 +1.00, 0, 1.05 +3.73, 0, 2.39 +1.00, 0, 1.05 +θ = π +4 +1.00, 0, 0.785 +3.41, 0, 2.76 +1.00, 0, 0.785 +θ = π +5 +1.00, 0, 0.628 +3.52, 0, 2.77 +1.00, 0, 0.628 +TABLE II. Examples of (iX) +1 +2 gate generated by Eq. (A2) +(with n = 2 and choosing m = 1), such that the couplings +and detunings are of the same order. The fidelity of these +gates compared to the one-segments (iX) +1 +2 gate are shown in +Fig. 7(b). +Composite (iX) +1 +3 gate in 3 segments +solution +Ω1, ∆1, t1 +Ω2, ∆2, t2 +Ω3, ∆3, t3 +θ = +π +1.8 +1.00, 0, 1.74533 2.87939, 0, 2.78827 1.00, 0, 1.74533 +θ = +π +2.2 +1.00, 0, 1.428 +2.16722, 0, 3.99737 +1.00, 0, 1.428 +θ = +π +2.4 +1.00, 0, 1.309 +2.07313, 0, 4.29359 +1.00, 0, 1.309 +θ = +π +2.6 +1.00, 0, 1.2083 2.02659, 0, 4.49157 1.00, 0, 1.2083 +θ = +π +2.8 +1.00, 0, 1.122 +2.00562, 0, 4.62459 +1.00, 0, 1.122 +θ = π +3 +1.00, 0, 1.05 +2.00, 0, 4.71 +1.00, 0, 1.05 +θ = π +4 +1.00, 0, 0.785 +2.07, 0, 4.80 +1.00, 0, 0.785 +θ = π +5 +1.00, 0, 0.628 +2.21, 0, 4.65 +1.00, 0, 0.628 +θ = π +6 +1.00, 0, 0.524 +2.37, 0, 4.43 +1.00, 0, 0.524 +θ = π +7 +1.00, 0, 0.449 +2.53, 0, 4.19 +1.00, 0, 0.449 +TABLE III. Examples of (iX) +1 +3 gate generated by Eq. (A2) +(with n = 3 and choosing m = 1) such that the couplings +and detunings are of the same order. The fidelity of these +gates compared to the one-segments (iX) +1 +3 gate are shown in +Fig. 7(c). +Ω2 = Ω +2 , ∆2 = 0, t2 = 2 +� +2π − π +n +� +Ω +, +(A3b) +Ω3 = Ω, ∆3 = 0, t3 = π +Ω, +(A3c) +where Ω > 0 is a free real parameter. +2. +Ω1 = Ω, ∆1 = ∆, t1 = +2π +√ +Ω2 + ∆2 , +(A4a) +Ω2 = +�√ +Ω2 + ∆2� 3 +2 +2π∆2 +tan +� π +2n +� +, ∆2 = 0, +t2 = 2 +� +2π − +π +2n +� +Ω2 +, +(A4b) +Ω3 = Ω, ∆3 = −∆, t3 = +2π +√ +Ω2 + ∆2 , +(A4c) +where Ω > 0 and ∆ are free real parameters. +3. +Ω1 = Ω, ∆1 = ∆, +t1 = +2 +� +mπ − arctan +�� +1 + ∆2 +Ω2 tan +� π +2n +��� +√ +Ω2 + ∆2 +, +(A5a) +Composite iX gate in 4 segments with same coupling for each +solution +Ω1, ∆1, t1 +Ω2, ∆2, t2 +Ω3, ∆3, t3 +Ω4, ∆4, t4 +ξ ≈ 0.46097 1.00, 0, 4.71 1.00, 0.460966, 5.70612 1.00, -0.460966, 5.70612 1.00, 0, 4.71 +ξ ≈ 6.03285 1.00, 0, 4.71 1.00, 6.03285, 1.02748 +1.00, -6.03285, 1.02748 1.00, 0, 4.71 +TABLE IV. The two 4-segments composite iX gates given +by Eq. +(A6). +The fidelity of these gates compared to the +one-segments iX gate are shown in Fig. 7(d). +Ω2 = +� +Ω2 + ∆2� +sin +� π +2n +� +2Ω sin +� π +2n +� +− t1∆2 cos +� π +2n +�, ∆2 = 0, t2 = +π +nΩ2 , +(A5b) +Ω3 = Ω, ∆3 = −∆, +t3 = +2 +� +mπ − arctan +�� +1 + ∆2 +Ω2 tan +� π +2n +��� +√ +Ω2 + ∆2 +, +(A5c) +where Ω > 0 and ∆ are free real parameters, and m is +a free integer parameter (with the constraint t1 > 0). +4. +iX gate in 4 segments with constant coupling +A first-order solution for the iX gate in 4 segments with +equal couplings: +Ω1 = Ω, ∆1 = 0, t1 = 4 arctan +� +1 + +√ +2 +� +Ω +, +(A6a) +Ω2 = Ω, ∆2 = ξΩ, t2 = +2π +� +Ω2 + ∆2 +2 +, +(A6b) +Ω3 = Ω, ∆3 = −ξΩ, t3 = +2π +� +Ω2 + ∆2 +2 +, +(A6c) +Ω4 = Ω, ∆4 = 0, t4 = 4 arctan +� +1 + +√ +2 +� +Ω +, +(A6d) +where Ω is a free real parameter and ξ is one of the two +positive solutions of the equation +� +2πξ2�2 = (1 + ξ2)3 (i.e., +ξ ≈ 0.461, 6.033). +These two solutions are summarized in +Table IV, and their fidelities are shown in Fig. 7(d). +One +can see clearly how the fidelity improves with the composite +design. +5. +The fidelity of the composite gates +In Fig. 7, we present the results of the composite gates. We +show the fidelity of our composite gates compared to regular +uniform ones. The parameters of the segments were given in +previous subsections of this section. Note that in these plots +we consider the fidelity as a deterministic object and plot its +values as a function of the values of the error. + +12 +FIG. 7. +(a) The fidelity of examples of the composite iX +gate in 3 segments given by Eq. (A1), compared to the one- +segments case as a function of the detuning error. +These +parameters are given in Table I. (b) The fidelity of exam- +ples of the composite (iX) +1 +2 gate in 3 segments given by Eq. +(A2), compared to the one-segments case. The parameters +are given in Table II. (c) The fidelity of examples of the com- +posite (iX) +1 +3 gate in 3 segments given by Eq. (A2), compared +to the one-segments case. The parameters are given in Table +III. (d) The fidelity of the two 4-segment composite iX gates +given by Eq. (A6), compared to the one-segments case. The +parameters are given in Table IV. We see the robustness of +the segmented design in comparison to the uniform one. +Composite gates in 3 segments +Gate +Ω1, ∆1, t1 +Ω2, ∆2, t2 +Ω3, ∆3, t3 +X +1.06, 1.784, 1.521 2.029, -0.005, 1.547 1.048, -1.776, 1.516 +X +1 +2 +2.043, 0.2884, 2.0 5.763, -1.8525, 2.0 +2.043, 0.2885, 2.0 +X +1 +3 +3.629, 0.2737, 7.0 3.607, -0.4956, 7.0 +3.6319, 0.259, 7.0 +H +4.773, -0.978, 2.0 1.1855, 0.5415, 2.0 1.7075, -0.31135, 2.0 +TABLE V. Examples of non-perturbative solutions using the +fully correlated detuning error model. +Appendix B: Non-Perturbative Solutions for Fully +Correlated Detuning Errors +Using the parameters optimization algorithm described in +Section C, we generated optimization solutions for the follow- +ing gates: X, X +1 +2 , X +1 +3 , H, resulting in the parameters given +in Table V. Note that each of the gates constructed using +these parameters is multiplied by a global phase. The fidelity +of the resulting gates is compared to the one-segments gates +in Fig. 8. +FIG. 8. +The fidelity of examples of the composite gates in 3 +segments generated by the optimization algorithm compared +to the one-segments case, as a function of the detuning error. +In (a) the ideal gate is X, in (b) the ideal gate is the Hadamard +gate, in (c) the ideal gate is X +1 +2 and in (d) the ideal gate is +X +1 +3 . All gates are calculated up to the global phase. These +examples are given in Table V. We see the robustness of the +segmented design in comparison to the uniform one. +Appendix C: Detailed explanation regarding the +numeric approach methodology +In this section, we will describe in further details the op- +timization process of the numeric, non-perturbative method, +and we will examplify the process using the detuning error +model. As stated before, the non-perturbative approach gen- +erates optimized parameters by using a loss function which is +composed of two subfunctions: +1. Value range loss subfunction — returns the sum: +N−1 +� +i=0 +max(0, −ti) + max(0, Ωmax − Ωi)+ +max(0, Ωi − Ωmin) + max(0, ∆max − ∆i)+ +max(0, ∆i − ∆min). +(C1) +This subfunction ensures that the parameters we obtain +are within their legal value range (ti values measure the +length of the waveguides, which means they cannot be +negative; the detuning and coupling parameters have a +range of feasible values they can be in). +2. Robust fidelity loss — returns the fidelity between the +error-less matrix and a range of matrices created by +using current parameters Ωi, ∆i, ti, i ∈ {0, 1, ..., N −1}. +This robust fidelity loss is calculated in the following way: +1. Set vector X to be a vector of n numbers evenly spaced +between −3σ to 3σ; X = [−3σ, −3σ + +6σ +n−1, −3σ + +2 6σ +n−1, ..., 3σ − +6σ +n−1, 3σ], +where n is the number of error values used for the opti- +mization (varies between optimization processes, in the +range of 2,500 to 10,000). +2. Set vector Dist to be the given error distribution vector; +for example, a Gaussian distributed vector is calculated +as Dist = [a−n/2+1, a−n/2+2, ..., a0, ..., an/2−2, an/2−1], +where ai = +1 +σ· +√ +2π · e +−x2 +i +2σ2 . +3. Use the current parameters Ωi, ∆i, ti, i ∈ {0, 1, ..., N − +1} and create n waveguide matrices, differing in the + +(a) +(b) +1.000 +1.000 +0.998 +0.998 +Fidelity +Fidelity +uniform +0=T/2.8 +0.996 +0.996 +0=π/2.2 +θ=π/3 +uniform +E ~ 6.0328 +0=T/2.4 +θ=π/4 +E ~ 0.461 +0=T/2.6 +=T/5 +0.994 +0.994 +0.10 +-0.05 +0.00 +0.05 +0.10 +-0.10 +-0.05 +0.00 +0.05 +0.10 +U/3 +ε/Q +(c) +(d) +1.000 +1.000 +uniform +@=π/3 +0.998 +0.998 +Fidelity +Fidelity +0=π/1.8 +θ=/4 +θ=π/2.2 +θ=π/5 +0.996 +0=T/2.4 +θ=T/6 +0.996 +uniform +△1=1Q +θ=T/2.6 +@=π/7 +△1=0.52 +A1=1.102 +0=T/2.8 +A1=0.752 +△1=1.202 +0.994 +0.994 +-0.10 +-0.05 +0.00 +0.05 +0.10 +-0.10 +-0.05 +0.00 +0.05 +0.10 +=/Q +=/Q(a) +(b) +1.000 +1.0000 +0.995 +0.9998 +delity +0 0.990 +0.985 +0.9994 +uniform +segmented +uniform +segmented +0.20-0.15 -0.10 -0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.20 -0.15 +5 -0.10 -0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +e/Q +U/3 +(c) +(d) +1.000 +1.000 +0.998 +0.999 +匠 0.994 + 0.997 +0.992 +0.996 +uniform +segmented +1 +uniform +segmented +0.20 -0.15 -0.10 -0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +-0.20 -0.15 -0.10 -0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +U/3 +U/313 +value of the error δ∆; δ∆ of the matrix Mi is the ele- +ment xi of the vector X: +Mi = U3(Ω0, ∆0 + xi, t0, Ω1, ∆1 + xi, t1, Ω2, ∆2 + xi, t2). +4. Calculate the fidelity loss of all these matrices and store +them in vector F: +Fi = Floss(Uideal, Mi), +where Floss(Uideal, U) += +1 − Fnorm(Uideal, U) and +Fnorm(Uideal, U) = |Tr(U † +idealU)|/2 +5. Return F · Dist (scalar product between the vectors). +Minimizing these subfunctions increases the overall fidelity +robustness while keeping the parameters in their previously +approved range. Generally, the optimizer used was the Adam +optimizer with a learning rate of 10−3, but some gates were +more delicate (for instance, X0.5) and required a smaller +learning rate. +The optimization also worked well with a +stochastic gradient descent optimizer. +Appendix D: The parameters of directional couplers +as a function of distance and widths +In order to estimate the detuning and coupling coeffi- +cients corresponding to the geometric parameters, we used +Lumerical, a commercially available finite difference eigen- +mode solver. We solved for the fields and effective mode in- +dices E(w), H(w), n(w) for different widths w. +With these +solutions, we were able to approximate the dynamics param- +eters, using the coupled-mode theory perturbative approxi- +mation [51]: +mi ≜ ω +4 +� � +[ϵ(x, y) − ϵ(i)(x, y)] +� +⃗E⊥(wi) +�2 +dxdy, +(D1a) +∆ = ∆β(w1, w2, g) ≈ 2π +λ (n1 − n2) + M1 − M2, +(D1b) +Ω = κ(w1, w2, g) ≈ +ω +4 +� � +[ϵ(x, y) − ϵ(2)(x, y)] ⃗E⊥(w1) · ⃗E⊥(w2)dxdy, +(D1c) +where ϵ(i) is defined as the permittivity distribution in space +when only waveguide i exists. m1 and m2 represent small cor- +rections to the propagation constants, β1 and β2, respectively, +because of the presence of the second waveguide. +To be able to use this in the perturbative method or in +gradient-based optimization algorithms, we performed this +evaluation for a large number of geometries within our range +of interest, and a multidimensional interpolation function was +derived. The use of the coupled mode theory approximation +enabled us to do so with a number of simulations that grows +linearly with the number of different widths, and as a constant +with respect to the number of gaps, instead of a number that +grows as the number of widths squared times the number of +gaps, as was needed for a more precise supermodes solution. +The widths we took are between 300 and 400 nm, and the +gaps we took are between 800 and 1200 nm. +By calculating Eq. (D1) for different width values for the +two waveguides, we obtain a grid of width values, which are +correlated to a grid of detuning and coupling coefficients, as +can be seen in figure 9. After obtaining this grid, we used +FIG. 9. Grid of waveguide widths mapped to coupling and +detuning coefficient values +fitting algorithms in order to fit polynomial and exponential +functions to the data. +For the detuning coefficient, a Taylor series in the form +below was used (where wi is the width of waveguide i): +∆ = +4 +� +i=0 +ai · wi +1 + bi · wi +2 +In figure 10, we can see how the approximate function be- +haves similarly to the values generated from the Lumerical +simulations, where the average difference between the ma- +trices is 0.00022, which translated to ∼ 1% of the detuning +coefficient. +FIG. 10. Comparison between the fit function estimation and +the original detuning grid. +For the Coupling coefficient, the following exponential func- +tion was used: +Ω = a0 + a1 · (w1 + w2) · ea2·(w1+w2) +In figure 11, we can see how the estimating function behaves +similarly to the values generated from the CMT approxima- +tion, where the average difference between the matrices is +0.00017, which translated to ˜0.5% of the coupling coefficient. +Lastly, we corrected the inaccuracies resulting from the cou- +pled mode theory approximation as follows: We used the in- +terpolation function to find the optimal composite design. We +then calculated the expected rotation angle for each segment, +θ = ΩgL, where Ωg = +√ +Ω2 + ∆2 is the generalized coupling +FIG. 11. Comparison between the fit function estimation and +the original coupling grid. + +0.500 +Coupling values +0.500 +Detuning values +0.475 +0.0425 +0.475 + 0.15 +0.450 +0.0400 +0.450 + 0.10 +[un] +0.0375 + 0.05 +0.0350 +N 0.400 +apir +0.00 +0.0325 +0.375 +AeM +0.05 +0.350 +0.0300 +0.350 +0.10 +0.325 +0.0275 +0.325 +0.15 +0.300 +0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 +0.300 + +0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 +Waveguide 1 width [um] +Waveguide 1 width [um]Origin parameters +Estimated parameters +Difference matrix +0.500 +0.500 +0.500 +0.0012 +0.475 +0.15 +0.475 +0.15 +0.475 +0.10 +0.10 +0.0010 +0.425 +0.05 +0.425 +0.05 + width +0.425 +0.0008 +2 +0.400 +0.00 +2 +0.400 - +0.00 +0.400 +lide +0.0006 +0.05 +0.375 +0.05 +0.0004 +0.350- +0.10 +0.350 - +0.10 +0.350 +0.0002 +0.325 +0.15 +0.325 +0.15 +0.325 +0.300+ +0.300 +0.300+ +0.30 +0.35 +0.40 +0.45 +0.50 +0.30 +0.35 +0.40 +0.45 +0.50 +0.30 +0.35 +0.40 +0.45 +0.50 +Waveguide 1 width [um] +Waveguide 1 width [um] +Waveguide 1 width [um]Origin parameters +Estimated parameters +Difference matrix +0.500 +0.500 +0.500 +0.475 +0.0425 +0.475 +0.0425 +0.475 +0.0008 +0.0400 +0.0400 +0.425 +0.0375 +0.0375 +0.0006 +0.400 +0.0350 +2 0.400 +0.0350 +~ 0.400 +ide +ide +0.0004 +0.375 +0.0325 +0.0325 +0.375 +ave +0.350 +0.0300 +0 0.350 - +0 0.350 +0.0300 +0.0002 +0.325 +0.0275 +0.325 +0.0275 +0.325 +0.300 +0.300 +0.300- +0.30 +0.35 +0.40 +0.45 +0.50 +0.30 +0.35 +0.40 +0.45 +0.50 +0.30 +0.35 +0.40 +0.45 +0.50 +Waveguide 1 width [um] +Waveguide 1 width [um] +Waveguide 1 width [um]14 +coefficient. Then, in Lumerical, we solved for the generalized +coupling coefficient of the selected segment using the more +precise supermodes method [51]. In this method, we solved +for the modes of both waveguides together, unlike the coupled +mode theory, where we solve for each mode separately and as- +sume that the coupling is weak. The length of the segment +is then determined from the desired rotation angle and the +precise generalized coupling coefficient. +Appendix E: The geometrical parameters of the +robust segmented gates +Here we present selected solutions achieved from both ap- +proaches described in Sec. III B. The parameters generated by +the perturbative approach is presented in table VI, and pa- +rameters generated by the non-perturbative approach is pre- +sented in table VII. The gap between the two waveguides for +all these solutions is 1.2µm. +Composite gates in 3 segments based on the model of correlated errors of widths +the gate +wauni, wbuni, zuni [µm] +wa1, wb1, z1 [µm] +wa2, wb2, z2 [µm] +wa3, wb3, z3 [µm] +−iX +0.450, 0.45,79.44 +0.4857, 0.4345,47.1117 0.4057, 0.4896, 40.5109 0.4857, 0.4345, 47.1117 +(iX) +1 +2 +0.4, 0.4,118.972 +0.426, 0.387,79.892 +0.318, 0.499, 55.8067 +0.426, 0.387, 79.892 +iH +0.426, 0.460, 58.8037 +0.379, 0.486,29.6481 +0.5, 0.31, 53.75 +0.379, 0.486, 29.6481 +� +1 +3I − i +� +2 +3X +0.450, 0.45,35.278 +0.358, 0.457,21.890 +0.485, 0.34, 28.0211 +0.358, 0.457, 21.890 +TABLE VI. Selected robust segmented gates achieved by the +perturbative approach. These gates are robust against corre- +lated errors in the widths. +Composite gates in 3 segments based on the model of correlated errors of widths +the gate +wauni, wbuni, zuni [µm] +wa1, wb1, z1 [µm] +wa2, wb2, z2 [µm] +wa3, wb3, z3 [µm] +−iX +0.450, 0.45,79.44 +0.375, 0.425, 49.254 0.429, 0.363, 52.608 +0.391, 0.45, 46.63 +(iX) +1 +2 +0.4, 0.4,20.0872 +0.48, 0.326, 15.28 +0.32, 0.478, 28.402 +0.48, 0.324, 15.18 +iH +0.426, 0.460, 58.8037 +0.430, 0.452, 70.29 0.422, 0.325, 33.522 0.430, 0.452, 70.328 +� +1 +3I − i +� +2 +3X +0.450, 0.45,35.278 +0.351, 0.46, 20.468 +0.459, 0.34, 34.078 +0.349, 0.46, 20.261 +TABLE VII. Selected robust segmented gates achieved by the +non-perturbative approach. These gates are robust against +correlated errors in the widths. + diff --git a/x9AyT4oBgHgl3EQfa_dv/content/tmp_files/load_file.txt b/x9AyT4oBgHgl3EQfa_dv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..839bbacbd5202b272ae55b6b549ea21857e11168 --- /dev/null +++ b/x9AyT4oBgHgl3EQfa_dv/content/tmp_files/load_file.txt @@ -0,0 +1,1423 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf,len=1422 +page_content='Segmented Composite Design of Robust Single-Qubit Quantum Gates Ido Kaplan ∗,1 Muhammad Erew ∗,2 Yonatan Piasetzky,2 Moshe Goldstein,2 Yaron Oz,2 and Haim Suchowski2 1School of Electrical Engineering, the Iby and Aladar Fleischman Faculty of Engineering, Tel-Aviv University, Tel-Aviv 6997801, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' 2Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 6997801, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (Dated: January 3, 2023) Error mitigation schemes and error-correcting codes have been the center of much effort in quan- tum information processing research over the last few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' While most of the successful proposed schemes for error mitigation are perturbative in the noise and assume deterministic systematic er- rors, studies of the problem considering the full noise and errors distribution are still scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In this work, we introduce an error mitigation scheme for robust single-qubit unitary gates based on composite segmented design, which accounts for the full distribution of the physical noise and er- rors in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We provide two optimization approaches to construct these robust segmented gates: perturbative and non-perturbative, that addresses all orders of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We demonstrate our scheme in the photonics realm for the dual-rail directional couplers realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We show that the 3-segmented composite design for the fundamental single-qubits unitary operations reduces the er- ror by an order of magnitude for a realistic distribution of errors, and that the two approaches are compatible for small errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' This is shown to significantly reduce the overhead of modern error cor- rection codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Our methods are rather general and can be applied to other realizations of quantum information processing units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' INTRODUCTION The potential exponential speedup for solving hard computational problems and the possible real-time ca- pability to decrypt classical encryption protocols are the driving forces behind the tremendous research effort in- vested in quantum information processing (QIP) and quantum computing [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Over the last several decades, major theoretical breakthroughs have been achieved, de- veloping quantum algorithms that serve a variety of problems and fields, including algorithms for combinato- rial optimization, quantum machine learning, decryption protocols, and variational quantum algorithms to find the ground state energy of Hamiltonian systems such as molecules [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Yet, the realization of a quantum infor- mation processor is still far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The major obstacles lie in the inherent systematic errors and stochastic noise of the physical building blocks, which influence state prepa- ration through the measurement process or the unitary operations (gates), the basic ingredients of any quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The problem of errors and noise is usually dealt with by error mitigation schemes or error-correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In the former, one attempts to reduce the error using various algorithmic schemes, typically with a small over- head [7–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In the latter, one constructs logical qubits (or quantum gates) using many physical qubits, with re- dundancy and significant overhead that ensures that the logical qubit significantly outperforms the physical qubit [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Most relevant error-correcting codes are stabilizer codes [7], a prime example being the surface code, having relative tolerance to local errors [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Yet, the capabil- ity of fault-tolerant quantum computation of the surface These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' code is conditioned: The probability of errors has to be under certain thresholds for each operation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=', single- qubit gates or double-qubit gates [7–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' High-fidelity physical gate are thus extremely important for realizing a useful error-correcting code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' An important step to- wards fault-tolerant quantum computation is to increase the fidelity of single quantum operations, the single uni- tary gates, which are fundamental building blocks of QIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' This is challenging in the experimental realizations of QIP, where the slightest fabrication defects or an inac- curate coupling strength can lead to errors that include deviations from target driving amplitudes and frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In the past few years, several studies suggested schemes for enhancing the robustness of state-to-state processes [21–31], and for devising robust unitary gates in various realizations of QIPs [32–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' One of the leading concepts of robust designs is based on the principles of compos- ite pulse sequences used in atomic physics and nuclear magnetic resonances, which utilize a sequence composed of constant pulses to minimize the errors through the evolution of the quantum systems [21–24, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' These techniques basically use a perturbative expansion of the gate’s operation in small deterministic systematic errors, and mitigate the errors order by order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Usually, these schemes deal with varying one parameter of the Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Recently, an expansion of the technique have been pro- vided to include the full parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Specifically, in integrated photonic based QIP, which utilizes photons as low-noise carriers of quantum information in the dual-rail representation, fabrication may cause geometrical errors that influence mainly the Hamiltonian’s diagonal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' A recent proposed robust solution for state-to-state direc- tional couplers based on composite segmented couplers of different widths [31] was experimentally demonstrated arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content='00253v1 [quant-ph] 31 Dec 2022 2 [39], showing that the design approach does not require modifications to the fabrication protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' However, all these proposed composite schemes deal with deterministic errors and noise, whereas noise is ran- dom by its nature, with randomness inherited from the quantum world, thermodynamic fluctuations, and from errors in manufacturing, preparation, and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' These issues become even more acute when dealing with a realization of robust unitary gate operations needed to allow full operation and control of quantum informa- tion processors, with a high-enough accuracy to comply with a specific target design for each physical realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' For photonic based realization, for example, this target accuracy is the fourth decimal point [41–44], a target that places stringent fabrication tolerances on process pa- rameters such as etching depth, waveguide widths, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=', which are challenging to meet in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' While the current known perturbative schemes have succeeded to construct robust gates for the realization of robust uni- tary gate operations, treatment of the statistical nature of noise and errors is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Here we present a scheme for robust unitary opera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The scheme, based on segmented composite de- sign, offers a way to design high-fidelity single-qubit quantum unitary gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In contrast to previous demon- strations of robust unitary gate designs, we provide pro- tocols to devise robust unitary gate operations consid- ering the statistical nature of noise and errors in phys- ical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We show that by knowing the appropri- ate model of systematic errors of a specific physical re- alization of single-qubit gates, robust gates can be con- structed by composing them rather than by correcting them through an operation before or after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In devising ro- bust unitary gates, we follow two design paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The first one is based on a perturbative approach, where we reduce the fully correlated error order by order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The second is a non-perturbative method, where we search for the local maxima of the fidelity cost func- tion so that we optimize while accounting for all orders of errors or their variances simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' As an example, we apply both methods to the photonic dual-rail real- ization, providing robust high-fidelity unitary solutions to different single-qubit gates, including the fundamen- tal X, X 1 2 , X 1 3 and H gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We demonstrate that the unitary segmented solutions are effective and compatible in practical scenarios of directional-couplers realizations, and are far more robust to systematic errors as com- pared to uniform couplers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Furthermore, we present the advantage of utilizing optimized segmented couplers in reducing the logical error in the quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' While we take the integrated photonics path-encoded qubits re- alization as an example to illustrate the strengths of the scheme on-chip building blocks for quantum applications, the method is rather general and can be applied to any other realization of a QIP device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Our paper is organized as follows: In Section II we present the single qubit gates and our methods for de- signing robust ones for a general statistical error model, and illustrate them for an example error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In Sec- tion III, we describe how single qubit gates are physi- cally realized in integrated photonics, describe the error model in the integrated photonics realization, and, using our methods, find and design several robust gates ac- cording to a statistical model of fabrication errors in the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We further show how the logical error in a surface code, as a consequence, would behave given our solutions and an error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In Section IV, we summarize and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In the appendixes, we provide details of the calculations and further infor- mation on various solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' METHOD AND ILLUSTRATION ON A REDUCED ERROR MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Single Qubit Quantum Gates and Fidelity The time evolution of a general qubit system {|0⟩ , |1⟩} is governed by the Schr¨odinger equation: i∂t � c1 (t) c2 (t) � = � −∆ (t) Ω∗ (t) Ω (t) ∆ (t) � � c1 (t) c2 (t) � , (1) where c1 (t) and c2 (t) are the probability amplitudes at time t of the states |0⟩ and |1⟩ respectively, Ω (t) is the (complex) Rabi frequency, ∆ (t) is the (real) detuning, and we set ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The unitary propagator of such a system is: U (t, 0) = T � exp � −i � t 0 � −∆ (t′) Ω∗ (t′) Ω (t′) ∆ (t′) �� dt′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (2) When Ω and ∆ are independent of time, the propagator simplifies to: U (t, 0) = exp � −it � −∆ Ω∗ Ω ∆ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (3) Using physical systems that follow such SU (2) dynam- ics, one can implement various single qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' How- ever, when one considers noise in the physical system, the implemented gate deviates from the desired one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In or- der to quantify how far the noisy gate is from the desired one, we will consider the metric provided by the fidelity F of the gate U(ϵ), which is defined as F(Uideal, U(ϵ)) = 1 2|Tr(U † idealU(ϵ)|, (4) where Uideal is the desired ideal unitary gate given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (3), and U(ϵ) is its actual physical realization, which depends on a set of jointly distributed random errors, ϵ = {ϵa}m a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' This fidelity takes values in the interval [0, 1], where 1 corresponds to the case of no errors, and 0 corresponds to the case of maximal deviation from the desired matrix, such as getting iY instead of iX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The goal in this work is to increase the expectation value of the fidelity: ¯F = Eϵ[F(Uideal, U(ϵ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (5) 3 The relevant statistical error model should be taken de- pending on the specific physical realization of the gates, where one considers the quantum errors, thermodynamic errors, and the errors of manufacturing, preparation, and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Maximizing the mean fidelity over a wide error range is crucial for fault-tolerant computation, as mentioned in the introduction, since a certain thresh- old for the resulting physical error probability has to be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Constructing Robust Composite Gates The method that we employ to design robust gates is to compose pulses or segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The reasoning behind this approach is the natural assumption that the relevant errors are highly correlated, and this correlation can be applied to cancel errors with appropriately tuned designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Consider an ideal unitary gate Uideal, as well as its ac- tual noisy segmented realization U (N) = �N k=1 Uk(ϵk), where ϵk is the random error vector of the kth segment, which includes m errors: ϵk = {ϵa k}m a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' All the errors are jointly distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Each segment Uk without errors is as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The goal is to increase the expectation value of the fidelity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In our analysis, we employ two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The first one is perturbative in the error random variables, where we consider them to be fully correlated and design the segmented gate such that we cancel the errors order by order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' More specifically, we con- struct analytical solutions of 3-segmented designs that cancel the first order error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Clearly, cancellation of higher order error terms requires a larger number of segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The second method is non-perturbative, where we consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (5) as a cost function to be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' While these two methods are compatible for small errors or small variances of errors, as will be seen, the non- perturbative approach also offer a path for addressing large values of the random error variances, where the optimization take into account all orders in the errors simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Example: A detuning Error Model In order to illustrate our methods in a relatively sim- ple case, we consider first a physical system which allows only real Ω’s, and we assume a single error random vari- able ϵk = ϵ, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=', N, which is a systematic error in ∆, and neglect the error in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We further assume that the errors of the different segments are fully correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' This assumption describes well the errors in several quantum and classical systems that follow such SU (2) dynamics, such as gates of trapped ions [25, 26], sum-frequency gen- eration [45], atomic systems [27–29], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The N-segmented gate reads: U (N) = N � k=1 U (Ωk, ∆k, tk, ϵ) , (6) where U (Ω, ∆, t, ϵ) = e−it(ΩX−∆Z−ϵZ) , (7) and tk, Ωk, ∆k are the length of the kth-segment, its cou- pling, and its detuning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The N-segmented gate fidelity (4) is F(U (N)(0), U (N)(ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Perturbative Method In the perturbative approach, we consider the error in the quantum gate E(ϵ) = U(ϵ) − U(0) = � k>0 Ekϵk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (8) The task is to design an N-segmented gate U such that E = O(ϵn) for a given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In many practical realizations, it is sufficient to take n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Note that since E(ϵ) is a function of a random variable instance, removing the linear order term is not simply removing the expectation value of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Let us take for example U = X, and construct the gate up to an overall phase, that is, iX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We need to find N and {Ωk}N k=1, {∆k}N k=1, {tk}N k=1 such that U (N) (0) = iX, ∂E(N)(ϵ) ∂ϵ ��� ϵ=0 = 0 , ∂2E(N)(ϵ) ∂ϵ2 ��� ϵ=0 = 0, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Using the uni- tarity of each propagator one can simplify the equations and reduce their complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In Appendix A we present analytical robust solutions of the equations for the gates of the form (iX) 1 n , where n is a positive integer, employ- ing three segments, and two solutions of the iX gate using four segments with the same coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We also compare in this appendix our solutions’ fidelity to that of a single segment for n = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Non-Perturbative Method In the non-perturbative approach, we search for a max- imum of a cost function (minimum of a loss function) by optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In order to simplify the optimization pro- cess, we split the loss function into two subfunctions, the invalid range loss subfunction and the robust fidelity loss subfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The former ensures that the parameters we obtain are within their allowed range (for instance, the length of the waveguides cannot be negative) by strongly penalizing deviations from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The robust fidelity loss subfunction calculates the fidelity for a range of N error values between −3σ to 3σ, σ being the standard devia- tion, and weighs these fidelities according to the assumed error distribution (for instance, a normal distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' An overall minus sign is added in order for the algorithm 4 to minimize this value and thus maximize the overall fi- delity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' For example, the loss function used for errors that have a normal distribution is: Loss = 1 − 3σ � x=−3σ e− x2 2σ2 Distsum F(Uideal, Uoptimization)(x) +µ · N−1 � i=0 m−1 � j=0 max(0, pj,i − pMax j ) + max(0, pMin j − pj,i) , (9) The former sum in the loss function is discrete: be- tween each pair of subsequent x values there’s an interval of 1 n, where n + 1 is the number of samples used to esti- mate the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The value Distsum = �x=3σ x=−3σ e− x2 2σ2 is used to normalize the distribution function, which guar- antees that the robust fidelity loss subfunction’s minimal value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' µ defines the weight ratio between the loss subfunctions (set to be in the range of [5, 100]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' There are N segments used, with m parameters per segment (for instance, in this model, m = 3 because there are three parameters: Ω, ∆, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' p is the selected parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' pMin j and pMax j are chosen by physical limitations (for instance, tMin = 0, since the length of the waveguides cannot be negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' By minimizing these two subfunctions, we obtain phys- ically feasible parameters which minimize the fidelity loss for errors between −3σ to 3σ weighted by the given error distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Furthermore, the optimizer we used is the Adam optimizer with a learning rate of 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Examples of non-perturbative solutions for the detun- ing error model and their simulations can be seen in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' 1(a),(b) one solution on the Bloch sphere compared to the uniform gate, as well as how errors affect the result of the gate for two different initial states for each case (uniform and composite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Further details regarding the optimization process are described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' ROBUST SEGMENTED GATES IN INTEGRATED PHOTONICS As there are several realizations of quantum gates and each one has an appropriate statistical model of errors, we choose, the following, to apply our methods to the photonic realm [42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' This realm, which utilizes pho- tons as excellent low-noise carriers of quantum informa- tion, requires that unitary gate comply with a target de- sign to a fourth decimal point accuracy[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (c) ( d ) 𝑤2 𝑔 𝑡 𝑡1 𝑡2 𝑡3 𝑔 𝑤𝑎,1 𝑤𝑎,2 𝑤𝑏,1 𝑤𝑏,2 𝑤𝑐,1 𝑤𝑐,2 𝑤1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Na¨ıve vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' composite gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (a-b) Bloch sphere repre- sentation of a robust composite −iX Gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The plot provides a schematic description of two different states on Bloch sphere and the trajectories they follow under the uniform −iX gate (in continuous red), and under the segmented gate presented in Table V (in continuous blue, turquoise, and purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In dashed lines, we show the trajectories the states follow when an error ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content='17Ω occurs simultaneously in all detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In black, we depict the torque vector of the uniform gate around which the states rotate under the −iX operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' One can see the robustness of the segmented gate against such errors compared to the uniform regular gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We show this for two different initial states, |0⟩ and cos � π 8 � |0⟩ + sin � π 8 � |1⟩, to em- phasize that the whole gate is robust, not only the complete population transfer between |0⟩ and |1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' In other words, the −iX gate is robust for any initial state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' not only the mag- nitude of the element U12 of the representative matrix of the operation is robust (against errors in the physical system) but also its phase, as well as the phase of the element U11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (c-d) Dual rail photonic realization of unitary gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Schematic 2D top view illustrations of standard and composite gates respec- tively, based on the directional couplers realization of gates in integrated photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' We denote the waveguides widths w1, w2, the length t and the gap g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' Directional Couplers as Gates and their Error Model According to the coupled-mode theory, the propaga- tion of the pair of electrical fields E1,2 in a directional coupler of a fixed cross-section is described exactly by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (1) and (3), where the actual matrix elements that describe the dynamics along the two waveguides are the mode propagation constants’ mismatch ∆β and the inter- action coupling κ between the two waveguides [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The coupling coefficient κ between the waveguides is equiv- alent to the off-diagonal term Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The mode mismatch between the mode index ∆β = β1 − β2 is equivalent to the diagonal term ∆, and the propagation length z, is equivalent to the evolution time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' The coupling κ is largely determined by the distance between the cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AyT4oBgHgl3EQfa_dv/content/2301.00253v1.pdf'} +page_content=' (b) (a)